Two homogeneous measures of noncompactness β and γ on an infinite dimensional Banach space X are called “equivalent” if there exist positive constants b and c such that bβ(S) ≤ γ (S) ≤ cβ(S) for all bounded sets S ⊂ X . If such constants do not exist, the measures of noncompactness are “inequivalent.”Weask a foundational questionwhich apparently has not previously been considered: For what infi...

We say that a Banach space X is the unique predual of its dual (more precisely the unique isometric predual of its dual) in case it is isometric to any Banach space whose dual is isometric to the dual of X. (We say that two Banach spaces Y and Z are isomorphic if there is a bounded linear bijective operator T : Y → Z with bounded inverse T; if moreover ‖T (y)‖ = ‖y‖ for all y ∈ Y we say that Y ...

and Applied Analysis 3 Lemma 2.7 Goebel-Kirk . Let X be a Banach space. For each ε ∈ ε0 X , 2 , one has the equality δX 2 − 2δX ε 1 − ε/2. Lemma 2.8 Ullán . Let X be a Banach space. For each 0 ≤ ε2 ≤ ε1 < 2 the following inequality holds: δX ε1 − δX ε2 ≤ ε1 − ε2 / 2 − ε1 . Using these lemmas we obtain: Theorem 2.9. Let X be a Banach space which satisfies δX 1 > 0, that is, ε0 X < 1. Then X is P...

Let N and M be von Neumann algebras. It is proved that L p (N) does not Banach embed in L p (M) for N infinite, M finite, 1 ≤ p < 2. The following considerably stronger result is obtained (which implies this, since the Schatten p-class Cp embeds in L p (N) for N infinite). Theorem. Let 1 ≤ p < 2 and let X be a Banach space with a spanning set (x ij) so that for some C ≥ 1, (i) any row or column...

where g:(0, 1)-»X is an essentially bounded strongly measurable function. In this paper we examine analogues of the Radon-Nikodym Property for quasiBanach spaces. If 0 < p < 1, there are several possible ways of defining "differentiable" operators on Lp, but they inevitably lead to the conclusion that the only differentiable operator is zero. For example, a differentiable operator on L\ has the...

We investigate the concepts of linear convexity and C-convexity in complex Banach spaces. The main result is that any C-convex domain is necessarily linearly convex. This is a complex version of the Hahn-Banach theorem, since it means the following: given a C-convex domain Ω in the Banach space X and a point p / ∈Ω, there is a complex hyperplane through p that does not intersect Ω. We also prov...

We investigate the concepts of linear convexity and C-convexity in complex Banach spaces. The main result is that any C-convex domain is necessarily linearly convex. This is a complex version of the Hahn-Banach theorem, since it means the following: given a C-convex domain Ω in the Banach space X and a point p / ∈Ω, there is a complex hyperplane through p that does not intersect Ω. We also prov...

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...

We show that a separable proximinal subspace of X, say Y is strongly proximinal (strongly ball proximinal) if and only if Lp(I, Y ) is strongly proximinal (strongly ball proximinal) in Lp(I,X), for 1 ≤ p <∞. The p =∞ case requires a stronger assumption, that of ’uniform proximinality’. Further, we show that a separable subspace Y is ball proximinal in X if and only if Lp(I, Y ) is ball proximin...

Throughout this paper X denotes a bicompact space, B(X) denotes the Banach space of all bounded real-valued functions on X, C(X) denotes the closed linear subspace of B(X) of continuous real-valued functions on X, B(N) is denoted by m (where N is the set of positive integers). The space h(X) is the Banach space of all absolutely summable real-valued functions on X and k denotes k(N). The sequen...