A separable reflexive Banach space having no finite dimensional Čebyšev subspaces

On uniform continuity of convex bodies with respect to measures in Banach spaces

“…Applying Remark 2.19 we get that the new sphere is finite universal with r = 1 2 . This construction of ||| ||| is similar to a construction in [9]. Example.…”

Epigraphical and Uniform Convergence of Convex Functions

“…Indeed, we show that the nonparenthetical result holds in spaces not containing 1 whereas the parenthetical result holds precisely when Mackey and weak-star convergence agree sequentially in the dual space. Thus we provide a convex function extension of the theorem in [19], and we recapture (withoutŠmulyan's criterion) some differentiability results of [10]. Unfortunately, as examples show, these results do not extend in a fashion to allow one to obtain Attouch-Wets convergence from a natural weaker form of epi-convergence in spaces not containing 1 .…”

“…This is because Mackey and norm convergence coincide sequentially in the dual space of any space not containing 1 (see [10,19]). This leads us to the third section where we study when a sequence of convex functions converging uniformly on weakly compact sets (resp.…”

“…To show this, we recall that a space is said to be sequentially reflexive if norm and Mackey convergence coincide sequentially in the dual space. In [19] it was shown that a Banach space X is sequentially reflexive if and only if X ⊃ 1 (by X ⊃ 1 we mean X does not contain an isomorphic copy of 1 ). Because the manuscript [19] is unpublished, we refer the reader to Theorem 5 in the appendix of [10] for a proof of this result.…”

“…In [19] it was shown that a Banach space X is sequentially reflexive if and only if X ⊃ 1 (by X ⊃ 1 we mean X does not contain an isomorphic copy of 1 ). Because the manuscript [19] is unpublished, we refer the reader to Theorem 5 in the appendix of [10] for a proof of this result. We will not pursue questions of stability of W-convergence under addition however, the next theorem has ramifications in this area (see the proof of Theorem 3.1 and the discussion following it).…”