Reflexive space

Reflexive space

Locally Convex Topological Vector Spaces

Published in Kenneth Kuttler, Modern Analysis, 2017

Proof: Let {xn}⊆B≡B(0,1)¯. Let Y be the closure of the linear span of {xn}. Thus Y is a separable, hence weakly separable Banach space. It is reflexive because it is a closed subspace of a reflexive space. (See Problem 7 of this chapter.) By the Alaoglu theorem, the closed unit ball B* in Y’ is weak* compact. Also by Theorem 6.20, B* is a metric space with a suitable metric. Thus B* is complete and totally bounded with respect to this metric and it follows that B* with the weak* topology is separable. Since Y is reflexive, B* is also weakly compact and separable. By Corollary 6.21 B**, the closed unit ball in Y” is weak* sequentially compact. Then by Lemma 6.22 B, the unit ball in Y, is weakly sequentially compact. It follows there exists a subsequence xnk, of the sequence {xn} and a point x ∈ Y, such that for all f ∈ Y′,

On nonlinear perturbations of a periodic integrodifferential equation with critical exponential growth

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Journal Information

Published in Applicable Analysis, 2023

Yane Araújo, Eudes Barboza, Gilson de Carvalho

We would like to point out that space has suitable properties which give to problem () a variational framework. More specifically, it is uniformly convex Banach space (see Lemma 2.2), and therefore is a reflexive space. Moreover, is dense in (see Lemma 2.4).

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Locally Convex Topological Vector Spaces

On nonlinear perturbations of a periodic integrodifferential equation with critical exponential growth