Chapter 1

GEOMETRICAL BACKGROUND OF METRIC FIXED POINT THEORY

Stanisław Prus

1. Introduction

The interplay of the geometry of Banach spaces and fixed point theory has been very strong and fruitful. In particular, geometrical properties play key roles in metric fixed point problems. In this text we discuss the most basic of these properties. Since many fixed point results have a quantitative character, we put special emphasis on the scaling coefficients and functions corresponding to the considered properties. The material is far from exhausting the subject and basically we do not go into applications. They can be found elsewhere in this volume. The reader may also consult [5], [44] and [1].

We consider Banach spaces over the real field only. Unless otherwise stated the metric and topological notions are referred to the metric given by a norm. Our notation and terminology is standard. It coincides for instance with that of [5]. However, for convenience of the reader we recall some notation. Let X be a Banach space. By B_{X} and S_{X} we denote the closed unit ball and the unit sphere, respectively. Next, let A be a nonempty subset of X. Then co(A) stands for the convex hull of A and span(A) denotes the linear subspace spanned by A. Given x ∈ X, by d(x, A) we denote the distance of x to A.

2. Strict convexity and smoothness

Convexity is the most elementary property of a norm. In the simplest case when we deal with points of a line in a space X we can actually apply the well-known facts concerning convex functions of a real variable.

Remark 2.1 For any points x, y of a Banach space X the function t ↦ ∥x + ty∥ is convex on ℝ.

In some applications it is necessary to impose conditions stronger than convexity on a norm. Strict convexity is one of these conditions.

Definition 2.2 A Banach space X is strictly convex (or rotund) if

∥\frac{x + y}{2} ∥ < 1

whenever x, y ∈ S_{X}, x ≠ y.

Condition 3 of our first theorem gives a simple geometric interpretation of strict convexity.

Theorem 2.3 Let X be a Banach space. The following conditions are equivalent.

1. X is strictly convex.

2. If 1 < p < ∞ and x, y ∈ X, x ≠ y, then ∥λx + (1 − λ)y∥^{p} < λ∥x∥^{p} + (1 − λ)∥y∥^{p} for every 0 < λ < 1.

3. The sphere S_{X} does not contain a nontrivial segment.

4. If x, y ∈ X and ∥x + y∥ = ∥x∥ + ∥y∥, then x = 0 or y = 0 or y = αx for some α > 0.

5. If x^{∗} ∈ X^{∗}, x^{∗} ≠ 0, then x^{∗}(x) = ∥x^{∗}∥ for at most one point x ∈ S_{X}.

Proof. Let 1 < p < ∞. Considering the function t ↦ t^{p}, we see that

(\frac{a + b}{2})^{p} < \frac{1}{2}(a^{p} + b^{p})  (2.1)

for all a, b ≥ 0, a ≠ b. Assume that X is strictly convex and x, y ∈ X, x ≠ y. If ∥x∥ = ∥y∥, then ∥(x + y)/2∥ < ∥x∥. Consequently,

∥\frac{x + y}{2} ∥^{p} < \frac{1}{2}(∥x∥^{p} + ∥y∥^{p}).

From (2.1) we see that this inequality holds also in the case when ∥x∥ ≠ ∥y∥. Take now 0 < λ < 1. If 0 < λ ≤ \frac{1}{2}, then

∥λx + (1 − λ)y∥^{p} ≤ 2λ ∥\frac{x + y}{2} ∥^{p} + (1 − 2λ)∥y∥^{p} < λ∥x∥^{p} + (1 − λ)∥y∥^{p}.

The proof is similar if \frac{1}{2} < λ < 1. This shows that 1 ⇒ 2.

The implication 2 ⇒ 3 is obvious. To prove that 3 ⇒ 4, suppose ∥x+y∥ = ∥x∥+∥y∥ for some nonzero vectors x, y ∈ X such that y ≠ αx for all α > 0. We can assume that 0 < ∥x∥ ≤ ∥y∥ = 1. Consider the functions f(t) = ∥tx/∥x∥ + y∥ and g(t) = t + 1, t ≥ 0. Then f(0) = g(0), f(∥x∥) = g(∥x∥), and f(t) ≤ g(t) for every t ≥ 0. From Remark 2.1 we see that f(t) = g(t) for every t ≥ 0 which shows that the segment with the endpoints x/∥x∥, y is contained in S_{X}.

To show that 4 ⇒ 5, assume there is a functional x^{∗} ∈ X^{∗}, x^{∗} ≠ 0 for which x^{∗}(x) = x^{∗}(y) = ∥x^{∗}∥ where x, y ∈ S_{X}, x ≠ y. Then

∥x^{∗}∥ = x^{∗} (\frac{x + y}{2} ) ≤ ∥x^{∗}∥ ∥\frac{x + y}{2} ∥ ≤ ∥x^{∗}∥.

This gives us the equality ∥x + y∥ = 2 = ∥x∥ + ∥y∥, which contradicts condition 4.

Assume now that there exist x, y ∈ S_{X}, x ≠ y, such that ∥(x + y)/2∥ = 1. Choose a functional x^{∗} ∈ S_{X^{∗}} so that x^{∗}((x + y)/2) = ∥(x + y)/2∥ = 1. It is easy to show that x^{∗}(x) = 1 = x^{∗}(y), and 5 ⇒ 1 is proved. ■

In some cases it is more convenient to use metric segments instead of affine ones (see for instance [1]). Let x, y be points of a Banach space X. The set I(x, y) of all z ∈ X such that ∥z −x∥ ≤ (1−t)∥y −x∥ and ∥z −y∥ ≤ t∥y −x∥ for some t ∈ [0, 1] is called the metric segment with the endpoints x, y. It is easy to see that equality actually holds in the above inequalities. Setting t = 1/2 in this definition, we obtain the so-called metric midpoint of x, y. Obviously, I(x, y) contains the affine segment co({x, y}). In general those two sets may be different but in strictly convex spaces they coincide.

Proposition 2.4 A Banach space X is strictly convex if and only if I(x, y)=co({x, y}) for any x, y ∈ X.

Proof. Assume that X is not strictly convex. By Theorem 2.3 the sphere S_{X} contains a segment co({x, y}) with x ≠ y. Then x, y are in the metric segment I(0, x + y) but not in co({0, x + y}). Suppose in turn that X is strictly convex and z ∈ I(x, y) where x ≠ y. Then ∥z − x∥ = (1 − t)∥y − x∥ and ∥z − y∥ = t∥y − x∥ for some t ∈ [0, 1]. Consequently,

∥z − x∥ + ∥z − y∥ = ∥y − x∥ = ∥(z − x) + (y − z)∥.

From Theorem 2.3 we see that z = x or z = y or y − z = α(z − x) for some α > 0. In each case z ∈ co({x, y}). ■

We will show two more characterizations of strict convexity. The first of them is related to fixed point theory. Let C be a nonempty subset of a Banach space X. A mapping T : C → X is called nonexpansive if

∥T x − T y∥ ≤ ∥x − y∥

for all x, y ∈ C. By Fix(T) we denote the set of all fixed points of T, i.e. points x ∈ C such that T x = x. Clearly, Fix(T) may be empty.

Theorem 2.5 A Banach space X is strictly convex if and only if for every nonempty convex set C ⊂ X and every nonexpansive mapping T : C → C the set Fix(T) is convex.

Proof. Assume that X is strictly convex and x, y ∈ Fix(T) for some nonexpansive mapping T : C → C. If z = tx + (1 − t)y where t ∈ [0, 1], then ∥T z − x∥ ≤ ∥z − x∥ = (1−t)∥y−x∥ and similarly, ∥T z−y∥ ≤ t∥y−x∥. Hence T z ∈ I(x, y) and from Theorem 2.4 we see that T z ∈ co({x, y}). It is easy to check that T z = tx + (1 − t)y, so T z = z.

To show the converse, we assume that X is not strictly convex. By Theorem 2.3 the sphere S_{X} contains a segment co({x, y}) with x ≠ y. Consider the convex set

C = \{α\frac{x + y}{2} + β \frac{x − y}{2} : α ∈ [−1, 1], β ∈ [0, 1]\} .

Let x^{∗} ∈ S_{X^{∗}} be such that x^{∗}((x + y)/2) = 1. Then x^{∗}(x) = 1 = x^{∗}(y) and in particular x^{∗}(u) ∈ [−1, 1] for every u ∈ C. Given such u, we put T u = x^{∗}(u)x if x^{∗}(u) ≥ 0 and T u = x^{∗}(u)y if x^{∗}(u) < 0. It is easy to see that T (C) ⊂ C is not convex and Fix(T) = T (C). In order to show that T is nonexpansive we take u, v ∈ C such that x^{∗}(u) ≥ 0 and x^{∗}(v) < 0. Since co({x, y}) ⊂ S_{X},

∥T u − T v∥ = ∥x^{∗}(u)x − x^{∗}(v)y∥ = x^{∗}(u) − x^{∗}(v) ≤ ∥x − y∥.
■

Another characterization of strict convexity is connected to approximation theory. Let A be a nonempty subset of a Banach space X. Given x ∈ X, by P_{A}x we denote the set of all points in A nearest to x, i.e. points y ∈ A such that ∥x − y∥ = d(x, A). Clearly P_{A}x may be empty. The set-valued mapping P_{A} is called the metric projection of X onto A. As an easy consequence of Theorem 2.3 we obtain the following result.

Theorem 2.6 A Banach space X is strictly convex if and only if for each nonempty convex set A ⊂ X and each x ∈ X the set P_{A}x contains at most one point.

We now pass to the property which is partially dual to strict convexity.

Definition 2.7 A Banach space X is smooth if for each x ∈ S_{X} there is a unique functional x^{∗} ∈ S_{X^{∗}} such that x^{∗}(x) = 1.

From Theorem 2.3 we derive the relations between smoothness and strict convexity.

Theorem 2.8 Let X be a Banach space. Then

1. If X^{∗} is smooth, then X is strictly convex.

2. If X^{∗} is strictly convex, then X is smooth.

Of course, if X is reflexive, then also the opposite implications hold, but in general this is not so (see [5] p. 61). It is also worth while to mention that each separable space X admits an equivalent norm such that X is both strictly convex and smooth with respect to this norm. Moreover, the same is true for reflexive spaces (see [27], p. 148).

Definition 2.7 means that for each x ∈ S_{X} there is only one hyperplane supporting the ball B_{X} at x. This is strongly connected to differentiability of the norm. Let X be a Banach space. A function f : X → ℝ is Gâteaux differentiable at x ∈ X if for each h ∈ X the limit

f′(x)(h) = lim_{t→0} \frac{f(x + th) − f(x)}{t}

exists and f′(x) ∈ X^{∗}. Then f′(x) is called the Gâteaux derivative of f at x. If in addition the above limit is uniform in h ∈ S_{X} (uniform in x, h ∈ S_{X}), we say that f is Fréchet differentiable at x (resp. uniformly Fréchet differentiable). In this case f′(x) is called the Fréchet derivative of f at x.

As an example consider the function f(x) = ∥x∥. We say that the norm is Gâteaux differentiable if f is Gâteaux differentiable at each nonzero point u ∈ X. Fréchet differentiability is defined in a similar way. Notice that by homogenity it is enough to check these conditions for u ∈ S_{X}. By Remark 2.1 f has one-sided directional derivatives at each nonzero point u. This leads us to the following observation.

Remark 2.9 A norm ∥ · ∥ of a space X is Gâteaux differentiable if and only if

lim_{t→0−} \frac{1}{t}(∥u + ty∥ − ∥u∥) = lim_{t→0+} \frac{1}{t}(∥u + ty∥ − ∥u∥)

for any vectors u, y ∈ X, u ≠ 0.

Example 2.10 1. Let X be a Hilbert space and Φ(t) = t^{2} for t ≥ 0. Then it is easy to check that the function x ↦ Φ(∥x∥) is uniformly Fréchet differentiable with Φ′(∥x∥) = 2x for every x ∈ X. Consequently, the function f(x) = ∥x∥ is also uniformly Fréchet differentiable and f′(x) = x/∥x∥ for each x ≠ 0. Here we identify X^{∗} with X.

2. Consider the space L^{p}(Ω) where 1 < p < ∞ and Ω is a measure space. Put Φ(t) = t^{p} for t ≥ 0. Then the function Φ(∥x∥) is uniformly Fréchet differentiable with Φ′(∥x∥) = p|x(ω)|^{p−1} sgn x(ω) for every x ∈ L^{p}(Ω) (see [26], p. 184). It follows that the function f(x) = ∥x∥ is also uniformly Fréchet differentiable with f′(x) = ∥x∥^{1−p}|x(ω)|^{p−2}x(ω) for every x ≠ 0.

There is a tool that can be used without any differentiability assumption. Let f : X → ℝ be a convex function and x ∈ X. The set

∂f(x) = {x^{∗} ∈ X^{∗} : x^{∗}(y − x) ≤ f(y) − f(x) for all y ∈ X}

is said to be the subdifferential of f at x. Elements x^{∗} of ∂f(x) correspond to hyperplanes supporting the epigraph epi f = {(y, t) ∈ X × ℝ : f(y) ≤ t}. Namely, the condition x^{∗} ∈ ∂f(x) means that epi f is entirely on one side of the hyperplane H = {(y, t) ∈ X × ℝ : x^{∗}(y) − t = x^{∗}(x) − f(x)} and intersects H at the point (x, f(x)). If f is continuous, then ∂f(x) ≠ ∅ for every x ∈ X. Moreover, f is Gâteaux differentiable at x if and only if ∂f(x) is a singleton. In this case ∂f(x) = {f′(x)} (see [98]).

Given a function ϕ : [0, +∞) → [0, +∞) and x ∈ X, we put

J_{ϕ}x = {x^{∗} ∈ X^{∗} : x^{∗}(x) = ∥x^{∗}∥∥x∥, ∥x^{∗}∥ = ϕ(∥x∥)}.

The function ϕ is said to be a weight if it is continuous, strictly increasing, ϕ(0) = 0, and lim_{t→+∞} ϕ(t) = +∞. In this case the set-valued mapping x ↦ J_{ϕ}x is called a duality mapping with the weight ϕ. This concept is strongly related to subdifferential.

Theorem 2.11 Let X be a Banach space and ϕ : [0, +∞) → [0, +∞) be a continuous nondecreasing function. Then

∂Φ(∥x∥) = J_{ϕ}x

for every nonzero x ∈ X where Φ(t) = ∫_{0}^{t} ϕ(s) ds, t ≥ 0.

Proof. Clearly, Φ′(t) = ϕ(t) for all t ≥ 0, so the function Φ is convex. Let x ∈ X, x ≠ 0 and assume that x^{∗} ∈ X^{∗} satisfies the conditions x^{∗}(x) = ∥x^{∗}∥∥x∥, ∥x^{∗}∥ = ϕ(∥x∥). We shall show that x^{∗} ∈ ∂Φ(∥x∥), i.e. x^{∗}(y − x) ≤ Φ(∥y∥) − Φ(∥x∥) for every y ∈ X. If ∥y∥ > ∥x∥, then

∥x^{∗}∥ = ϕ(∥x∥) = Φ′(∥x∥) ≤ \frac{Φ(∥y∥) − Φ(∥x∥)}{∥y∥ − ∥x∥}.

Hence Φ(∥y∥) − Φ(∥x∥) ≥ ∥x^{∗}∥(∥y∥ − ∥x∥) ≥ x^{∗}(y) − x^{∗}(x). The proof in the other case is similar.

Assume now that x^{∗} ∈ ∂Φ(∥x∥). Then x^{∗}(y) ≤ x^{∗}(x) for every y ∈ X with ∥y∥ = ∥x∥. It follows that x^{∗}(x) = ∥x∥∥x^{∗}∥. Next,

Φ(t∥x∥) − Φ(∥x∥) ≥ x^{∗}(tx − x) = (t − 1)∥x∥∥x^{∗}∥

for every t > 0. If t > 1, we therefore get

∥x^{∗}∥ ≤ \frac{Φ(t∥x∥) − Φ(∥x∥)}{t∥x∥ − ∥x∥}

which shows that ∥x^{∗}∥ ≤ Φ′(∥x∥) = ϕ(∥x∥). Similarly we obtain the opposite inequality. ■

If ϕ is a weight, then both sets J_{ϕ}x and ∂Φ(∥x∥) are equal to {0} at x = 0. In this case it is therefore not necessary to exclude this point. On the other hand, the subdifferential of the norm at zero is the whole ball B_{X}. Theorem 2.11 with ϕ(t) = 1 for all t ≥ 0 gives us the formula for this subdifferential at nonzero points.

Corollary 2.12 Let X be a Banach space and f(x) = ∥x∥ for x ∈ X. Then

∂f(x) = {x^{∗} ∈ X^{∗} : ∥x^{∗}∥ = 1, x^{∗}(x) = ∥x∥}

for every nonzero x ∈ X.

As a consequence we obtain the following characterization of smoothness (for the direct proof see [9], p. 179).

Theorem 2.13 A Banach space X is smooth if and only if its norm is Gâteaux differentiable.

The function Φ in Theorem 2.11 is differentiable. From Theorem 2.13 we therefore see that if X is smooth, then J_{ϕ}x is a singleton for every nonzero x.

3. Finite dimensional uniform convexity and smoothness

Several uniform versions of strict convexity and smoothness can be found in the literature. They usually correspond to some moduli defined as the least upper or/and greatest lower bounds of some quantities depending on norms of elements of a space. The following remark describes the construction and basic properties of such moduli in a more precise way.

Remark 3.1 Let a > 0 and f : [0, a) → [0, +∞) be a function obtained by taking the least upper or/and greatest lower bounds of a family consisting of convex uniformly bounded functions g : [0, a) → [0, +∞) such that g(0) = 0. Then f is Lipschitzian on each interval [0, b] with b < a and the function t ↦ f(t)/t is nondecreasing.

Of course, the function f need not be convex. Nevertheless, one can construct a convex function which is equivalent to f. Let I = [0, a], where 0 < a < +∞, or I = [0, +∞). Given a function f : I → [0, +∞) with f(0) = 0, by f^{∗} we denote its dual Young’s function, i.e.

f^{∗}(t) = sup{st − f(s) : s ∈ I}

for all t ≥ 0. Clearly, f^{∗} is a convex nonnegative function with f^{∗}(0) = 0. Moreover, the restriction of f^{∗∗} to I is the maximal convex function minorizing f (see [35], Theorem I.4.1).

Lemma 3.2 Let us assume in addition that the function t ↦ f(t)/t is nondecreasing. Then

1. f(t/2) ≤ f^{∗∗}(t) ≤ f(t) for every t ∈ I.

2. lim_{s→0} f(s)/s = sup{t ≥ 0 : f^{∗}(t) = 0}.

Proof.

1. Consider the function f̃: I → [0, +∞) given by the formula

f̃(t) = ∫_{0}^{t} \frac{f(s)}{s} ds.

It is easy to see that f̃ is convex and f(t/2) ≤ f̃(t) ≤ f(t) for every t ∈ I. Consequently, f̃(t) ≤ f^{∗∗}(t) ≤ f(t) for all t ∈ I.

2. Let α = lim_{s→0} f(s)/s and t ≥ 0 be such that f^{∗}(t) = 0. Then f(s) ≥ st for every s ∈ I. Hence α ≥ t which shows that α ≥ sup{t ≥ 0 : f^{∗}(t) = 0}. Next, observe that f(s) ≥ αs for every s ∈ I. Therefore f^{∗}(α) ≤ 0. But f(0) = 0 which implies that f^{∗} is nonnegative. It follows that α ≤ sup{t ≥ 0 : f^{∗}(t) = 0}.

■

We review those moduli which are most frequently applied in metric fixed point theory.

Definition 3.3 Let X be a Banach space. Given z ∈ S_{X} and ϵ ∈ [0, 2], we put

δ_{X}(z; ϵ) = inf \{1 − ∥\frac{x + y}{2} ∥ : x, y ∈ B_{X}, ∥x − y∥ ≥ ϵ, x − y = tz for some t ≥ 0\} .

The space X is called uniformly convex (or uniformly rotund) in every direction (UCED for short) if δ_{X}(z; ϵ) > 0 for every z ∈ S_{X} and every ϵ ∈ (0, 2].

The class of all UCED spaces is quite large. For instance every separable Banach space X admits an equivalent norm with respect to which it is UCED (see [109]). Here we will establish some properties of the modulus δ_{X}(z; ϵ).

Proposition 3.4 Let X be a Banach space with dim(X) ≥ 2, z ∈ S_{X} and ϵ ∈ [0, 2]. Then

δ_{X}(z; ϵ) = inf \{1 − ∥\frac{x + y}{2} ∥ : x, y ∈ S_{X}, x − y = ϵz\} .

Proof. Take 0 < ϵ < 2. It suffices to show that for any x, y ∈ B_{X} with ∥x − y∥ ≥ ϵ and x − y ∈ span({z}) there exist u, v ∈ S_{X} such that u − v = ϵz and ∥(x + y)/2∥ ≤ ∥(u + v)/2∥. We can clearly suppose that x − y = ϵz. Moreover, translating x and y in the direction of (x + y)/2, we can also assume that x ∈ S_{X}. Then we choose norm-one vectors u, v in a two dimensional subspace containing x, y for which u − v = x − y and y, u, v are contained in one of the half planes determined by the line joining x with −x. Then λ(x + y)/2 = βu + (1 − β)x for some λ ≥ 1 and β ≥ 0. Observe that

\frac{λ}{2}(u + v) = (β + λ)u + (1 − β − λ)x,

so the points u, λ(x + y)/2, λ(u + v)/2 lie on the same ray with the initial point x. Moreover, ∥x∥ = 1 = ∥u∥, λ∥(u+v)/2∥ ≥ 1 and λ(x+y)/2 lies between x and λ(u+v)/2 on the ray. From Remark 2.1 we therefore see that ∥(x + y)/2∥ ≤ ∥(u + v)/2∥. ■

Example 3.5 1. Let X be a Hilbert space. Using the parallelogram law, one can easily show that

δ_{X}(z; ϵ) = 1 − \sqrt{1 − (\frac{ϵ}{2})^{2}}

for every z ∈ S_{X} and every ϵ ∈ [0, 2].

2. Let X be the plane ℝ^{2} with the norm:

∥(x, y)∥ = max \{|x|, |\frac{1}{3}x − \frac{4}{3}y| , |\frac{1}{4}x + y|\}

where (x, y) ∈ ℝ^{2}. Direct computation gives

δ_{X}(z; ϵ) = \begin{cases}\frac{1}{8}ϵ & if 0 ≤ ϵ ≤ 1, \\ \frac{3}{8}ϵ − \frac{1}{4} & if 1 < ϵ ≤ \frac{6}{5}, \\ \frac{1}{6}ϵ & if \frac{6}{5} < ϵ ≤ 2 \end{cases}

where z = (1, 0). This example shows that δ_{X}(z; ϵ) need not be a convex function.

Our next result shows that the modulus δ_{X}(z; ϵ) fits the general scheme described at the beginning of this section. Consequently, it satisfies the conclusion of Remark 3.1.

Proposition 3.6 Let X be a Banach space with dim(X) ≥ 2 and z ∈ S_{X}. Then there exists a family F of convex functions f : [0, 2] → [0, 1] such that f(0) = 0 and

δ_{X}(z; ϵ) = inf{f(ϵ) : f ∈ F}

for every ϵ ∈ [0, 2].

Proof. Given u ∈ S_{X}, ϵ ∈ [0, 2], we set δ(u, z; ϵ) = inf{1 − ∥(x + y)/2∥} where the infimum is taken over all x, y ∈ B_{X} such that x + y = sz for some s ≥ 0 and x − y = tz for some t ≥ 0. One can easily check that the family F of all functions δ(u, z; ϵ) where u ∈ S_{X} has the desired properties. ■

UCED can be also characterized without using the modulus δ_{X}(z; ϵ).

Theorem 3.7 A Banach space X is UCED if and only if lim_{n→∞} ∥x_{n} − y_{n}∥ = 0 whenever (x_{n}), (y_{n}) are sequences in X such that lim_{n→∞} ∥x_{n}∥ = 1, lim_{n→∞} ∥y_{n}∥ = 1, lim_{n→∞} ∥x_{n} + y_{n}∥ = 2 and there is z ∈ S_{X} with x_{n} − y_{n} ∈ span({z}) for each n.

Proof. Suppose that a space X is UCED and there exist sequences (x_{n}), (y_{n}) in X with the following properties: lim_{n→∞} ∥x_{n}∥ = 1, lim_{n→∞} ∥y_{n}∥ = 1, lim_{n→∞} ∥x_{n} + y_{n}∥ = 2, lim inf_{n→∞} ∥x_{n} − y_{n}∥ > c > 0 and there is z ∈ S_{X} such that x_{n} − y_{n} ∈ span({z}) for each n. Given 0 < γ, we can assume that x_{n}, y_{n} ∈ (1 + γ)B_{X}, ∥(x_{n} + y_{n})/2∥ ≥ 1 − γ and ∥x_{n} − y_{n}∥ ≥ c for all n. Then

δ_{X} (z; \frac{c}{1 + γ} ) ≤ 1 − \frac{1}{1 + γ} ∥\frac{x_{n} + y_{n}}{2} ∥ ≤ \frac{2γ}{1 + γ} .

Since γ > 0 is arbitrary and δ_{X}(z; ϵ) is continuous, δ_{X}(z; c) = 0, contrary to our assumption.

Assume now that X is not UCED. Then δ_{X}(z; ϵ) = 0 for some z ∈ S_{X} and ϵ > 0. From the definition of δ_{X}(z; ϵ) we obtain sequences (x_{n}), (y_{n}) in B_{X} such that ∥x_{n} − y_{n}∥ ≥ ϵ and 1 − 1/n ≤ ∥(x_{n} + y_{n})/2∥ for every n. Clearly,

2 − \frac{2}{n} ≤ ∥x_{n}∥ + ∥y_{n}∥ ≤ ∥x_{n}∥ + 1 ≤ 2

for all n. This shows that lim_{n→∞} ∥x_{n}∥ = 1. Similarly, lim_{n→∞} ∥y_{n}∥ = 1, which completes our proof. ■

The modulus δ_{X}(z; ϵ) has local character since it depends on the direction z. Taking the infimum over z, we obtain the global modulus of convexity. This is however a special case of more general idea of measuring convexity of balls. To describe this idea we need to recall concept of a k-dimensional volume. Let X be a Banach space. Given x_{1}, …, x_{k+1} ∈ X, we put

A(x_{1}, …, x_{k+1}) = sup \{det |\begin{matrix}1 & … & 1 \\ x_{1}^{∗}(x_{1}) & … & x_{1}^{∗}(x_{k+1}) \\ ⋮ & ⋮ & ⋮ \\ x_{k}^{∗}(x_{1}) & … & x_{k}^{∗}(x_{k+1}) \end{matrix}| : x_{1}^{∗}, …, x_{k}^{∗} ∈ B_{X^{∗}} \} . (3.1)

Clearly, A(x_{1}, x_{2}) = ∥x_{1} − x_{2}∥ for any x_{1}, x_{2} ∈ X. Moreover, in the case when X is the k-dimensional Euclidean space, \frac{1}{k!}A(x_{1}, …, x_{k+1}) is equal to the volume of co({x_{1}, …, x_{k+1}}) and A(x_{1}, …, x_{k+1}) is equal to the volume of the parallelepiped drawn on the vectors x_{1}, …, x_{k+1}. Formula (3.1) can be therefore seen as a definition of k-dimensional volume of such parallelepiped for an arbitrary Banach space. We put

μ_{X}^{k} = sup{A(x_{1}, …, x_{k+1}) : x_{1}, …, x_{k+1} ∈ B_{X}}.

Let now X be a Banach space with dim(X) ≥ k + 1. This will be the standing assumption when dealing with the following modulus of k-convexity of X.

Definition 3.8

δ_{X}^{k}(ϵ) = inf \{1 − ∥\frac{1}{k + 1} ∑_{i=1}^{k+1} x_{i}∥ : x_{1}, …, x_{k+1} ∈ B_{X}, A(x_{1}, …, x_{k+1}) ≥ ϵ\}

where ϵ ∈ [0, μ_{X}^{k}). The space X is said to be k-uniformly rotund (k-UR for short) if δ_{X}^{k}(ϵ) > 0 for every 0 < ϵ < μ_{X}^{k}.

Additionally, we say that all spaces of dimension less than k + 1 are k-UR. The modulus with k = 1 was introduced as the first one and in this case slightly different notation and terminology is used. Namely, we write δ_{X} or just δ instead of δ_{X}^{1}. Clearly,

δ_{X}(ϵ) = inf \{1 − ∥\frac{x + y}{2} ∥ : x, y ∈ B_{X}, ∥x − y∥ ≥ ϵ\}

where ϵ ∈ [0, 2]. The function δ_{X} is called the modulus of convexity of X and instead of 1-uniform rotundity terms uniform convexity (UC for short) or uniform rotundity (UR for short) are used.

Remark 3.9 1. In the definition of δ_{X}^{k}(ϵ) the inequality in “A(x_{1}, …, x_{k+1}) ≥ ϵ” can be replaced by the equality.

2. The function δ_{X}^{k} : [0, μ_{X}^{k}) → [0, 1] is nondecreasing.

3. We have δ_{X}^{k}(ϵ) = inf δ_{E}^{k}(ϵ) where the infimum is taken over all subspaces E of X with dim(E) = k + 1.

The same reasoning as in the proof of Proposition 3.4 applies to the modulus δ.

Proposition 3.10 Let X be a Banach space with dim(X) ≥ 2 and ϵ ∈ [0, 2]. Then

δ(ϵ) = inf \{1 − ∥\frac{x + y}{2} ∥ : x, y ∈ S_{X}, ∥x − y∥ = ϵ\} .

The modulus of convexity δ need not be a convex function (see [78] or [44]). On the other hand, the idea of the proof of Proposition 3.6 can be extended even to δ_{X}^{k}. Indeed, let a > 0 and k ∈ ℕ. We say that a function f : [0, a) → [0, 1] is k-convex if the function ϵ ↦ f(ϵ^{k}) is convex. In [71] the following result was proved.

Proposition 3.11 Let X be a Banach space. There exists a family F of k-convex functions f : [0, μ_{X}^{k}) → [0, 1] such that f(0) = 0 and

δ_{X}^{k}(ϵ) = inf{f(ϵ) : f ∈ F}

for every ϵ ∈ [0, μ_{X}^{k}).

Notice that in general δ need not be continuous at 2 (see [44]). Nevertheless, Proposition 3.11 and Remark 3.1 enable us to establish continuity of δ_{X}^{k} inside its domain.

Corollary 3.12 The function δ_{X}^{k} is Lipschitzian on each interval (a, b) where 0 < a < b < μ_{X}^{k} and the function δ_{X}^{k}(ϵ^{k})/ϵ is nondecreasing. In particular, δ_{X}^{k} is continuous on [0, μ_{X}^{k}).

Example 3.13 1. Let X be a Hilbert space. In [11] it was shown that

δ_{X}^{k}(ϵ) = 1 − (1 − \frac{k}{(k + 1)^{1+1/k}} ϵ^{2/k})^{1/2}

for every ϵ in the domain of this function. If k = 1, this is just an easy consequence of the parallelogram law (compare to Example 3.5.1).

2. Let 1 < p < ∞ and X = L^{p}([0, 1]). In this space we have the following Clarkson’s and Hanner’s inequalities (see [22], [50]). If p ≥ 2, then

2(∥x∥^{p} + ∥y∥^{p}) ≤ ∥x + y∥^{p} + ∥x − y∥^{p} ≤ (∥x∥ + ∥y∥)^{p} + |∥x∥ − ∥y∥|^{p}

for all x, y ∈ X. Moreover, in case 1 < p ≤ 2 the opposite inequalities hold. From these inequalities one can derive the following formulae (see [50]). If p ≥ 2, then

δ_{X}(ϵ) = 1 − (1 − (\frac{ϵ}{2})^{p})^{1/p} ,  (3.2)

and if 1 < p ≤ 2, then

(1 − δ_{X}(ϵ) + \frac{ϵ}{2})^{p} + |1 − δ_{X}(ϵ) − \frac{ϵ}{2}|^{p} = 2  (3.3)

where ϵ ∈ [0, 2].

Analysis of the proof in [50] shows that in the above example the interval [0, 1] can be replaced by any measure space Ω such that L^{p}(Ω) is at least two dimensional. Notice that L^{p}(Ω) is n dimensional only if the measure is purely atomic with n atoms (see [5], p. 30). Then this space is isometrically isomorphic to ℝ^{n} with the l^{p}-norm.

There is a general way of showing that two spaces have the same modulus of convexity. Let X and Y be Banach spaces. We say that Y is finitely representable in X whenever for every finite dimensional subspace F of Y and every γ > 0 there exists a linear one-to-one operator T : F → X such that ∥T ∥∥T^{−1}∥ < 1 + γ.

Remark 3.14 Let 1 ≤ p < ∞ and Ω be a measure space such that L^{p}(Ω) is infinite dimensional. Then each of the spaces L^{p}(Ω), L^{p}([0, 1]) is finitely representable in the other one (see [56], p. 60).

It is easy to see that if Y is finitely representable in X, then δ_{X}^{k}(ϵ) ≤ δ_{Y}^{k} (ϵ) for every k and every ϵ ∈ [0, μ_{Y}^{k}). Remark 3.14 gives us therefore another way of extending formulae (3.2) and (3.3) to the space X = L^{p}(Ω) where 1 < p < ∞ and Ω is an arbitrary measure space such that L^{p}(Ω) is infinite dimensional.

By Dvoretzky’s theorem (see [34] or [75]), l^{2} is finitely representable in any infinite dimensional Banach space X. Consequently,

δ_{X}^{k}(ϵ) ≤ δ_{l^{2}}^{k}(ϵ) = 1 − (1 − \frac{k}{(k + 1)^{1+1/k}} ϵ^{2/k})^{1/2} .

In case k = 1 this estimate holds even under the weaker assumption that dim(X) ≥ 2. This result is known as the Day-Nordlander theorem (see [87] or [27], p. 60).

A Banach space X is said to be superreflexive if each Banach space finitely representable in X is reflexive. In particular, every superreflexive space is reflexive. Moreover, a space X is superreflexive if and only if X^{∗} has the same property (see [26], p.152). The coefficient

ϵ_{0}^{k}(X) = sup{ϵ : δ_{X}^{k}(ϵ) = 0}

is called the characteristic of k-convexity of the space X. If k = 1, we denote it by ϵ_{0}(X) and call simply the characteristic of convexity of X. The condition ϵ_{0}(X) < 2 characterizes the so-called uniformly nonsquare spaces (see [53]). The following result was proved in [12].

Theorem 3.15 Let X be a Banach space and k ∈ ℕ. If ϵ_{0}^{k}(X) < 2^{k}, then X is superreflexive.

This shows that uniformly convex spaces and even uniformly nonsquare spaces are superreflexive. On the other hand, superreflexive spaces can be renormed in such a way that they become uniformly convex (see [36]). Actually, for each superreflexive space X there exist p ≥ 2, K > 0, and an equivalent norm ∥ · ∥ in X such that

δ(ϵ) ≥ Kϵ^{p}

for any ϵ ∈ [0, 2], where δ is the modulus corresponding to the norm ∥ · ∥ (see [92] or [26], p. 154).

A modification of the proof of Theorem 3.7 gives us its counterpart for k-UC.

Theorem 3.16 A Banach space X is k-UC if and only if lim_{n→∞} A(x_{n}^{1}, …, x_{n}^{k+1}) = 0 whenever (x_{n}^{1}), …, (x_{n}^{k+1}) are sequences in X such that lim_{n→∞} ∥x_{n}^{i}∥ = 1 for i = 1, …, k + 1 and lim_{n→∞} ∥x_{n}^{1} + ⋯ + x_{n}^{k+1}∥ = k + 1.

Clearly, spaces,UCED spaces are spaces,UC and Theorem 2.3 shows that UC spaces are spaces,strictly convex. By Theorem 3.16 with k = 1 and the compactness argument, UC is actually equivalent to strict convexity for finite dimensional spaces. We now pass to a result which gives a uniform version of condition 2 in Theorem 2.3.

Theorem 3.17 A Banach space X is UC if and only if for any 1 < p < ∞ and r > 0 there is a convex strictly increasing function g_{r} : [0, +∞) → [0, +∞) such that g_{r}(0) = 0 and

∥λx + (1 − λ)y∥^{p} ≤ λ∥x∥^{p} + (1 − λ)∥y∥^{p} − λ(1 − λ)g_{r}(∥x − y∥)  (3.4)

for all x, y ∈ rB_{X} and all 0 ≤ λ ≤ 1.

Proof. Assume that X is UC and let 1 < p < ∞. Clearly, it is enough to prove the conclusion for r = 1. From (2.1) it follows that

μ_{0}(t) = inf {2^{p−1}(∥x∥^{p} + ∥y∥^{p}) − ∥x + y∥^{p} : x, y ∈ B_{X}, ∥x − y∥ ≥ t} ≥ 0

for every 0 < t ≤ 2.

Suppose that μ_{0}(t) = 0 for some t > 0. Then there exist sequences (x_{n}), (y_{n}) in B_{X} such that ∥x_{n} − y_{n}∥ ≥ t for all n and

lim_{n→∞} 2^{p−1}(∥x_{n}∥^{p} + ∥y_{n}∥^{p}) − ∥x_{n} + y_{n}∥^{p} = 0.

Passing to subsequences, we can assume that the limits

a = lim_{n→∞} ∥x_{n}∥ , b = lim_{n→∞} ∥y_{n}∥ and c = lim_{n→∞} ∥x_{n} + y_{n}∥

exist. For these values of a and b equality in inequality (2.1) holds. Consequently, a = b > 0 and so c^{p} = 2^{p}a^{p}. From Theorem 3.16 we see that lim_{n→∞} ∥x_{n} − y_{n}∥ = 0, which is a contradiction. This gives us the inequality μ_{0}(t) > 0 for every 0 < t ≤ 2.

We set

μ(t) = inf{(λ(1 − λ))^{−1}(λ∥x∥^{p} + (1 − λ)∥y∥^{p} − ∥λx + (1 − λ)y∥^{p})}

where the infimum is taken over all x, y ∈ B_{X} with ∥x − y∥ ≥ t and 0 < λ < 1. A similar reasoning as in the proof of Theorem 2.3 shows that μ(t) ≥ 2^{1−p}μ_{0}(t) > 0 for every 0 < t ≤ 2 and it suffices to take as g_{1} the double dual Young’s function μ^{∗∗}.

Assume now that (3.4) is satisfied. If x, y ∈ B_{X}, ∥x − y∥ = ϵ, then

∥\frac{x + y}{2} ∥ ≤ 1 − \frac{1}{4}g_{1}(ϵ).

Thus we obtain the estimate δ(ϵ) ≥ \frac{1}{4}g_{1}(ϵ) which shows that the space X is UC. ■

Consider the case when X is a Hilbert space. Then the generalized parallelogram law

∥λx + (1 − λ)y∥^{2} = λ∥x∥^{2} + (1 − λ)∥y∥^{2} − λ(1 − λ)∥x − y∥^{2}

holds for all x, y ∈ X and 0 ≤ λ ≤ 1. We can therefore put g_{r}(t) = t^{2} for every r.

A different general way of measuring convexity of balls in Banach spaces was developed by Milman [84]. Here we recall only its particular case. Let X be a Banach space with dim(X) ≥ k + 1 and E_{k} be the family of all k dimensional subspaces of X.

Definition 3.18

Δ_{X}^{k}(ϵ) = inf_{x∈S_{X}} inf_{E∈E_{k}} sup_{y∈S_{E}} ∥x + ϵy∥ − 1

where ϵ ≥ 0. A Banach space X is called k-uniformly convex (k-UC for short) if Δ_{X}^{k}(ϵ) > 0 for every ϵ > 0.

In addition we say that all spaces X with dim(X) < k + 1 are k-UC. Using Remarks 2.1 and 3.1, one can easily obtain the following properties of the above modulus.

Remark 3.19 Let f denote the function Δ_{X}^{k}. Then

1. f(0) = 0 ≤ f(ϵ) for every ϵ ≥ 0.

2. The function ϵ ↦ f(ϵ)/ϵ is nondecreasing on (0, +∞).

3. f satisfies the Lipschitz condition with the constant 1.

The next two facts are also worth noting. We have

Δ_{X}^{k}(ϵ) = inf Δ_{E}^{k}(ϵ)

for every ϵ ≥ 0 where the infimum is taken over all subspaces E of X with dim(E) ≥ k + 1. Secondly, Δ_{X}^{k}(ϵ) ≤ Δ_{X}^{k+1}(ϵ) for every ϵ ≥ 0 and consequently k-UC implies (k + 1)-UC. Of course,

Δ_{X}^{1}(ϵ) = inf_{x,y∈S_{X}} max{∥x + ϵy∥, ∥x − ϵy∥} − 1

for every ϵ ≥ 0. It turns out that there is a simple relation between this modulus and δ_{X}. Namely,

Δ_{X}^{1} (\frac{ϵ}{2(1 − δ_{X}(ϵ))}) = \frac{δ_{X}(ϵ)}{1 − δ_{X}(ϵ)}  (3.5)

for every ϵ ∈ [0, 2). This formula was established in [37] (see also [28]). It gives us a special instance of the following general result proved in [72].

Theorem 3.20 Let k ∈ ℕ. A Banach space X is k-UR if and only if X is k-UC.

In order to characterize the property dual to k-UC we need to introduce one more modulus. Let X be a Banach space with dim(X) ≥ k + 1.

Definition 3.21 The modulus β_{X}^{k} of k-uniform smoothness is given by the formula

β_{X}^{k}(t) = sup_{x∈S_{X}} sup_{E∈E_{k}} inf_{y∈S_{E}} \frac{1}{2}(∥x + ty∥ + ∥x − ty∥) − 1

where t ≥ 0. A Banach space X is called k-uniformly smooth (k-US for short) if

lim_{t→0} \frac{β_{X}^{k}(t)}{t} = 0.

In addition we say that all spaces X with dim(X) < k+1 are k-US. The function f = β_{X}^{k} has the properties listed in Remark 3.19. Moreover, β_{X}^{k+1}(t) ≤ β_{X}^{k}(t) for every t ≥ 0 and consequently k-US implies (k + 1)-US. It is also clear that β_{X}^{k}(t) = inf β_{E}^{k}(t) for every t ≥ 0 where the infimum is taken over all subspaces E of X with dim(E) ≥ k + 1. The following duality theorem can be found in [5] (see also [80]).

Theorem 3.22 Let k ∈ ℕ and X be a Banach space. X (X^{∗}) is k-UC if and only if X^{∗} (resp. X) is k-US.

From Theorems 3.22 and 3.15 we see that k-US spaces are superreflexive. In case k = 1 a different notation and terminology is used. Namely, instead of β_{X}^{1} we write ρ_{X} or simply ρ. We called this function the modulus of uniform smoothness of the space X and instead of 1-uniform smoothness we just use the term uniform smoothness (US for short). Clearly,

ρ_{X}(t) = sup_{x,y∈S_{X}} \frac{1}{2}(∥x + ty∥ + ∥x − ty∥) − 1.

Let u, v be nonzero vectors of a Banach space X. The function f(t) = ∥u + tv∥ + ∥u − tv∥ is convex on ℝ. This shows that the modulus ρ_{X} is also a convex function. Moreover, f(−t) = f(t) for every t ∈ ℝ. Consequently, f is nondecreasing on the interval [0, +∞). It easily follows that in the definition of ρ_{X} one can replace the condition “x, y ∈ S_{X}” by “x, y ∈ B_{X}”.

Example 3.23 1. Let X be a Hilbert space with dim(X) ≥ 2. Then

ρ_{X}(t) = \sqrt{1 + t^{2}} − 1

for every t ≥ 0.

2. Let 1 < p < ∞ and Ω be a measure space such that the space X = L^{p}(Ω) is at least two dimensional. Using Clarkson’s and Hanner’s inequalities and Remark 3.14, one can show that if 1 < p ≤ 2, then

ρ_{X}(t) = (1 + t^{p})^{1/p} − 1,

and if p > 2, then

ρ_{X}(t) = (\frac{(1 + t)^{p} + |1 − t|^{p}}{2} )^{1/p} − 1

for every t ≥ 0 (see [74]).

The proof of Theorem 3.22 does not give any satisfactory relations between moduli of convexity and smoothness. Such relations, known as Lindenstrauss’ formulae, were obtained for k = 1.

Theorem 3.24 Let X be a Banach space and t ≥ 0. Then

(i) ρ_{X^{∗}}(t) = sup{tϵ/2 − δ_{X}(ϵ) : 0 ≤ ϵ ≤ 2},

(ii) ρ_{X}(t) = sup{tϵ/2 − δ_{X^{∗}}(ϵ) : 0 ≤ ϵ ≤ 2}.

Proof. We will prove only formula (i). We have

2ρ_{X^{∗}}(t) = sup{∥x^{∗} + ty^{∗}∥ + ∥x^{∗} − ty^{∗}∥ − 2 : x^{∗}, y^{∗} ∈ S_{X^{∗}}}
= sup{x^{∗}(x) + ty^{∗}(x) + x^{∗}(y) − ty^{∗}(y) − 2 : x, y ∈ S_{X}, x^{∗}, y^{∗} ∈ S_{X^{∗}}}
= sup{∥x + y∥ + t∥x − y∥ − 2 : x, y ∈ S_{X}}
= sup{∥x + y∥ + tϵ − 2 : x, y ∈ S_{X}, ∥x − y∥ = ϵ, 0 ≤ ϵ ≤ 2}
= sup{tϵ − 2δ_{X}(ϵ) : 0 ≤ ϵ ≤ 2}.

Equality (ii) can be obtained in a similar way. ■

The coefficient

ρ_{0}(X) = lim_{t→0} \frac{ρ_{X}(t)}{t}

is called the characteristic of smoothness of a Banach space X. Lindenstrauss’ formulae means that 2ρ_{X^{∗}} and 2ρ_{X} are dual Young’s functions of 2δ_{X} and 2δ_{X^{∗}}, respectively. By Lemma 3.2 we therefore obtain the following equalities.

Corollary 3.25 Let X be a Banach space. Then 2ρ_{0}(X^{∗}) = ϵ_{0}(X) and 2ρ_{0}(X) = ϵ_{0}(X^{∗}).

This and Theorem 3.15 show that if ρ_{0}(X) < 1, then X is superreflexive. From Lindenstrauss’ formulae and the corresponding theorems for the modulus of convexity we can also deduce the next two results. If dim(X) ≥ 2, then

ρ_{X}(t) ≥ \sqrt{1 + t^{2}} − 1  (3.6)

for every t ≥ 0. Moreover, for each superreflexive space X there exist an equivalent norm ∥ · ∥ and constants C > 0, q ≤ 2 such that ρ(t) ≤ Ct^{q} for every t ≥ 0 where the modulus ρ is evaluated with respect to ∥ · ∥ (see [26], p. 157).

Uniform smoothness is strictly related to differentiability of a norm.

Theorem 3.26 A Banach space X is US if and only if the norm of X is uniformly Fréchet differentiable.

Proof. Let X be a Banach space and f(x) = ∥x∥ for x ∈ X. Clearly,

0 ≤ \frac{1}{t}(f(x + ty) − 1) − \frac{1}{−t}(f(x − ty) − 1) ≤ 2\frac{ρ_{X}(t)}{t}  (3.7)

for all x, y ∈ S_{X} and t > 0. In light of Remark 2.9 this shows that if X is US, then the norm of X is Gâteaux differentiable. Moreover, from Remark 2.1 we see that the directional derivative f′(x)y lies between the divided differences which appear in (3.7). It follows that they tend to f′(x)y as t → 0 uniformly with respect to x, y ∈ S_{X}.

To show the opposite implication we observe that

2\frac{ρ_{X}(t)}{t} ≤ sup_{x,y∈S_{X}} |\frac{1}{t}(f(x + ty) − 1) − f′(x)y| + sup_{x,y∈S_{X}} |\frac{1}{−t}(f(x − ty) − 1) − f′(x)y|

for every t > 0. If f is uniformly Fréchet differentiable, then the right-hand side expression tends to 0 as t → 0. Consequently, lim_{t→0} ρ_{X}(t)/t = 0. ■

4. Infinite dimensional geometrical properties

In the sequel τ will denote a Hausdorff topology on a Banach space X. We will always assume that

1. scalar multiplication is sequentially continuous with respect to τ, i.e. if a sequence of scalars (t_{n}) converges to t and a sequence (x_{n}) converges to x in τ, then (t_{n}x_{n}) converges to tx in τ,

2. if a sequence (x_{n}) converges to x in τ and y ∈ X, then (x_{n} + y) converges to x + y in τ,

3. the norm of X is sequentially lower semicontinuous with respect to τ, i.e.

∥x∥ ≤ lim inf_{n→∞} ∥x_{n}∥

whenever (x_{n}) converges to x in τ.

Of course, if a topology τ is linear, then it satisfies conditions 1 and 2. The basic examples of Hausdorff topologies which satisfies 1, 2, and 3 are τ = w, i.e. the weak topology of X and τ = w^{∗}, i.e. the weak^{∗} topology if X is a dual space. Another important example is the topology of convergence locally in measure (clm for short) in L^{p} spaces. Namely, let Ω be a σ-finite measure space with a measure μ and 1 ≤ p < ∞. We fix a partition {Ω_{n}} of Ω into subsets of positive finite measures. The clm topology can be defined via the metric

d(f, g) = ∑_{n=1}^{∞} \frac{1}{2^{n}μ(Ω_{n})} ∫_{Ω_{n}} \frac{|f − g|}{1 + |f − g|} dμ

where f, g ∈ L^{p}(Ω). Each clm-convergent sequence contains a subsequence that converges a.e. to the same limit. This and Fatou’s lemma show that the norm of L^{p}(Ω) is clm-sequentially lower semicontinuous. It is also easy to see that clm-topology is weaker than the norm topology in L^{p}(Ω).

If μ(Ω) < ∞, then instead of d we can consider the simpler metric

d_{0}(f, g) = ∫_{Ω} \frac{|f − g|}{1 + |f − g|} dμ.

In this case the clm topology reduces to the topology of convergence in measure. The set ℕ with the counting measure is another special case. Then we obtain the space l^{p} and clm convergence of a sequence is just coordinatewise convergence. For bounded sequences it is therefore equivalent to weak convergence if p > 1 and to weak^{∗} convergence if p = 1 and l^{1} is seen as the dual space of c_{0}.

Definition 4.1 A Banach space X (or its norm) has the Kadec-Klee property with respect to a topology τ (KK(τ) for short) provided that if (x_{n}) is a sequence in B_{X} converging to x ∈ X in τ and lim inf_{n→∞} ∥x_{n} − x∥ > 0, then ∥x∥ < 1.

This definition means that if a sequence (x_{n}) in S_{X} converges to x ∈ S_{X} in the topology τ, then it converges to x in norm. In particular, if a space X has KK(w) property, then weak convergence of a sequence in S_{X} to a limit in S_{X} is equivalent to convergence in norm.

We will define a uniform version of KK(τ) property. For brevity the limit of a sequence (x_{n}) with respect to a topology τ will be denoted by τ- lim_{n→∞} x_{n}. We say that (x_{n}) is τ-null if the limit is equal to 0. Next, by N_{1}(τ) (N^{1}(τ)) we denote the set of all τ-null sequences (x_{n}) such that ∥x_{n}∥ ≥ 1 (resp. ∥x_{n}∥ ≤ 1) for all n. The condition N_{1}(τ) = ∅ characterizes the spaces for which the convergence of sequences with respect to τ is equivalent to the norm convergence. In case τ = w this is the so-called Schur property. Obviously all finite dimensional spaces have this property and by Schur’s theorem the same is true for l^{1}. We set ν_{X,τ} = sup inf_{n∈ℕ} ∥x_{n} − x∥ where the supremum is taken over all sequences (x_{n}) ∈ B_{X} with x = τ- lim_{n→∞} x_{n}.

Definition 4.2 Let X be a Banach space with N_{1}(τ) ≠ ∅. Given ϵ ∈ [0, ν_{X,τ}), we put

K_{X,τ} (ϵ) = inf{1 − ∥x∥}

where the infimum is taken over all points x such that x = τ- lim_{n→∞} x_{n} for some sequence (x_{n}) in B_{X} with ∥x_{n} − x∥ ≥ ϵ for all n. The space X has the uniform Kadec-Klee property with respect to τ (UKK(τ) for short) if K_{X,τ} (ϵ) > 0 for every ϵ ∈ (0, ν_{X,τ}).

Note that since N_{1}(τ) ≠ ∅, there exists a τ-null sequence in S_{X}. It follows that 1 ≤ ν_{X,τ} and the set over which we take the infimum in the formula for K_{X,τ} (ϵ) is not empty. In addition, we say that spaces X with N_{1}(τ) = ∅ also have UKK(τ) property. In case τ = w we drop the name of the topology in our notation. Thus KK, UKK and the modulus K_{X} refer to the weak topology.

Clearly, UKK(τ) property implies KK(τ) property and in general they are not equivalent. For instance each separable space and each reflexive space can be renormed to have KK property (see [26]) which is not the case for UKK property (see [93]).

Proposition 4.3 There exists a family F of convex functions f : [0, ν_{X,τ}) → [0, 1] such that f(0) = 0 for every f ∈ F and

K_{X,τ} (ϵ) = inf{f(ϵ) : f ∈ F}

for every ϵ ∈ [0, ν_{X,τ}).

Proof. Take u ∈ S_{X} and a τ-null sequence (u_{n}) in S_{X}. Having ϵ ∈ [0, ν_{X,τ}), by K(ϵ; u, (u_{n})) we denote the infimum of all numbers t ∈ [0, 1] for which there is a scalar sequence (s_{n}) such that ∥(1 − t)u + s_{n}u_{n}∥ ≤ 1 and s_{n} ≥ ϵ for all n. It is easy to check that K(ϵ; u, (u_{n})) is a convex function of ϵ and K_{X,τ} (ϵ) = inf K(ϵ; u, (u_{n})) where the infimum is taken over all u ∈ S_{X} and all τ-null sequences (u_{n}) in S_{X}. ■

Corollary 4.4 The modulus K_{X,τ} satisfies the conclusion of Remark 3.1.

Our next aim is to establish a relation between K_{X} and δ_{X}^{k}. For this purpose we need the following lemma.

Lemma 4.5 Let (x_{n}) be a weakly null sequence in a Banach space X and (x_{n}^{∗}) be a bounded sequence in X^{∗}. For every ϵ > 0 there exists an increasing sequence (n_{k}) such that |x_{n_{k}}^{∗}(x_{n_{i}})| < ϵ if i ≠ k.

Proof. We can assume that X is a separable space. Passing to a subsequence, we can also assume that (x_{n}^{∗}) converges weak^{∗} to some y^{∗} ∈ B_{X^{∗}}. The desired subsequence can be chosen by induction. Given ϵ > 0, we find n_{1} so that |y^{∗}(x_{n})| < ϵ/2 for all n ≥ n_{1}. Next, having n_{1} < ⋯ < n_{k−1}, we pick n_{k} > n_{k−1} with |x_{n_{i}}^{∗}(x_{n_{k}})| < ϵ and |(x_{n_{k}}^{∗} − y^{∗})(x_{n_{i}})| < ϵ/2 for i = 1, …, k − 1. Then |x_{n_{k}}^{∗}(x_{n_{i}})| < ϵ. ■

Theorem 4.6 Let X be a Banach space and k ∈ ℕ. Then

δ_{X}^{k}(ϵ^{k}) ≤ K_{X}(ϵ)

for every ϵ ∈ [0, ν_{X}).

Proof. We fix ϵ ∈ [0, ν_{X}) and take a sequence (x_{n}) in B_{X} such that it converges weakly to x and inf_{n∈ℕ} ∥x_{n} − x∥ ≥ ϵ. For each n there is x_{n}^{∗} ∈ S_{X^{∗}} such that x_{n}^{∗}(x_{n} − x) = ∥x_{n} − x∥.

Take now γ > 0. In view of Lemma 4.5 we can assume that |x_{n}^{∗}(x_{m} − x)| < γ if m ≠ n. Then

A(x, x_{n_{1}}, …, x_{n_{k}}) ≥ det |\begin{matrix}x_{n_{1}}^{∗}(x_{n_{1}} − x) & … & x_{n_{1}}^{∗}(x_{n_{k}} − x) \\ ⋮ & ⋮ & ⋮ \\ x_{n_{k}}^{∗}(x_{n_{1}} − x) & … & x_{n_{k}}^{∗}(x_{n_{k}} − x) \end{matrix}| ≥ ϵ^{k} − (k! − 1)γ^{k}

whenever n_{1} < ⋯ < n_{k}. Hence

∥x∥ ≤ lim inf_{n→∞} \frac{1}{k + 1} ∥x + ∑_{i=1}^{k} x_{i+n}∥ ≤ 1 − δ_{X}^{k}(ϵ^{k} − (k! − 1)γ^{k}).

Using Corollary 3.12, we see that δ_{X}^{k}(ϵ^{k}) ≤ 1−∥x∥. This clearly gives us the conclusion. ■

Theorem 4.6 shows that k-UR spaces have UKK property. However, UKK property is much weaker than k-UR. Spaces with UKK property need not be even reflexive (see Example 4.23.2). One may therefore look for a more direct infinite dimensional generalization of uniform convexity.

Let (x_{n}) be a sequence in a space X. The number

sep(x_{n}) = inf{∥x_{m} − x_{n}∥ : m ≠ m}

is called the separation constant of (x_{n}).

Definition 4.7 A Banach space X is nearly uniformly convex (NUC for short) provided that for every ϵ > 0 there is δ > 0 such that if (x_{n}) is a sequence in B_{X} with sep(x_{n}) ≥ ϵ, then

inf{∥x∥ : x ∈ co({x_{n}})} ≤ 1 − δ.

NUC implies reflexivity. This is a consequence of a result due to James. Before formulating this theorem, we need to recall some additional terminology. A sequence (x_{n}) in a Banach space X is called a (Schauder) basis of X if each x ∈ X has a unique expansion

x = ∑_{n=1}^{∞} α_{n}x_{n}

for some scalars α_{1}, α_{2}, . . .. We say that (x_{n}) is a basic sequence if it is a basis of the closed linear span of the set {x_{n}} (see [76]). The following result can be derived from [54].

Theorem 4.8 If a Banach space X is not reflexive, then for every 0 < γ < 1 there exists a basic sequence (x_{n}) in B_{X} such that sep(x_{n}) ≥ γ and

inf{∥x∥ : x ∈ co({x_{n}})} ≥ γ.

Now we are in a position to establish a relation between NUC and UKK property.

Theorem 4.9 A Banach space X is NUC if and only if X is reflexive and has UKK property.

Proof. Assume that a Banach space X is NUC. Theorem 4.8 shows that X is reflexive. Let (x_{n}) be a sequence in B_{X} converging weakly to x for which lim inf_{n→∞} ∥x_{n} − x∥ ≥ ϵ > 0. Using weak lower semicontinuity of the norm, one can easily find a subsequence (x_{n_{k}}) so that sep(x_{n_{k}}) ≥ ϵ/2. Since X is NUC, we can assume that there exist elements y_{k} ∈ co({x_{i}}_{i=n_{k}+1}^{n_{k+1}}), k = 1, 2, …, such that ∥y_{k}∥ ≤ 1 − δ for every k, where δ > 0 depends only on ϵ. Clearly, (y_{k}) also converges weakly to x, so ∥x∥ ≤ 1 − δ. This shows that X has UKK property.

Let now X be a reflexive space with UKK property. Take a sequence (x_{n}) in B_{X} such that sep(x_{n}) ≥ ϵ > 0. Passing to a subsequence, we can assume that (x_{n}) converges weakly to some x and lim inf_{n→∞} ∥x_{n}−x∥ ≥ ϵ/2. Then ∥x∥ ≤ 1−K_{X}(ϵ/2) and Mazur’s theorem gives us y ∈ co({x_{n}}) for which ∥y∥ ≤ 1 − K_{X}(ϵ/2)/2. ■

From Theorems 4.6 and 4.9 we immediately obtain the following corollary.

Corollary 4.10 If a Banach space X is k-UC for some k, then X is NUC.

We introduce another modulus corresponding to UKK(τ) property. Let X be a Banach space with N_{1}(τ) ≠ ∅. Given ϵ ≥ 0, we put

d_{X,τ} (ϵ) = inf {lim inf_{n→∞} ∥x + ϵx_{n}∥ − 1}

where the infimum is taken over all x ∈ X with ∥x∥ ≥ 1 and all sequences (x_{n}) ∈ N_{1}(τ). The function d_{X,τ} is nonnegative and satisfies the Lipschitz condition with the constant 1 on [0, ∞). For the moduli d_{X,τ} and K_{X,τ} we have a formula similar to (3.5).

Theorem 4.11 Let X be a Banach space. Then

d_{X,τ} (\frac{ϵ}{1 − K_{X,τ} (ϵ)}) = \frac{K_{X,τ} (ϵ)}{1 − K_{X,τ} (ϵ)}

for every ϵ ∈ [0, 1).

Proof. Given ϵ, γ ∈ (0, 1), we find x ∈ S_{X} and (y_{n}) ∈ N_{1}(τ) such that sup_{n∈ℕ} ∥x + ϵy_{n}∥ − 1 < d_{X,τ} (ϵ) + γ. We put u = (1 + d_{X,τ} (ϵ) + γ)^{−1}x, u_{n} = (1 + d_{X,τ} (ϵ) + γ)^{−1}ϵy_{n} and z_{n} = u + u_{n} for n ∈ ℕ. Then (z_{n}) is a sequence in B_{X} converging to u with respect to τ and ∥z_{n} − u∥ ≥ ϵ(1 + d_{X,τ} (ϵ) + γ)^{−1} for every n. Therefore

K_{X,τ} (\frac{ϵ}{1 + d_{X,τ} (ϵ) + γ} ) ≤ 1 − ∥u∥ = 1 − \frac{1}{1 + d_{X,τ} (ϵ) + γ} .

Since K_{X,τ} is continuous,

K_{X,τ} (\frac{ϵ}{1 + d_{X,τ} (ϵ)}) ≤ \frac{d_{X,τ} (ϵ)}{1 + d_{X,τ} (ϵ)}.  (4.1)

From Corollary 4.4 we see that if 0 < ϵ < ν_{X,τ}, then K_{X,τ} (ϵ) < 1. We fix 0 < γ < 1 − K_{X,τ} (ϵ) and choose a sequence (z_{n}) in B_{X} so that (z_{n}) converges to x with respect to τ, ∥z_{n} − x∥ ≥ ϵ for all n and 1 − ∥x∥ < K_{X,τ} (ϵ) + γ. Put v = (1 − K_{X,τ} (ϵ) − γ)^{−1}x, v_{n} = ϵ^{−1}(z_{n} − x) for n ∈ ℕ. Then ∥v∥ > 1, ∥v_{n}∥ ≥ 1 for every n and (v_{n}) is τ-null. Hence

d_{X,τ} (\frac{ϵ}{1 − K_{X,τ} (ϵ) − γ} ) ≤ sup_{n∈ℕ} ∥v + \frac{ϵ}{1 − K_{X,τ} (ϵ) − γ} v_{n}∥ − 1 ≤ \frac{K_{X,τ} (ϵ) + γ}{1 − K_{X,τ} (ϵ) − γ} .

It follows that

d_{X,τ} (\frac{ϵ}{1 − K_{X,τ} (ϵ)}) ≤ \frac{K_{X,τ} (ϵ)}{1 − K_{X,τ} (ϵ)}  (4.2)

for every ϵ ∈ [0, ν_{X,τ}).

To show the inequality opposite to (4.2) we set ϕ(ϵ) = ϵ/(1−K_{X,τ} (ϵ)) where ϵ ∈ [0, 1). By (4.1),

d_{X,τ} (ϕ(ϵ)) ≥ (1 + d_{X,τ} (ϕ(ϵ)))K_{X,τ} (\frac{ϕ(ϵ)}{1 + d_{X,τ} (ϕ(ϵ))}) .

But from (4.2) we see that

\frac{ϕ(ϵ)}{1 + d_{X,τ} (ϕ(ϵ))} ≥ ϕ(ϵ)(1 − K_{X,τ} (ϵ)) ≥ ϵ.

Using Corollary 4.4 again, we obtain

d_{X,τ} (ϕ(ϵ)) ≥ \frac{K_{X,τ} (ϵ)}{ϵ}ϕ(ϵ) = \frac{K_{X,τ} (ϵ)}{1 − K_{X,τ} (ϵ)}.

This is the equality opposite to (4.2). ■

Corollary 4.12 A Banach space X has UKK(τ) property if and only if d_{X,τ} (ϵ) > 0 for every ϵ > 0.

We will study the property dual to NUC. For this purpose we need the next definition.

Definition 4.13 A Banach space X is nearly uniformly smooth (NUS for short) provided that for every ϵ > 0 there is η > 0 such that if 0 < t < η and (x_{n}) is a basic sequence in B_{X}, then there exists k > 1 for which

∥x_{1} + tx_{k}∥ ≤ 1 + ϵt.  (4.3)

Notice that if a space X is NUS, then only finitely many elements of a given sequence (x_{n}) may not satisfy condition (4.3). NUS can be therefore characterized in the following way.

Remark 4.14 An infinite dimensional space X is NUS if and only if lim_{t→0} b̃_{X}(t)/t = 0 where

b̃_{X}(t) = sup {lim sup_{n→∞} ∥x_{1} + tx_{n}∥ − 1}

and the supremum is taken over all basic sequences (x_{n}) in B_{X}.

Recall that if a weakly null sequence (x_{n}) does not converge to zero in norm, then there is a subsequence (x_{n_{k}}) which is a basic sequence. Moreover, every basic sequence in a reflexive space is weakly null (see [76]). It follows that in the case when X is reflexive the supremum in the definition of b̃_{X} can be taken over all sequences (x_{n}) ∈ N^{1}(w). This leads us to a more general modulus.

Let τ be a topology in a Banach space X and t ≥ 0. We put

b_{X,τ} (t) = sup {lim sup_{n→∞} ∥x + tx_{n}∥ − 1}

where the supremum is taken over all x ∈ B_{X} and all (x_{n}) ∈ N^{1}(τ). Notice that b_{X,τ} is a nonnegative convex function with b_{X,τ} (0) = 0.

Definition 4.15 A Banach space X is nearly uniformly smooth with respect to τ (NUS(τ) for short) if

lim_{t→0} \frac{b_{X,τ} (t)}{t} = 0.

Proposition 4.16 A Banach space X is NUS if and only if X is NUS(w) and reflexive.

Proof. If X is reflexive, then the moduli b_{X,w} and b̃_{X} are equal. To complete the proof it is therefore enough to show that NUS implies reflexivity. Assume that a space X is NUS but not reflexive. By Theorem 4.8 for every 0 < θ < 1 there exists a basic sequence (x_{n}) in B_{X} such that

θ ≤ ∥\frac{1}{1 + t}x_{1} + \frac{t}{1 + t}x_{k}∥

for all 0 < t < 1 and k. Hence θ(1 + t) ≤ 1 + b̃_{X}(t) and since 0 < θ < 1 is arbitrary, t ≤ b̃_{X}(t). This contradicts our assumption. ■

Recall that f^{∗} denotes dual Young’s function of f.

Theorem 4.17 Let X be a reflexive space. By b and d we denote the moduli b_{X,w} and d_{X^{∗},w}, respectively. Then

2b^{∗} (\frac{ϵ}{4}) ≤ d(ϵ) ≤ b^{∗}(2ϵ)

for every ϵ ∈ [0, 1).

Proof. Fix ϵ ∈ (0, 1). Next, take arbitrary x^{∗} ∈ X^{∗} with ∥x^{∗}∥ ≥ 1 and a sequence (x_{n}^{∗}) ∈ N_{1}(w). We find elements x, y_{1}, y_{2}, ⋯ ∈ B_{X} so that x^{∗}(x) = ∥x^{∗}∥ and x_{n}^{∗}(y_{n}) = ∥x_{n}^{∗}∥ for every n. We can assume that (y_{n}) converges weakly to some y ∈ B_{X}. Then (x_{n}) ∈ N^{1}(w) where x_{n} = \frac{1}{2}(y_{n} − y) for all n. Moreover, lim_{n→∞} x_{n}^{∗}(x_{n}) = \frac{1}{2} lim_{n→∞} ∥x_{n}^{∗}∥. Hence

lim inf_{n→∞} ∥x^{∗} + ϵx_{n}^{∗}∥ lim sup_{n→∞} ∥x + sx_{n}∥ ≥ lim inf_{n→∞}(x^{∗} + ϵx_{n}^{∗})(x + sx_{n})

= x^{∗}(x) + ϵs lim_{n→∞} x_{n}^{∗}(x_{n})

≥ 1 + \frac{ϵs}{2}

for every s ≥ 0. This shows that

1 + \frac{ϵs}{2} ≤ (1 + b(s))(1 + d(ϵ)) ≤ 1 + b(s) + d(ϵ) + ϵb(s).

Therefore ϵs/2 − 2b(s) ≤ d(ϵ) which gives us the inequality 2b^{∗}(ϵ/4) ≤ d(ϵ).

Take now x ∈ B_{X} and (x_{n}) ∈ N^{1}(w). Put s = d(ϵ)/ϵ. For each n there is y_{n}^{∗} ∈ S_{X^{∗}} such that ∥x + sx_{n}∥ = y_{n}^{∗}(x + sx_{n}). Passing to a subsequence, we can assume that (y_{n}^{∗}) converges weakly to some x^{∗} ∈ B_{X^{∗}} and the limit η = lim_{n→∞} ∥x_{n}^{∗}∥ exists where x_{n}^{∗} = y_{n}^{∗} − x^{∗}. Consider two cases.

I. ϵ < η. Then ϵ∥x^{∗}∥/η < 1. Assuming that x^{∗} ≠ 0, from Remark 2.1 we get

1 − ∥x^{∗}∥ ≥ ∥x^{∗} + x_{n}^{∗}∥ − 1 ≥ \frac{η}{ϵ∥x^{∗}∥} (∥x^{∗} + \frac{ϵ∥x^{∗}∥}{η}x_{n}^{∗}∥ − 1) .

for every n. Hence

1 ≥ ∥x^{∗}∥ + \frac{η}{ϵ} lim inf_{n→∞} (∥\frac{1}{∥x^{∗}∥}x^{∗} + \frac{ϵ}{η} x_{n}^{∗}∥ − 1) ≥ \frac{η}{ϵ} d(ϵ)

so ∥x^{∗}∥ ≤ 1 − sη. Consequently,

lim sup_{n→∞} ∥x + sx_{n}∥ = x^{∗}(x) + s lim sup_{n→∞} x_{n}^{∗}(x_{n}) ≤ 1 − sη + sη = 1.

Observe that if x^{∗} = 0, then η = 1 and the last inequality also holds.

II. ϵ ≥ η. Then

lim sup_{n→∞} ∥x + sx_{n}∥ ≤ 1 + sη ≤ 1 + sϵ.

In both cases we obtain b(s) ≤ sϵ. Hence d(ϵ) = sϵ ≤ 2sϵ − b(s) ≤ b^{∗}(2ϵ). ■

Theorem 4.17 and Lemma 3.2 enables us to establish duality between NUC and NUS.

Corollary 4.18 A Banach space X (X^{∗}) is NUS if and only if X^{∗} (resp. X) is NUC.

From Theorem 3.22 and Corollaries 4.10 and 4.18 we see that if a space X is k-US for some k, then X is NUS. The next geometrical infinite dimensional property we want to discuss is not so strongly related to convexity or smoothness as the previous ones.

Definition 4.19 A Banach space X satisfies the Opial property with respect to a topology τ if

lim inf_{n→∞} ∥x_{n}∥ < lim inf_{n→∞} ∥x + x_{n}∥

for every τ-null bounded sequence (x_{n}) in X and every x ≠ 0.

Replacing the strict inequality “<” with “≤”, we obtain the definition of the socalled nonstrict (or weak) Opial property with respect to the topology τ. Of course, in this case it is not necessary to exclude the point x = 0. If τ = w or τ = w^{∗}, then τ convergent sequences are bounded so the word “bounded” can be dropped from the definition.

Example 4.20 Let φ be a periodic function with period 2π such that

φ(t) = \begin{cases}1 & if 0 ≤ t ≤ \frac{4}{3}π, \\ −2 & if \frac{4}{3}π < t < 2π. \end{cases}

Put Φ_{n}(t) = φ(nt), n ∈ ℕ. Then (Φ_{n}) is a weakly null sequence of Rademacher-like functions in L^{p}([0, 2π]) for every 1 < p < ∞. Consider the function

λ_{p}(c) = lim_{n→∞} ∥Φ_{n} − c∥^{p} = ∫_{0}^{2π} |φ(t) − c|^{p} dt

where c ∈ ℝ is treated as the constant function. Since

λ′_{p}(0) = −p ∫_{0}^{2π} |φ(t)|^{p−1} sgn(φ(t)) dt = \frac{4}{3}πp(2^{p−2} − 1),

λ′_{p}(0) ≠ 0 whenever p ≠ 2. It follows that λ_{p}(0) is not a minimal value of λ_{p}, except for the case p = 2. Consequently, L^{p}([0, 2π]) does not satisfy even the nonstrict Opial property for any p ≠ 2.

Definition 4.21 Let X be a Banach space with N_{1}(τ) ≠ ∅. The Opial modulus of X with respect to the topology τ is the function r_{X,τ} : [0, ∞) → ℝ given by the formula

r_{X,τ} (c) = inf {lim inf_{n→∞} ∥x + x_{n}∥ − 1}

where c ≥ 0 and the infimum is taken over all elements x ∈ X with ∥x∥ ≥ c and all sequences (x_{n}) ∈ N_{1}(τ). The space X has the uniform Opial property with respect to τ if r_{X,τ} (c) > 0 for every c > 0.

In case τ = w the name of the topology is omitted from the terminology concerning Opial properties. The uniform Opial property with respect to a topology τ implies the Opial property with respect to τ. Moreover, for spaces with N_{1}(τ) ≠ ∅ the condition r_{X,τ} (0) ≥ 0 characterizes the nonstrict Opial property with respect to τ. Notice that

d_{X,τ} (t) + 1 = inf \{lim inf_{n→∞} ∥x + tx_{n}∥ : ∥x∥ ≥ 1, (x_{n}) ∈ N_{1}(τ)\} = t (r_{X,τ} (\frac{1}{t}) + 1)

for every t > 0. Consequently, X has the uniform Opial property with respect to τ if and only if d_{X,τ} (t) > t − 1 for all t > 0.

We will describe the general method of evaluating some of the considered moduli. Let τ be a topology in a Banach space X and ρ : [0, ∞) × [0, ∞) → [0, ∞) be a nondecreasing with respect to each variable continuous function. We say that X has property L(ρ, τ) if

lim inf_{n→∞} ∥x + x_{n}∥ = ρ (lim inf_{n→∞} ∥x_{n}∥, ∥x∥)

whenever x ∈ X and (x_{n}) is a bounded τ-null sequence. In spaces with property L(ρ, τ) we easily get the following formulae.

Proposition 4.22 Let X be a Banach space with property L(ρ, τ). Then

d_{X,τ} (t) = b_{X,τ} (t) = ρ(t, 1) − 1, r_{X,τ} (t) = ρ(1, t) − 1

for every t ≥ 0.

Example 4.23 1. Let Ω be a measure space with a measure μ and 1 ≤ p < ∞. We will show that if f ∈ L^{p}(Ω) and (f_{n}) is a bounded sequence in L^{p}(Ω) converging to zero a.e., then

lim_{n→∞} ∥f + f_{n}∥^{p} = ∥f∥^{p} + lim_{n→∞} ∥f_{n}∥^{p}  (4.4)

provided that the above limits exist. First observe that for every ϵ > 0 there is C > 0 such that

||a + b|^{p} − |a|^{p}| ≤ ϵ|a|^{p} + C|b|^{p}

for all a, b ∈ ℝ. Hence

||f(ω) + f_{n}(ω)|^{p} − |f_{n}(ω)|^{p} − |f(ω)|^{p}| ≤ ϵ|f_{n}(ω)|^{p} + (C + 1)|f(ω)|^{p}

for any n ∈ ℕ, ω ∈ Ω. Let g_{n} denote the nonnegative part of the function ||f + f_{n}|^{p} − |f_{n}|^{p} − |f|^{p}| − ϵ|f_{n}|^{p}. Then g_{n} ≤ (C + 1)|f|^{p} for every n and the sequence (g_{n}) converges to zero a.e. By the dominated convergence theorem, lim_{n→∞} ∫_{Ω} g_{n} dμ = 0. Moreover,

|∥f +f_{n}∥^{p}−∥f_{n}∥^{p}−∥f∥^{p}| ≤ ∫_{Ω} ||f +f_{n}|^{p}−|f_{n}|^{p}−|f|^{p}| dμ ≤ ∫_{Ω} g_{n} dμ+ϵ ∫_{Ω} |f_{n}|^{p} dμ.

Passing to the limit with n → ∞, we see that

| lim_{n→∞} ∥f + f_{n}∥^{p} − lim_{n→∞} ∥f_{n}∥^{p} − ∥f∥^{p}| ≤ ϵM

where M = lim sup_{n→∞} ∥f_{n}∥^{p}. Since ϵ > 0 is arbitrary, this gives us formula (4.4). Let now Ω be σ-finite. Each clm-null sequence in X = L^{p}(Ω) has a subsequence converging to zero a.e. It follows that X has property L(ρ_{p}, clm) with

ρ_{p}(s, t) = (s^{p} + t^{p})^{1/p}.

By Proposition 4.22

d_{X,clm}(t) = b_{X,clm}(t) = r_{X,clm}(t) = (1 + t^{p})^{\frac{1}{p}} − 1

for every t ≥ 0. In contrast to the case of weak topology, for each 1 ≤ p < ∞ the space L^{p}(Ω) has therefore the uniform Opial property with respect to the clm topology. It has also UKK(clm) property and if p > 1, then it is NUS(clm).

2. Let 1 < p < ∞ and (X_{n}) be a sequence of Banach spaces with the Schur property. Consider the space

X = (∑_{n=1}^{∞} X_{n})_{l^{p}} ,

i.e. X is the space of all sequences x = (x(k)) such that x(k) ∈ X_{k} for every k and

∥x∥ = (∑_{k=1}^{∞} ∥x(k)∥_{X_{k}}^{p})^{\frac{1}{p}} < ∞.

We put P_{m}x = (x(1), …, x(m), 0, 0, …) where x = (x(k)) with x(k) ∈ X_{k} for every k. This formula gives us the projection P_{m} : X → X and we set R_{m} = Id−P_{m}. Take an element x ∈ X and a weakly null sequence (x_{n}) in X. Since weak convergence is equivalent to norm convergence in each X_{n}, lim_{n→∞} ∥P_{m}x_{n}∥ = 0 for every m. Consequently,

lim inf_{n→∞} ∥x + x_{n}∥^{p} = lim inf_{m→∞} lim inf_{n→∞}(∥P_{m}(x + x_{n})∥^{p} + ∥R_{m}(x + x_{n})∥^{p})
= lim inf_{m→∞}(∥P_{m}x∥^{p} + lim inf_{n→∞} ∥R_{m}x + x_{n}∥^{p})
=∥x∥^{p} + lim inf_{n→∞} ∥x_{n}∥^{p}.

This shows that X has property L(ρ_{p}, w). Consider now some particular cases. If X_{n} = l^{1} for every n, then X is not reflexive but it has UKK property in the nontrivial way. If in turn X_{n} is the space ℝ^{n} endowed with the maximum norm, then l^{∞} is finitely representable in the resulting space X. Consequently, X is not superreflexive. On the other hand, X is reflexive, so it is NUC.

3. Let X be a Hilbert space, x ∈ X and (x_{n}) be a weakly null sequence in X. Then lim_{n→∞}⟨x_{n}, x⟩ = 0, so

lim inf_{n→∞} ∥x + x_{n}∥^{2} = ∥x∥^{2} + lim inf_{n→∞} ∥x_{n}∥^{2}.

This shows that X has property L(ρ, w) with ρ(s, t) = (s^{2} + t^{2})^{1/2}.

Let J_{ϕ} be a duality mapping on a space X. We say that J_{ϕ} is sequentially τcontinuous if J_{ϕ} is single-valued and sequentially continuous as the function from X with the topology τ to X^{∗} with the weak^{∗} topology.

Proposition 4.24 Let X be a Banach space with a sequentially τ-continuous duality mapping J_{ϕ}. If x ∈ X and (x_{n}) is a bounded τ-null sequence, then

lim inf_{n→∞} Φ(∥x + x_{n}∥) = Φ(∥x∥) + lim inf_{n→∞} Φ(∥x_{n}∥)

where Φ(t) = ∫_{0}^{t} ϕ(s) ds, t ≥ 0.

Proof. Take x ≠ 0 and a bounded τ-null sequence (x_{n}). Next, fix n and consider the function g(t) = Φ(∥tx + x_{n}∥) where t ∈ ℝ. By our assumption and Theorem 2.11, g is differentiable and g′(t) = J_{ϕ}(tx + x_{n})(x) for every t ∈ ℝ. Since Φ is convex and increasing on [0, ∞), g is also convex. Hence g′ is continuous and consequently,

Φ(∥x + x_{n}∥) − Φ(∥x_{n}∥) = ∫_{0}^{1} J_{ϕ}(tx + x_{n})(x) dt.  (4.5)

From our assumption and the dominated convergence theorem we obtain

lim_{n→∞} ∫_{0}^{1} J_{ϕ}(tx + x_{n})(x) dt = ∫_{0}^{1} J_{ϕ}(tx)(x) dt = ∥x∥ ∫_{0}^{1} ϕ(∥tx∥) dt = Φ(∥x∥).

This and equality (4.5) give us the desired formula. ■

Corollary 4.25 Let X satisfy the assumption of Proposition 4.24. Then X has property L(ρ, τ) with ρ(s, t) = Φ^{−1}(Φ(s) + Φ(t)),

d_{X,τ} (t) = b_{X,τ} (t) = r_{X,τ} (t) = Φ^{−1}(Φ(1) + Φ(t)) − 1

for every t ≥ 0 and

K_{X,τ} (ϵ) = 1 − Φ^{−1}(Φ(1) − Φ(ϵ))

for all 0 < ϵ < 1.

Proof. The first part of the conclusion is a direct consequence of Propositions 4.22 and 4.24. In order to obtain the last formula we take a sequence (x_{n}) in B_{X} with τ- lim_{n→∞} x_{n} = x and ∥x_{n} − x∥ ≥ ϵ for all n. By Proposition 4.24,

Φ(∥x∥) + Φ(ϵ) ≤ Φ(∥x∥) + lim inf_{n→∞} Φ(∥x_{n} − x∥) ≤ Φ(1).

Hence 1−∥x∥ ≥ 1−Φ^{−1}(Φ(1)−Φ(ϵ)) which shows that K_{X,τ} (ϵ) ≥ 1−Φ^{−1}(Φ(1)−Φ(ϵ)).

Take now y ∈ S_{X} and a τ-null sequence (y_{n}) such that ∥y_{n}∥ = ϵ for every n. Since ϵ < 1, there exists s > 0 such that lim inf_{n→∞} ∥sy + y_{n}∥ = 1. Given γ > 0, we can therefore assume that ∥sy + y_{n}∥ ≤ 1 + γ for all n. Then

Φ(s) = Φ(∥sy∥) = lim inf_{n→∞} Φ(∥sy + y_{n}∥) − Φ(ϵ) = Φ(1) − Φ(ϵ).

On the other hand,

K_{X,τ} (\frac{ϵ}{1 + γ} ) ≤ 1 − \frac{∥sy∥}{1 + γ}

which shows that K_{X,τ} (ϵ) ≤ 1 − s. ■

Corollary 4.26 Let X satisfy the assumption of Proposition4.24.Then X has UKK(τ) property, the uniform Opial property with respect to τ and is NUS(τ).

Corollary 4.25 gives us another way to obtain the formulae for the moduli of the space X = L^{p}(Ω) in the case when 1 < p < ∞ and τ is the clm topology (see Example 4.23). In particular,

K_{X,clm}(ϵ) = 1 − (1 − ϵ^{p})^{\frac{1}{p}}

for every ϵ ≥ 0.

5. Normal structure

Normal structure plays essential role in some problems of metric fixed point theory, especially those concerning nonexpansive mappings (see [44] and [64]). Before formulating the definition, we need to set up some additional notation. Let A be a nonempty bounded set in a Banach space X. We put

r(A) = inf {sup_{y∈A} ∥x − y∥ : x ∈ A} .

This number is called the Chebyshev radius of A and the set Z(A) of all points x ∈ A for which the infimum is attained is called the Chebyshev center of A (with respect to A). In general Z(A) may be empty. Clearly, \frac{1}{2} diam(A) ≤ r(A) ≤ diam(A). A set A is said to be diametral if r(A) = diam(A) > 0.

Definition 5.1 Let B be a nonempty subset of a Banach space X and let F be a nonempty family of subsets of B. The family F is said to be normal provided that

r(A) < diam(A)

for every bounded set A ∈ F with diam(A) > 0. If there exists a constant 0 < c < 1 such that

r(A) ≤ c diam(A)

for every bounded set A ∈ F with diam(A) > 0, then F is said to be uniformly normal.

There are two basic special instances of normal families in a subset B of a space X.

1. The family F of all bounded closed convex subsets of B. In this case we say that B has normal structure, which we abbreviate to NS, (uniform normal structure which we abbreviate to UNS) whenever F is normal (resp. uniformly normal).

2. Let τ be a topology in the space X. Take F to be the family of all bounded convex τ-sequentially compact subsets of B. Then B is said to have normal structure with respect to τ (τ-NS for short) if F is normal. Normal structure with respect to the weak topology is called weak normal structure (WNS for short). Clearly, NS implies WNS and for reflexive spaces these two properties are equivalent.

Theorem 5.2 Let X be a Banach space. A set B ⊂ X does not have NS if and only if there is a bounded sequence (x_{n}) in B such that

lim_{n→∞} ∥x_{n} − x∥ = diam({x_{n}}) > 0  (5.1)

for every x ∈ co({x_{n}}).

Proof. Assume that B contains a sequence (x_{n}) for which condition (5.1) holds. Then the closure of co({x_{n}}) is diametral, so B does not have normal structure.

Suppose in turn that B contains a bounded convex closed diametral subset A. Put d = diam(A). By induction we construct a sequence (x_{n}) in A so that ∥y−x_{n}∥ ≥ d−2/n for every n and every y ∈ co({x_{k}}_{k=1}^{n−1}). As x_{1} we take a arbitrary element of A. Next, having x_{1}, …, x_{n−1}, we choose a 1/n-net C = {y_{1}, …, y_{m}} in co({x_{k}}_{k=1}^{n−1}) and set

z = \frac{1}{m} ∑_{i=1}^{m} y_{i}.

We can clearly assume that m ≥ n. Since A is diametral, there exists x_{n} ∈ A such that ∥z − x_{n}∥ ≥ d − 1/m^{2}. Then

d − \frac{1}{m^{2}} ≤ ∥\frac{1}{m} ∑_{i=1}^{m} y_{i} − \frac{1}{m} ∑_{i=1}^{m} x_{n}∥

≤\frac{1}{m}∥y_{k} − x_{n}∥ + \frac{1}{m} ∑_{i≠k} ∥y_{i} − x_{n}∥

≤\frac{1}{m}∥y_{k} − x_{n}∥ + \frac{m − 1}{m}d

for each k = 1, …, n. This gives us the inequality ∥y_{k} − x_{n}∥ ≥ d − 1/m ≥ d − 1/n. Consequently, ∥y − x_{n}∥ ≥ d − 2/n for every y ∈ co({x_{k}}_{k=1}^{n−1}).

Take now y ∈ co({x_{k}}). Then

d ≤ lim inf_{n→∞} ∥y − x_{n}∥ ≤ lim sup_{n→∞} ∥y − x_{n}∥ ≤ diam({x_{k}}) ≤ d

which shows that (x_{n}) satisfies (5.1). ■

The sequence (x_{n}) constructed in this proof does not have a Cauchy subsequence. It follows in particular that compact sets have NS.

Proposition 5.3 Let B be a subset of a Banach space X such that every τ-sequentially compact set A ⊂ B is separable. If B does not have τ-NS, then there is a bounded sequence (x_{n}) in B such that (x_{n}) converges to x with respect to τ and

lim_{n→∞} ∥x_{n} − x∥ = lim_{n→∞} ∥x_{n} − x_{m}∥ > 0

for every m. In case τ = w the separability assumption is not necessary.

Proof. To show the first part assume that A is a bounded convex τ-sequentially compact diametral subset of B. Let a countable set {u_{k}} ⊂ A be dense in A. We repeat the construction in the proof of Theorem 5.2 with the net C replaced by {u_{1}, …, u_{n}}. This gives us a sequence (x_{n}) which satisfies (5.1) for every x ∈ A and it suffices to take its τ-convergent subsequence. The second part follows directly from the proof of Theorem 5.2 and Mazur’s theorem. ■

Our next theorem is a simple consequence of Proposition 5.3.

Theorem 5.4 Let X be a Banach space in which every τ-sequentially compact set is separable. If X has the Opial property with respect to τ, then X has τ-NS. In case τ = w the separability assumption is not necessary.

UCED is another geometrical property which implies NS.

Proposition 5.5 Let A be a bounded convex subset of a Banach space X with

diam(A) > 0.

Then there exists z ∈ S_{X} such that

r(A) ≤ (1 − δ_{X}(z; 1)) diam(A).

Proof. Put d = diam(A). Given 0 < ϵ < d, we choose x, y ∈ A so that ∥x − y∥ ≥ d − ϵ and set z = (x − y)/∥x − y∥. For each u ∈ A we have ∥u − x∥ ≤ d, ∥u − y∥ ≤ d and (u − y) − (u − x) = ∥x − y∥z. Hence

∥u − \frac{x + y}{2} ∥ ≤ d (1 − δ_{X} (z; \frac{d − ϵ}{d})) .

It follows that r(A) ≤ d(1 − δ_{X}(z; 1 − ϵ/d)). In view of continuity of δ_{X}(z; ϵ) this completes the proof. ■

Proposition 5.5 shows that if δ_{X}(z; 1) > 0 for every z ∈ S_{X}, then X has NS. In particular, we obtain the following result.

Corollary 5.6 If a Banach space X is UCED, then X has NS.

Several coefficients related to normal structure were defined. We will discuss some of them. Let (x_{n}) be a bounded sequence in a space X. Using Ramsey’s theorem (see [33], p. 235), one can find a subsequence (x_{n_{k}}) such that the double limit of ∥x_{n_{k}} −x_{n_{i}}∥ over k, i → ∞, k ≠ i, exists. We denote this limit by lim_{k≠i} ∥x_{n_{k}} − x_{n_{i}}∥.

Definition 5.7 Let X be a Banach space with N_{1}(τ) ≠ ∅. The τ-convergent sequences coefficient of X is defined as

τCS(X) = inf \{\frac{lim_{n≠m} ∥x_{n} − x_{m}∥}{lim_{n→∞} ∥x_{n}∥}\}

where the infimum is taken over all bounded τ-null sequences in X such that both limits exist and lim_{n→∞} ∥x_{n}∥ ≠ 0.

In the particular case when τ = w we obtain the weakly convergent sequences coefficient W CS(X). Clearly, 1 ≤ τCS(X) ≤ 2 and Proposition 5.3 gives us the following result.

Proposition 5.8 Let X be a Banach space in which every τ-sequentially compact set is separable. If 1 < τCS(X), then X has τ-NS. In case τ = w the separability assumption is not necessary.

Obviously,

τCS(X) = inf { lim_{n≠m} ∥x_{n} − x_{m}∥}

where the infimum is taken over all bounded τ-null sequences in X such that the above limit exists and lim_{n→∞} ∥x_{n}∥ = 1. This observation shows that if a space X has L(ρ, τ) property, then τCS(X) = ρ(1, 1). In particular, we have τCS(X) = Φ^{−1}(2Φ(1)) in the case when X satisfies the assumption of Proposition 4.24. We see for instance that W CS(l^{p}(Γ)) = 2^{1/p} for every infinite set Γ and every 1 < p < ∞. As the next example we take X = L^{p}(Ω) where Ω is a σ-finite measure space and 1 ≤ p < ∞. Then clmCS(X) = 2^{1/p}. As we shall see in Theorem 5.19 the space L^{p}(Ω) has UNS whenever 1 < p < ∞. Consequently, it has clm-NS. In [69] it was shown that clm-sequentially compact sets in L^{1}(Ω) are separable. From Proposition 5.8 we therefore deduce that also L^{1}(Ω) has clm-NS. On the other hand, the closed convex hull of the Rademacher functions is a diametral set in L^{1}([0, 1]) (see the proof of Theorem 5.19). It follows that L^{1}([0, 1]) does not have even WNS.

Assuming the uniform Opial property, we can strengthen the conclusion of Theorem 5.4. Namely, we have the following obvious estimates.

Proposition 5.9 Let X be a Banach space. Then

1 + d_{X,τ} (1) = 1 + r_{X,τ} (1) ≤ τCS(X).

Corollary 5.10 Let X be a Banach space. If X has UKK(τ) property or the uniform Opial property with respect to τ, then τCS(X) > 1.

This and Proposition 5.8 show in particular that UKK property implies WNS. From Theorem 4.9 we therefore see that NUC spaces have NS.

Let X be a Banach space. We set

N(X) = inf \{\frac{diam(A)}{r(A)}\} ,

where the infimum is taken over all bounded convex closed sets A ⊂ X with diam(A) > 0. The number N(X) is called the normal structure coefficient of X. Clearly, the condition N(X) > 1 characterizes spaces X with UNS.

Proposition 5.11 Let X be a Banach space. Then N(X) ≤ W CS(X).

Proof. Let (x_{n}) be a weakly null sequence in X such that the limits d = lim_{n≠m} ∥x_{n} − x_{m}∥ and c = lim_{n→∞} ∥x_{n}∥ > 0 exist. Given 0 < γ < c/2, we can assume that ∥x_{n} − x_{m}∥ ≤ d + γ and ∥x_{n}∥ ≥ c − γ for all n, m. We find functionals x_{n}^{∗} ∈ S_{X^{∗}} so that x_{n}^{∗}(x_{n}) = ∥x_{n}∥ for every n. By Lemma 4.5 we can assume that |x_{n}^{∗}(x_{m})| < γ whenever m ≠ n. If x ∈ co({x_{k}}), then |x_{n}^{∗}(x)| < γ for n sufficiently large. Hence

∥x_{n} − x∥ ≥ x_{n}^{∗}(x_{n} − x) = ∥x_{n}∥ − γ ≥ c − 2γ.

This shows that r(A) ≥ c − 2γ where A is the closure of co({x_{k}}). Consequently,

N(X) ≤ \frac{diam(A)}{r(A)} ≤ \frac{d + γ}{c − 2γ}

which gives us the inequality N(X) ≤ d/c. Now we can conclude the proof by taking the infimum. ■

A reasoning similar to that in the proof of Proposition 5.5 leads us to the following estimate.

Theorem 5.12 Let X be a Banach space. Then

N(X) ≥ \frac{1}{1 − δ_{X}(1)}.

Theorem 5.12 shows that if δ_{X}(1) > 0 (or equivalently ϵ_{0}(X) < 1), then X has UNS. This result was extended in [3].

Theorem 5.13 Let X be a Banach space and k ∈ ℕ. Then

N(X) ≥ (max \{1 − \frac{1 − ϵ}{k!k}, 1 − δ_{X}^{k}(ϵ)\})^{−1} .

In particular, if ϵ_{0}^{k}(X) < 1, then X has UNS.

To obtain an estimate for the normal structure coefficient one can also use methods of convex analysis. A useful criterion for a point to be a minimum point of a convex function can be given in terms of subdifferentials (see [98]).

Theorem 5.14 Let A be a nonempty convex subset of a Banach space X and f : X → ℝ be a continuous convex function. The function f attains its minimal value on A at a point x_{0} ∈ A if and only if there is a functional x^{∗} ∈ ∂f(x_{0}) such that

x^{∗}(x − x_{0}) ≥ 0

for every x ∈ A.

Let A ≠ ∅ be a bounded convex subset of a space X. Given y ∈ A, we put f_{y}(x) = ∥x−y∥ for x ∈ X. The function f_{y} is convex, continuous, and weakly sequentially lower semicontinuous. Consequently, the same is true for the function f(x) = sup{f_{y}(x) : y ∈ A}. Assume that A is weakly compact. Since f is weakly sequentially lower semicontinuous, it attains its minimal value on A. Clearly, this minimal value equals r(A) and it is attained at the points of the Chebyshev center Z(A). We therefore see that if A is weakly compact, then Z(A) ≠ ∅. This is in particular the case when A is compact. We need now a formula for the subdifferential of the function f.

Theorem 5.15 Let T be a compact metric space and X be a finite dimensional Banach space. Assume that {f_{t}}_{t∈T} is a family of convex functions defined on X such that for each x ∈ X the function t ↦ f_{t}(x) is continuous on T. Put f = sup_{t∈T} f_{t}. Then

∂f(x_{0}) = co ( ⋃_{t∈T (x_{0})} ∂f_{t}(x_{0}))

for every x_{0} ∈ X where T (x_{0}) = {t ∈ T : f_{t}(x_{0}) = f(x_{0})}.

We can return to the problem of estimating the normal structure coefficient. From Theorems 5.14 and 5.15, and Corollary 2.12 we obtain the following result.

Theorem 5.16 Let A be a nonempty compact convex subset of a finite dimensional Banach space X and x_{0} ∈ A. If x_{0} ∈ Z(A), then there exist elements y_{1}, …, y_{n} ∈ A, functionals x_{1}^{∗}, …, x_{n}^{∗} ∈ B_{X^{∗}}, and nonnegative scalars t_{1}, …, t_{n} such that ∑_{i=1}^{n} t_{i} = 1,

x_{i}^{∗}(x_{0} − y_{i}) = ∥x_{0} − y_{i}∥ = r(A)

for i = 1, …, n, and

∑_{i=1}^{n} t_{i}x_{i}^{∗}(x − x_{0}) ≥ 0

for every x ∈ A.

It turns out that the assumptions appearing in Theorem 5.16 are not restraining in our case. Indeed, if a space X is not reflexive, then N(X) = 1 (see [79] or [6]) and for reflexive spaces we can modify the formula for N(X).

Lemma 5.17 If a space X is reflexive, then

N(X) = inf \{\frac{diam(K)}{r(K)}\} ,  (5.2)

where the infimum is taken over all convex hulls K of finite subsets of X such that diam(K) > 0.

Proof. Let X be a reflexive space and N′(X) denote the right hand side expression in (5.2). We take a bounded convex closed set A ⊂ X with diam(A) > 0. If 0 < r < r(A), then

⋂_{y∈A} B̄(y, r) ∩ A = ∅

where B̄(y, r) = {x ∈ X : ∥x − y∥ ≤ r}. Since B̄(y, r) ∩ A are weakly closed subsets of the weakly compact set A, there is a finite set F ⊂ A such that

⋂_{y∈co(F)} B̄(y, r) ∩ co(F) ⊂ ⋂_{y∈F} B̄(y, r) ∩ A = ∅.

It follows that r ≤ r(co(F)) and consequently,

N′(X) ≤ \frac{diam(co(F))}{r(co(F))} ≤ \frac{diam(A)}{r}.

But 0 < r < r(A) is arbitrary, so we get the inequality N′(X) ≤ diam(A)/r(A). This gives us the estimate N′(X) ≤ N(X) and the opposite one is trivial. ■

We will calculate the values of N(L^{p}(Ω)) and W CS(L^{p}(Ω)). For this purpose we need the following result from [105].

Lemma 5.18 Let 1 < p < ∞ and 1/p+1/q = 1. If y_{1}, …, y_{n} ∈ L^{p}([0, 1]) and t_{1}, …, t_{n} are nonnegative numbers with ∑_{i=1}^{n} t_{i} = 1, then

∑_{i,j=1}^{n} t_{i}t_{j}∥y_{i} − y_{j}∥^{s} ≥ 2 (∑_{i=1}^{n} t_{i}∥ ∑_{k=1}^{n} t_{k}y_{k} − y_{i}∥)^{s}

where s = max{p, q}.

Notice that if Ω is a purely atomic measure space, then L^{p}(Ω) is isometrically isomorphic to l^{p}(Γ) for some set Γ. As we have already observed W CS(l^{p}(Γ)) = 2^{1/p} if 1 < p < ∞ and the set Γ is infinite.

Theorem 5.19 Let 1 < p < ∞ and Ω be a measure space such that the space X = L^{p}(Ω) is infinite dimensional. Then

N(X) = min \{2^{\frac{1}{p}} , 2^{1− \frac{1}{p}} \}

and W CS(X) = N(X) if p ≥ 2 or Ω is not purely atomic.

Proof. We will first establish the formula for N(X). Since X is reflexive, in view of Lemma 5.17 and Remark 3.14 it suffices to consider the case when X = L^{p}([0, 1]). Let A be a convex hull of a finite subset of X such that diam(A) > 0. Then A is contained in a finite dimensional subspace of X. In particular, A is compact, so there is x_{0} ∈ Z(A). From Theorem 5.16 we obtain elements y_{1}, …, y_{n} ∈ A, functionals x_{1}^{∗}, …, x_{n}^{∗} ∈ B_{X^{∗}}, and nonnegative scalars t_{1}, …, t_{n} such that ∑_{i=1}^{n} t_{i} = 1, x_{i}^{∗}(x_{0} −y_{i}) = ∥x_{0} −y_{i}∥ = r(A) for i = 1, …, n, and

∑_{i=1}^{n} t_{i}x_{i}^{∗} (∑_{k=1}^{n} t_{k}y_{k} − x_{0}) ≥ 0.

By Lemma 5.18

2 (∑_{i=1}^{n} t_{i}∥ ∑_{k=1}^{n} t_{k}y_{k} − y_{i}∥)^{s} ≤ ∑_{i,j=1}^{n} t_{i}t_{j}∥y_{i} − y_{j}∥^{s} ≤ d^{s}

where s = max{p, q}, 1/p + 1/q = 1. On the other hand,

∑_{i=1}^{n} t_{i}∥ ∑_{k=1}^{n} t_{k}y_{k} − y_{i}∥ ≥ ∑_{i=1}^{n} t_{i}x_{i}^{∗}( ∑_{k=1}^{n} t_{k}y_{k} − y_{i}) ≥ ∑_{i=1}^{n} t_{i}x_{i}^{∗}(x_{0} − y_{i}) = r(A).

Consequently, diam(A)/r(A) ≥ 2^{1/s}. In light of Lemma 5.17 this shows that N(X) ≥ 2^{1/s}.

Clearly, X = L^{p}([0, 1]) contains a subspace isometrically isomorphic to l^{p}. Hence N(X) ≤ W CS(X) ≤ W CS(l^{p}) = 2^{1/p}, so N(X) = W CS(X) = 2^{1/p} if p ≥ 2. Assume now that 1 < p < 2. To estimate N(X) from below we consider the sequence (r_{n}) of the Rademacher functions. It is a normalized weakly null sequence with ∥r_{n} − r_{m}∥ = 2^{1/q} whenever m ≠ n. Therefore diam(A) = 2^{1/q} where A is the closure of co{r_{n}}. Recall that (r_{n}) is an orthonormal system in L^{2}([0, 1]). Hence

lim inf_{n→∞} ∥r_{n} − f∥ ≥ lim_{n→∞} ∫_{0}^{1} r_{n}(t)(r_{n}(t) − f(t)) dt = 1

for every f ∈ span({r_{n}}). This shows in particular that sup{∥f − r_{n}∥ : n ∈ ℕ} ≥ 1 for every f ∈ A and consequently, r(A) ≥ 1. It follows that N(X) ≤ 2^{1/q} and finally N(X) = 2^{1/q} if 1 < p < 2.

To complete the proof it suffices to evaluate W CS(X) for X = L^{p}(Ω) where 1 < p < 2 and Ω is not purely atomic. This last assumption implies that an infinite Rademacher system (r_{n}) can be found in X (see [5], p. 32). Hence W CS(X) ≤ lim_{m≠n} ∥r_{m} − r_{n}∥ = 2^{1/q}. This and Proposition 5.11 show that W CS(X) = N(X) = 2^{1/q} if 1 < p < 2. ■

Every infinite dimensional Hilbert space is isometrically isomorphic to a space l^{2}(Γ) for some infinite set Γ. Theorem 5.19 gives us therefore the values of considered coefficients for such a space.

Theorem 5.20 Let H be an infinite dimensional Hilbert space. Then

N(H) = W CS(H) = \sqrt{2}.

These formulae can be also obtained in a more direct way (see [99] and [79]). From Dvoretsky’s theorem, Lemma 5.17, and Theorem 5.20 we see that N(X) ≤ N(l^{2}) = \sqrt{2} for any infinite dimensional Banach space X. We recall two more estimates for the normal structure coefficient. The first of them was established in [3] but it can be also proved with help of the subdifferential technique (see [90]). It shows that all finite dimensional spaces have UNS.

Theorem 5.21 If X is a Banach space with dim(X) = n, then N(X) ≥ 1 + 1/n.

The second estimate is given in terms of the modulus of smoothness.

Theorem 5.22 Let X be a Banach space with dim(X) ≥ 2. Then

N(X) ≥ (inf \{1 + ρ_{X}(t) − \frac{t}{2} : t ≥ 0\})^{−1} .  (5.3)

Proof. Given a space X, we put ψ_{X}(t) = 1+ρ_{X}(t)−t/2 and a(X) = inf_{t≥0} ψ(t). Easy calculation gives a(l^{2}) = \sqrt{3}/2. Let now X be a Banach space X with dim(X) ≥ 2. From (3.6) we see that

a(X) ≥ a(l^{2}) > 2/3.  (5.4)

Moreover, if t > 4/3, then

\frac{ρ_{X}(t) − ρ_{X} (\frac{4}{3})}{t − \frac{4}{3}} ≥ \frac{3ρ_{X} (\frac{4}{3})}{4} ≥ \frac{3ρ_{l^{2}} (\frac{4}{3})}{4} = \frac{1}{2}

and consequently ψ_{X}(t) ≥ ψ_{X}(4/3). This shows that a(X) = inf_{0≤t≤4/3} ψ_{X}(t).

If X is not reflexive, then N(X) = 1 = 1/ψ_{X}(0) ≥ 1/a(X). We can therefore assume that X is reflexive. Let A be a convex hull of a finite subset of X with d = diam(A) > 0. We will show that d/r ≥ 1/a(X) where r = r(A). In view of (5.4) this is true if d/r > 3/2. Suppose now that d/r ≤ 3/2 and take x_{0} ∈ Z(A). Theorem 5.16 gives us corresponding elements y_{1}, …, y_{n} ∈ A, functionals x_{1}^{∗}, …, x_{n}^{∗} ∈ B_{X^{∗}}, and nonnegative scalars t_{1}, …, t_{n}. We put z_{i} = x_{0} − y_{i} for i = 1, …, n. Then

2(ρ_{X}(t) + 1) ≥ ∥\frac{1}{d}(z_{i} − z_{j}) + \frac{t}{r} z_{i}∥ + ∥\frac{1}{d}(z_{i} − z_{j}) − \frac{t}{r} z_{i}∥

≥ \frac{1}{d}x_{i}^{∗}(z_{i} − z_{j}) + \frac{t}{r} x_{i}^{∗}(z_{i}) − \frac{1}{d}x_{j}^{∗}(z_{i} − z_{j}) + \frac{t}{r} x_{j}^{∗}(z_{i})

= 2\frac{r}{d} + t − \frac{1}{d}x_{i}^{∗}(z_{j}) + (\frac{t}{r} − \frac{1}{d}) x_{j}^{∗}(z_{i})

for any i, j and every t ≥ 0. Hence

2(ρ_{X}(t) + 1) ≥ 2\frac{r}{d} + t + (\frac{2}{d} − \frac{t}{r} ) ∑_{j=1}^{n} t_{j}x_{j}^{∗}( ∑_{i=1}^{n} t_{i}y_{i} − x_{0}) ≥ 2\frac{r}{d} + t

if 2/d ≥ t/r. Since d/r ≤ 3/2, the last condition holds for all t ∈ [0, 4/3]. Consequently, d/r ≥ inf_{0≤t≤4/3} ψ_{X}(t) = 1/a(X). In light of Lemma 5.17 this gives us the desired estimate. ■

Theorem 5.22 can be given a form analogous to Theorem 5.12. Indeed, Theorem 3.24 (ii) shows that δ̃(ϵ) = sup{tϵ/2 − ρ_{X}(t) : t ≥ 0} is double dual Young’s function of δ_{X^{∗}} and estimate (5.3) can be written as

N(X) ≥ \frac{1}{1 − δ̃(1)}.

Using Theorem 5.22, we can show that if ρ_{0}(X) < 1/2, then X has UNS. Form Corollary 3.25 and Theorem 5.12 we therefore see that each of the conditions ρ_{0}(X) < 1/2 and ϵ_{0}(X) < 1 gives UNS both in X and X^{∗}. In contrast to Theorem 5.13 multidimensional uniform smoothness does not imply even NS.

Example 5.23 Consider Bynum’s spaces l^{2,1} and l^{2,∞} (see [19]). The first of them is the space l^{2} endowed with an equivalent norm given by the formula

∥x∥_{2,1} = ∥x^{+}∥ + ∥x^{−}∥

where x ∈ l^{2}, ∥ · ∥ stands for the standard norm in l^{2}, x^{+} denotes the positive part of x and x^{−} denotes the negative part of x. Similarly, l^{2,∞} is the space l^{2} with the norm

∥x∥_{2,∞} = max{∥x^{+}∥, ∥x^{−}∥}.

The space l^{2,∞} is dual to l^{2,1}.

In [102] it was proved that l^{2,1} is 2-UR. Theorem 3.22 therefore shows that the space l^{2,∞} is 2-US. It does not however have normal structure. Indeed, let (e_{n}) be the standard basis of l^{2}. It is easy to see that the closure of the set co({e_{n}}) is diametral.

6. Bibliographic notes

Books [9], [27] and [24] are our general references for geometry of Banach spaces. The basic notions of this theory, i.e. strict convexity and uniform convexity were introduced by Clarkson [22]. The monograph [52] is devoted to detailed study of the first property, but it also contains sections on the second one. Theorem 2.5 was obtained by Khamsi [59] and our proof simplifies his idea.

Mazur [83] obtained the formula for the derivative of the norm in L^{p} spaces. The notion of smoothness of a space was in turn introduced by Krein [65]. Various aspects of smoothness and differentiability are discussed in the book [26]. Duality mappings were studied for the first time in [14]. Theorem 2.11 is due to Asplund [4]. The book [21] provides the systematic treatment of duality mappings and further references.

There are numerous uniform versions of convexity in the literature. Many of them can be found in [27], [52], [26] and the survey papers [49], [86]. Uniform convexity of Clarkson is the most extensively studied and applied one. Clarkson [22] established also the inequalities, which are now named after him, and proved that the spaces L^{p}([0, 1]) are UC if 1 < p < ∞. The modulus of convexity was in turn introduced by Day [23]. He also introduced uniform smoothness and obtained Theorem 3.22 with k = 1. This result was next improved by Lindenstrauss. Namely, he defined the modulus of smoothness and proved Theorem 3.24 (see [74]). The exact values of the moduli of convexity and smoothness for L^{p} spaces were given by Hanner [50] and Lindenstrauss [74]. Our method of proving continuity of moduli has its origin in [43] (see also [44]). Part 1 of Lemma 3.2 was established in [82].

Several different scaling functions corresponding to UC and US were defined (see, for instance, [2] and [7]). In particular, the modulus considered in [10] is related to both these properties. The concept of UC can be extended to some metric spaces. Counterparts of UC for the hyperbolic metric on the open unit ball of a Hilbert space were successfully apply to the theory of fixed points of holomorphic mappings (see [47],

[45] and [66]). A general approach to UC in metric spaces was given in [100]. Theorem 3.17 is a special case of a more general result due to Zălinescu [108] (see also [106]). Uniform convexity in every direction was introduced by Garkavi [39] in connection with his study of Chebyshev centers.

Milman [84] was the first to undertake systematic study of multi-dimensional uniform convexity and smoothness. Another approach to finite dimensional uniform convexity was found by Sullivan [103] who defined the modulus of k-convexity. Theorem 3.20 was proved by Lin [74], but its partial cases with k = 1, 2 had been earlier obtained in [37] and [40]. The reader should be warned that there are different definitions of k-uniform convexity in the literature (see, for instance, [52], p. 73). The same is true for the Kadec-Klee property (see [26], p. 42). Moreover, some authors use the terms Radon-Riesz property or property (H) instead of the Kadec-Klee property (see [24]). In our terminology we follow Huff [51] who also introduced the uniform Kadec-Klee property and . It should be noticed that independently of Huff a property equivalent to NUC was introduced in [46] under the name of noncompact uniform convexity. Lennard [69] extended Huff’s concepts to an abstract topology τ (see also [13]). He used them to obtain a fixed point theorem in the space L^{1}(Ω) with the clm topology. Here we use the terminology from [55] which is slightly different from that of Lennard.

Corollary 4.10 was obtained independently in [107] and [63]. Our proof bases on Lemma 4.5 which was shown in [94]. Theorem 4.11 is a direct generalization of an analogous result obtained for the weak topology in [81]. Two different definitions of nearly uniform smoothness can be found in the literature (see [93] and [8]). Here we follow [93]. Corollary 4.18 was also obtained in that paper, but the present proof is patterned on a reasoning from [42]. A similar result was obtained in [30].

The Opial property originates in a fixed point theorem proved by Opial in [88] which is also the proper reference for Example 4.20. The uniform Opial property with respect to the weak topology was defined in [96] and the Opial modulus was introduced in [73]. The formula for r_{X,τ} given in Corollary 4.25 was also essentially obtained in [73]. Kadec-Klee and Opial properties were extended to the abstract hyperbolic metric setting in [60]. Refined versions of Opial properties can be found for instance in [101], [17] and [18]. Property L(τ, ρ) has its origin in [70] (see also [60] and [32]). Formula (4.4) is a special case of a more general result from [15].

The notion of normal structure was introduced by Brodskii and Milman in [16]. They also proved Theorem 5.2 and found the first application of normal structure to fixed point theory. Uniform normal structure was in turn introduced in [41]. It is an open problem whether UNS implies superreflexivity. In [49] a positive solution to this problem was announced but the proof turned out to be erroneous. Generalizations of NS to metric spaces were found in [61] and [89] (see also [58] and [64]). Normal structure with respect to an arbitrary topology was introduced in [62]. Here we follow the terminology of [55] and [32].

The idea of the proof of Theorem 5.2 was developed by Landes who obtained various characterizations of normal and weak normal structure and studied their hereditariness properties (see [67] and the survey paper [68]). More details on normal structure, its numerous modifications and applications can be found in survey papers [49], [86] and [104]. Theorem 5.4 has its origin in [48] and Corollary 5.6 was obtained independently in [25] and [109].

The coefficient W CS(X) and normal structure coefficients were introduced by Bynum [20]. He established their relations to normal structure and calculated the value of W CS(l^{p}). Theorem 5.12 is also due to Bynum. Several different formulae for W CS(X) were found (see [5], p. 120) and the definition of τCS(X) bases on one of them. This definition is contained in [55] and [32] where relations of τCS(X) to another quantities appearing in metric fixed point theory are studied.

Lemma 5.17 and Theorem 5.21 were proved by Amir [3]. The method used by him to obtain the second of these results is different from ours. We apply a technique due to Pichugov [90]. Theorem 5.19 was essentially proved in [94] (see also [91]). A different proof was given in [29]. Relation between the coefficient ρ_{0}(X) and UNS was studied in [57] but estimate (5.3) was given in [95].

Although multi-dimensional uniform smoothness does not imply normal structure, coefficients related to the modulus b_{X,τ} turned out to be useful in some fixed point problems (see [38], [31] and [32]).

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k-UC —011 k-UC —012 k-uniform convexity —033 k-US —013 k-US —021 N(L^{p}(Ω)) —030 N(X) —029 N(X) —030 W CS(l^{p}(Γ)) —030 W CS(L^{p}(Ω)) —030 W CS(l^{p}) —034 W CS(X) —034 τ-NS —026 τ-NS —026 τ-NS —027 τCS(X) —034

Index

Affine, 2 Ball, 4–5

closed unit, 1 open unit, 33

Banach space k-UC, 11, 18 k-uniformly convex (k-UC), 12 k-uniformly rotund (k-UR), 9 k-uniformly smooth (k-US), 13 k-UR, 12 finite dimensional, 29 infinite dimensional, 31 nearly uniformly convex (NUC), 17 nearly uniformly smooth (NUS), 19

Banach space

nearly uniformly smooth

with respect to τ (NUS(τ)), 19 not reflexive, 17 NUC, 17–18 NUS, 20 separable, 7 smooth, 4, 6 strictly convex (or rotund), 1 strictly convex, 2–3 superreflexive, 10–11 UC, 11 UCED, 8 uniformly convex in every direction (UCED), 7 US, 14 with dim(X) ≥ 2, 8–9 with property L(ρ,τ), 22

Basis Schauder, 17 standard, 32

Centers

Chebyshev, 33 Centre

Chebyshev, 25, 29 Characteristic

of k-convexity, 11 of convexity, 11 of smoothness, 14

Class

of all UCED spaces, 7 Coefficient

ρ_{0}(X), 34 τ-convergent sequences (τCS(X)), 27 normal structure (N(X)), 28 normal structure, 28–29, 31 weakly convergent sequences (WCS(X)), 27

Coefficients normal structure, 34 related to normal structure, 27 related to the modulus b_{X,τ}, 34 scaling, 1

Condition Lipschitz, 12, 18 Constant

separation, 17 Convexity, 1, 21

of balls in Banach spaces, 12

of balls, 8

strict, 1, 3–4, 6

partially dual to, 4 uniform versions of, 33

Derivative

of the norm, 33 Diametral, 25, 32 Differentiability, 4, 33 Directional derivative, 14 Directional derivatives

one-sided, 4 Epigraph, 5 Euclidean space k-dimensional, 8

Families

normal, 25 Family

normal, 25

of k-convex functions, 9

of all k dimensional subspaces, 12

of convex functions, 8

of convex uniformly bounded functions, 6 uniformly normal, 25 Finitely representable, 10, 23 Formulae

Lindenstrauss’, 14 Fréchet derivative of f at x, 4

Fréchet differentiability, 4 Fréchet differentiable uniformly, 4, 14

Function k-convex, 9 double dual Young’s, 12, 32 dual Young’s, 6, 20

Function

Fréchet differentiable

at x, 4 Lipschitzian, 6 weight, 5 Functions

dual Young’s, 14 Rademacher, 27, 31 Rademacher-like, 21 scaling, 33 Geometry

of Banach spaces, 1, 32 Global modulus of convexity, 8 Gâteaux derivative of f at x, 4

Gâteaux differentiable, 4 at x, 4–5 Half planes, 7 Hilbert space, 4, 7, 10, 12, 23, 33

infinite dimensional, 31 with dim(X) ≥ 2, 13

Homogenity, 4 Hyperplane, 4–5 Inequalities

Clarkson’s and Hanner’s, 10, 13 Kadec-Klee property

with respect to τ (KK(τ)), 15 Lemma

Fatou’s, 15 Lipschitzian, 9 Local character, 8 Mapping

duality, 23 sequentially τ-continuous, 23 with the weight ϕ, 5

nonexpansive, 3 Mappings

duality, 33 nonexpansive, 25

Measure space, 4, 10, 13, 22 σ-finite, 15, 27 purely atomic, 30

Measure counting, 15 purely atomic, 10

Metric space

compact, 29 Metric

abstract hyperbolic, 33 hyperbolic, 33

Midpoint

metric, 2 Minimum point

of a convex function, 28 Moduli, 6, 22, 24, 33

b_{X,w} and d_{X^{∗},w}, 20 b_{X,w} and b̃_{X}, 20 d_{X,τ} and K_{X,τ}, 18 continuity of, 33 of convexity and smoothness, 14

Modulus, 9, 12–13, 33 b_{X,τ}, 19 K_{X}, 16 K_{X,τ}, 16 δ, 9 δ_{X}(x; ϵ), 7 δ_{X}(z; ϵ), 8 ρ, 14 ρ_{X}, 13 corresponding to the norm, 11 corresponding to UKK(τ) property, 18 of k-convexity δ_{X}^{k}(ϵ), 9 of k-convexity, 33 of k-uniform smoothness β_{X}^{k}, 13 of convexity, 9–10, 14, 33 of smoothness, 31, 33 of uniform smoothness ρ_{X}, 13 Opial, 33

Modulus

Opial’s

with respect to τ, 21 Nearly uniform convexity, 33 Nearly uniform smoothness, 33 Net, 26 Norm

clm-sequentially lower semicontinuous, 15 differentiability of, 4, 14 equivalent, 4, 7, 11, 14, 32 Gâteaux differentiable, 4, 6, 14 uniformly Fréchet differentiable, 14

NS, 25–28, 32, 34

NUC, 17–18, 21, 23, 33

NUS, 19–21 NUS(w), 20 NUS(τ), 24 NUS(clm), 23 Operator

linear one-to-one, 10 Parallelepiped, 8 Parallelogram law, 7, 10 generalized, 12

Partition of Ω, 15 Plane, 7 Projection, 23 metric, 3 Properties

geometrical, 1 Property

L(ρ_{p},clm), 22 L(ρ,w), 23 L(ρ,τ), 22, 24, 27 (H), 33 dual to k-UC, 13 dual to NUC, 19 geometrical, 26 Kadec-Klee, 33 KK, 16 KK(w), 15 KK(τ), 16

Property

nonstrict (or weak) Opial with respect to τ, 21

nonstrict Opial, 21

with respect to τ, 22 Opial, 33

with respect to τ, 21–22, 26 Radon-Riesz, 33 Schur, 16, 23 UKK, 16–18, 23, 28 UKK(τ), 16, 18–19, 24, 27 UKK(clm), 23 uniform Kadec-Klee, 33 uniform Opial, 27

with respect to τ, 21–22, 24, 27 with respect to the clm topology, 23 with respect to the weak topology, 33

Radius

Chebyshev, 25 Ray, 7 Reflexive, 4, 10, 17, 19–20, 23, 29 Reflexivity, 17, 20 Segment, 3 affine, 2 metric, 2–3 nontrivial, 2

Segments metric, 2 Sequence basic, 17

Set diametral, 25, 27 of all fixed points, 3

Smooth, 6

Smoothness, 4–6, 21, 33

Space infinite dimensional NUS, 19

INDEX 41

reflexive, 16, 18–19, 29 separable, 4, 16 superreflexive, 11, 14

Spaces k-UR, 17 k-US, 13 Bynum’s, 32 finite dimensional, 11, 31 NUC, 28 reflexive, 4, 25, 29 strictly convex, 2, 11 superreflexive, 11, 13 UC, 11 UCED, 11 uniformly convex, 11 uniformly nonsquare, 11

Sphere, 3

unit, 1 Strict convexity, 11, 32 Structure

normal (NS), 25 normal, 25, 32, 34

with respect to τ (τ-NS), 25

with respect to an arbitrary topology, 34 uniform normal (UNS), 25 uniform normal, 34 weak normal (WNS), 25 weak normal, 34

Subdifferential, 5, 29, 31 of f at x, 5 of the norm at nonzero points, 5 of the norm at zero, 5

Subdifferentials, 28 Subset

diametral, 25 Superreflexive, 10–11, 14, 23 Superreflexivity, 34 System

infinite Rademacher, 31 Theorem

dominated convergence, 22, 24 duality, 13 fixed point, 33

inL^{1}(Ω) with clm the topology, 33 Mazur’s, 18, 26 Ramsey’s, 27 Schur’s, 16

Theory approximation, 3 fixed point, 1, 3, 34 metric fixed point, 6, 25, 34 of fixed points of Holomorphic mappings, 33

Topologies

Hausdorff, 15 Topology

clm, 15, 24, 33 Hausdorff τ, 15 linear, 15 norm, 15 of convergence in measure, 15 of convergence locally in measure (clm), 15 weak, 15–16, 23, 33 weak^{∗}, 15 UC, 11–12, 33

in metric spaces, 33 UCED, 7–8, 26–27 Uniform convexity (UC), 9 Uniform convexity, 32 finite dimensional, 33 in every direction, 33 infinite dimensional generelization, 17 multi-dimensional, 33 noncompact, 33 of Clarkson, 33

Uniform Kadec-Klee property

with respect to τ (UKK(τ)), 16 Uniform rotundity (UR), 9 Uniform smoothness, 13–14, 33 multi-dimensional, 32, 34

Uniformly convex, 11 UNS, 27–28, 31–32, 34 US, 13–14, 33 Volume

k-dimensional, 8 Weight, 5 WNS, 25, 27–28 clm-NS, 27