scholarly journals The Embedding Theorem of an L0-Prebarreled Module into Its Random Biconjugate Space

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
Xia Zhang ◽  
Ming Liu
Keyword(s):  
Embedding Theorem ◽  
Canonical Embedding ◽  
Linear Homeomorphism ◽  
Convex Topology ◽  
Convex Module ◽  
Locally Convex ◽  
Random Locally Convex Module

We first prove Mazur’s lemma in a random locally convex module endowed with the locally L0-convex topology. Then, we establish the embedding theorem of an L0-prebarreled random locally convex module, which says that if (S,P) is an L0-prebarreled random locally convex module such that S has the countable concatenation property, then the canonical embedding mapping J of S onto J(S)⊂(Ss⁎)s⁎ is an L0-linear homeomorphism, where (Ss⁎)s⁎ is the strong random biconjugate space of S under the locally L0-convex topology.

The β- and γ-duality of sequence spaces

1967 ◽  
Vol 63 (4) ◽  
pp. 963-981 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Linear Space ◽  
Sequence Spaces ◽  
Topological Properties ◽  
Product Topology ◽  
Convex Topology ◽  
Locally Convex Topology ◽  
Locally Convex

A K-space (E, τ) is a linear space E of sequences with a locally convex topology τ for which the inclusion map: (E, τ) → (ω, product topology) is continuous. In (2) topological properties of K-spaces were determined directly from properties of the space E and the topology τ. It is, however, very natural to consider duality properties of K-spaces and the purpose of this paper is to determine some of these properties.

An Embedding Theorem for Separable Locally Convex Spaces

1971 ◽  
Vol 14 (1) ◽  
pp. 119-120 ◽  
Author(s):  
Robert H. Lohman
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Space ◽  
Separable Banach Space ◽  
Locally Convex Space ◽  
Embedding Theorem ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Topological Vector Spaces ◽  
Locally Convex

A well-known embedding theorem of Banach and Mazur [1, p. 185] states that every separable Banach space is isometrically isomorphic to a subspace of C[0, 1], establishing C[0, 1] as a universal separable Banach space. The embedding theorem one encounters in a course in topological vector spaces states that every Hausdorff locally convex space (l.c.s.) is topologically isomorphic to a subspace of a product of Banach spaces.

scholarly journals A generalisation of Mackey's theorem and the uniform boundedness principle

1989 ◽  
Vol 40 (1) ◽  
pp. 123-128 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Uniform Boundedness ◽  
Mackey Topology ◽  
Convex Topology ◽  
Uniform Boundedness Principle ◽  
Bounded Sets ◽  
Locally Convex Topology ◽  
Locally Convex

We construct a locally convex topology which is stronger than the Mackey topology but still has the same bounded sets as the Mackey topology. We use this topology to give a locally convex version of the Uniform Bouudedness Principle which is valid without any completeness or barrelledness assumptions.

scholarly journals Barrelled spaces and dense vector subspaces

1988 ◽  
Vol 37 (3) ◽  
pp. 383-388 ◽  
Author(s):  
W.J. Robertson ◽  
S.A. Saxon ◽  
A.P. Robertson
Keyword(s):  
Simple Proof ◽  
Structure Theorem ◽  
Vector Subspace ◽  
Hamel Basis ◽  
Convex Topology ◽  
Barrelled Spaces ◽  
Locally Convex Topology ◽  
Locally Convex ◽  
Barrelled Space ◽  
Vector Subspaces

This note presents a structure theorem for locally convex barrelled spaces. It is shown that, corresponding to any Hamel basis, there is a natural splitting of a barrelled space into a topological sum of two vector subspaces, one with its strongest locally convex topology. This yields a simple proof that a barrelled space has a dense infinite-codimensional vector subspace, provided that it does not have its strongest locally convex topology. Some further results and examples discuss the size of the codimension of a dense vector subspace.

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