This is a survey of results on the classification of Banach spaces as metric spaces. It is based on a series of lectures I gave at the Functional Analysis Seminar in 1984-1985, and it appeared in the 1984-1985 issue of the Longhorn Notes. I keep receiving requests for copies, because some of the material here does not appear elsewhere and because the Longhorn Notes are not so easy to get. Havin...

It is a well-known result of Kadec that every two separable infinite dimensional Banach spaces are homeomorphic. Also in large classes of nonseparable Banach spaces (perhaps all) the density character of a Banach space is its only topological invariant (see the book [2] for details). The situation changes considerably if we consider uniform homeomorphisms. Several results are known which prove ...

Let X be a Banach space with closed unit ball B. Given k ∈ N, X is said to be k-β, repectively, (k + 1)-nearly uniformly convex ((k + 1)-NUC), if for every ε > 0, there exists δ, 0 < δ < 1, so that for every x ∈ B, and every ε-separated sequence (xn) ⊆ B, there are indices (ni) k i=1, respectively, (ni) k+1 i=1 , such that 1 k+1 ‖x + ∑k i=1 xni‖ ≤ 1 − δ, respectively, 1 k+1 ‖ ∑k+1 i=1 xni‖ ≤ 1−...

and Applied Analysis 3 If θ ≡ 0, the problem 1.4 reduces into the minimize problem, denoted by arg min φ , which is to find x ∈ C such that φ ( y ) − φ x ≥ 0, ∀y ∈ C. 1.7 The above formulation 1.5 was shown in 11 to covermonotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equil...

and Applied Analysis 3 It is easy to see that a quasi-nonexpansive mapping is an asymptotically quasi-nonexpansive mapping with the sequence {1}. T is said to be asymptotically nonexpansive in the intermediate sense if and only if it is continuous, and the following inequality holds: lim sup n→∞ sup x,y∈C (∥ ∥Tx − Tny∥∥ − ∥∥x − y∥∥) ≤ 0. 2.6 T is said to be asymptotically quasi-nonexpansive in ...

and Applied Analysis 3 nonexpansive mapping such that F S ∩ A−10/ ∅. Let {xn} be a sequence generated by x0 x ∈ C and un J−1 ( βnJxn ( 1 − βn ) JSJrnxn ) , Cn { z ∈ C : φ z, un ≤ φ z, xn } , Qn {z ∈ C : 〈xn − z, Jx0 − Jxn〉 ≥ 0}, xn 1 ΠCn∩Qnx0 1.6 for all n ∈ N ∪ {0}, where J is the duality mapping on E, {βn} ⊂ 0, 1 , and {rn} ⊂ a,∞ for some a > 0. If lim infn→∞ 1 − βn > 0, then {xn} converges s...

and Applied Analysis 3 Let X be a smooth Banach space. We always use φ : X × X → R to denote the Lyapunov functional defined by φ ( x, y ) ‖x‖ − 2〈x, Jy〉 ∥∥y∥∥2, ∀x, y ∈ X. 1.7 It is obvious from the definition of the function φ that (‖x‖ − ∥∥y∥∥)2 ≤ φ(x, y) ≤ (‖x‖ ∥∥y∥∥)2, ∀x, y ∈ X. 1.8 Following Alber 4 , the generalized projection ΠC : X → C is defined by ΠC x arg inf y∈C φ ( y, x ) , ∀x ∈ ...

In the paper it is shown that in a Banach space with a basis there is a norm with property (β) of S. Rolewicz if and only if there is a norm which is simultaneously nearly uniformly convex and nearly uniformly smooth. The Kuratowski measure of noncompactness of a set A in a Banach space is the infimum α(A) of those ǫ > 0 for which there is a covering of A by a finite number of sets Ai with diam...

and Applied Analysis 3 where J is the duality mapping from E into E∗. It is well known that if C is a nonempty closed convex subset of a Hilbert space H and PC : H → C is the metric projection of H onto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. It is obvious from the definition of function φ t...

and Applied Analysis 3 Theorem 2.1 Kirk 1 . LetU be a bounded open subset of a Hadamard space X and S : clU → X a nonexpansive mapping. Suppose that there exists p ∈ U such that every x in the boundary ofU does not belong to p, Sx \ {Sx}. Then, S has a fixed point in clU. Let C be a nonempty closed convex subset of a Hadamard space X. Then, for each x ∈ X, there exists a unique point yx ∈ C suc...