Deformation in locally convex topological linear spaces

2004 ◽  
Vol 47 (5) ◽  
pp. 687 ◽  
Author(s):  
Yanheng DING
Keyword(s):  
Linear Spaces ◽  
Topological Linear ◽  
Locally Convex

scholarly journals The Attouch-Wets topology and a characterisation of normable linear spaces

1991 ◽  
Vol 44 (1) ◽  
pp. 11-18 ◽  
Author(s):  
Ľubica Holá
Keyword(s):  
Metric Spaces ◽  
Continuous Functions ◽  
Linear Functions ◽  
Continuous Linear ◽  
Invariant Metric ◽  
Compact Open Topology ◽  
Connected Space ◽  
Linear Spaces ◽  
Topological Linear ◽  
Locally Convex

Let X and Y be metric spaces and C(X, Y) be the space of all continuous functions from X to Y. If X is a locally connected space, the compact-open topology on C(X, Y) is weaker than the Attouch-Wets topology on C(X, Y). The result is applied on the space of continuous linear functions. Let X be a locally convex topological linear space metrisable with an invariant metric and X* be a continuous dual. X is normable if and only if the strong topology on X* and the Attouch-Wets topology coincide.

scholarly journals Notes on differential calculus in topological linear spaces, III

1976 ◽  
Vol 21 (1) ◽  
pp. 88-95
Author(s):  
S. Yamamuro
Keyword(s):  
Real Number ◽  
Number Field ◽  
Open Subset ◽  
Differential Calculus ◽  
Hausdorff Spaces ◽  
Real Numbers ◽  
Linear Spaces ◽  
Topological Linear ◽  
Locally Convex ◽  
Real Number Field

Throughout this note, let E, F and G be locally convex Hausdorff spaces over the real number field R. We denote real numbers by Greek letters. The sets of all continuous semi-norms on E and F will be denoted by P(E) and P(F) respectively, and A will always stand for an open subset of E.

Weak Compactness and Separation

1964 ◽  
Vol 16 ◽  
pp. 204-206 ◽  
Author(s):  
Robert C. James
Keyword(s):  
Banach Space ◽  
Weak Compactness ◽  
The Other ◽  
Weakly Compact ◽  
Linear Spaces ◽  
Topological Linear ◽  
Locally Convex ◽  
Compact Subsets

The purpose of this paper is to develop characterizations of weakly compact subsets of a Banach space in terms of separation properties. The sets A and B are said to be separated by a hyperplane H if A is contained in one of the two closed half-spaces determined by H, and B is contained in the other; A and B are strictly separated by H if A is contained in one of the two open half-spaces determined by H, and B is contained in the other. The following are known to be true for locally convex topological linear spaces.

Notes on differential calculus in topological linear spaces, II

1975 ◽  
Vol 20 (2) ◽  
pp. 245-252 ◽  
Author(s):  
S. Yamamuro
Keyword(s):  
Real Number ◽  
Number Field ◽  
Open Subset ◽  
Differential Calculus ◽  
Hausdorff Spaces ◽  
Real Numbers ◽  
Linear Spaces ◽  
Topological Linear ◽  
Locally Convex ◽  
Real Number Field

Throughout this note, let E and F be locally convex Hausdorff spaces over the real number field R. We denote real numbers by Greek letters. The sets of all continuous semi-norms on E and F will be denoted by P(E) and P(F) respectively, and A will always stand for an open subset of E.

scholarly journals A factor theorem for locally convex differentiability spaces

1991 ◽  
Vol 43 (1) ◽  
pp. 101-113 ◽  
Author(s):  
Roger Eyland ◽  
Bernice Sharp
Keyword(s):  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Space Theory ◽  
Linear Spaces ◽  
Linear Maps ◽  
Factor Theorem ◽  
Continuous Convex Function ◽  
Topological Linear ◽  
Locally Convex ◽  
Differentiability Properties

The main result of this paper is that a continuous convex function with domain in a locally convex space factors through a normed space. In a recent paper by Sharp, topological linear spaces are categorised according to the differentiability properties of their continuous convex functions; we show that under suitable conditions the classification is preserved by linear maps. A technique for deducing results for locally convex spaces from Banach space theory is an immediate consequence. Examples are given and Asplund C(S) spaces are characterised.

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