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Topological dual of non-locally convex Orlicz-Bochner spaces

Marian Nowak (1999)

Commentationes Mathematicae Universitatis Carolinae

Let L ϕ ( X ) be an Orlicz-Bochner space defined by an Orlicz function ϕ taking only finite values (not necessarily convex) over a σ -finite atomless measure space. It is proved that the topological dual L ϕ ( X ) * of L ϕ ( X ) can be represented in the form: L ϕ ( X ) * = L ϕ ( X ) n ∼ ⊕ L ϕ ( X ) s ∼ , where L ϕ ( X ) n ∼ and L ϕ ( X ) s ∼ denote the order continuous dual and the singular dual of L ϕ ( X ) respectively. The spaces L ϕ ( X ) * , L ϕ ( X ) n ∼ and L ϕ ( X ) s ∼ are examined by means of the H. Nakano’s theory of conjugate modulars. (Studia Mathematica 31 (1968), 439–449). The well known results of the duality theory...

Topological groups and convex sets homeomorphic to non-separable Hilbert spaces

Taras Banakh, Igor Zarichnyy (2008)

Open Mathematics

Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover 𝒰 of X there is a sequence of maps (f n: X → X)nεgw such that each f n is 𝒰 -near to the identity map of X and the family f n(X)n∈ω is locally finite...

Topological Properties of Real Normed Space

Kazuhisa Nakasho, Yuichi Futa, Yasunari Shidama (2014)

Formalized Mathematics

In this article, we formalize topological properties of real normed spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. Then we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, image and inverse image are considered here. The fact that Lipschitz continuity operators preserve convergence of sequences is also refered...

Topological tensor products of a Fréchet-Schwartz space and a Banach space

Alfredo Peris (1993)

Studia Mathematica

We exhibit examples of countable injective inductive limits E of Banach spaces with compact linking maps (i.e. (DFS)-spaces) such that E ⊗ ε X is not an inductive limit of normed spaces for some Banach space X. This solves in the negative open questions of Bierstedt, Meise and Hollstein. As a consequence we obtain Fréchet-Schwartz spaces F and Banach spaces X such that the problem of topologies of Grothendieck has a negative answer for F ⨶ π X . This solves in the negative a question of Taskinen. We also give...

Topological type of weakly closed subgroups in Banach spaces

Tadeusz Dobrowolski, Janusz Grabowski, Kazuhiro Kawamura (1996)

Studia Mathematica

The main result says that nondiscrete, weakly closed, containing no nontrivial linear subspaces, additive subgroups in separable reflexive Banach spaces are homeomorphic to the complete Erdős space. Two examples of such subgroups in ℓ 1 which are interesting from the Banach space theory point of view are discussed.

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