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...

We continue the work of [14, 3, 1, 19, 16, 4, 12, 11, 20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to WKL0 over RCA0. We show that the separation theorem for separably closed convex sets is equivalent to ACA0 over RCA0. Our strategy for proving these geometrical Hahn–Banach...

We prove a large deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space, that is, the analog of Cramér theorem for random compact sets. Several works have been devoted to deriving limit theorems for random sets. For i.i.d. random compact sets in R, the law of large numbers was initially proved by Artstein and Vitale [1] and the central limit theorem by Cressie [3...

We prove that every separable metric space which admits an `1tree as a Lipschitz quotient, has a σ-porous subset which contains every Lipschitz curve up to a set of 1-dimensional Hausdorff measure zero. This applies to any Banach space containing `1. We also obtain an infinite-dimensional counterexample to the Fubini theorem for the σ-ideal of σ-porous sets.

A Wiener-Hopf operator on a Banach space of functions on R is a bounded operator T such that PS−aTSa = T , a ≥ 0, where Sa is the operator of translation by a. We obtain a representation theorem for the Wiener-Hopf operators on a large class of functions on R with values in a separable Hilbert space.

Let E be a Banach space. The concept of n-type over E is introduced here, generalizing the concept of type over E introduced by Krivine and Maurey. Let E′′ be the second dual of E and fix g′′ 1 , . . . ,g′′ n ∈ E′′. The function τ : E×Rn → R, defined by letting τ(x,a1, . . . ,an) = ‖x+∑ni=1aig′′ i ‖ for all x ∈ E and all a1, . . . ,an ∈R, defines an n-type over E. Types that can be represented ...

In the Banach space setting, the existence of viable solutions for differential inclusions with nonlinear growth; that is, x(·)(t) ∈ a.e. on I, x(t) ∈ S, ∀t ∈ I, x(0) = x₀ ∈ S, (∗), where S is a closed subset in a Banach space X, I = [0, T], (T > 0), F : I × S → X, is an upper semicontinuous set-valued mapping with convex values satisfying F(t, x) ⊂ c(t)(||x|| + ||x|| (p)K, ∀(t, x) ∈ I × S, whe...

We prove that every separable polyhedral Banach space X is isomorphic to a polyhedral Banach space Y such that, the set ext BY ∗ cannot be covered by a sequence of balls B(yi, 2i) with 0 < 2i < 1 and 2i → 0. In particular ext BY ∗ cannot be covered by a sequence of norm compact sets. This generalizes a result from [7] where an equivalent polyhedral norm ||| · ||| on c0 was constructed such that...

This work introduces the concept of an M-complete approximate identity (M-cai) for a given operator subspace X of an operator space Y. M-cai's generalize central approximate identities in ideals in C *-algebras, for it is proved that if X admits an M-cai in Y , then X is a complete M-ideal in Y. It is proved, using " special " M-cai's, that if J is a nuclear ideal in a C *-algebra A, then J is ...