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.

scholarly journals Inductive and projective limits of normed spaces

1968 ◽  
Vol 9 (2) ◽  
pp. 103-105 ◽  
Author(s):  
John S. Pym
Keyword(s):  
Inductive Limit ◽  
Vector Spaces ◽  
Continuous Linear ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Inductive System ◽  
Linear Maps ◽  
Inductive And Projective Limits ◽  
Projective Limits ◽  
Locally Convex

Let {Ui, Uij} be an inductive system of normed linear spaces Ui and continuous linear maps uij; Uj → Ui. (We write j ≺ i if uij: Uj → Ui.) An inductive limit of the system with respect to a class of spaces A in and maps f in is a space Uu in Uu and a system ui → Uu of maps in such that (i) whenever j ≺ i, and that (ii) if A is any space in and fi: Ui → A is any system of maps in for which then there is a unique map f: Uu → A in such that fi = fo ui for each i. If is the class of all vector spaces and is the class of linear maps, we obtain the algebraic inductive limit, which we denote simply by U. The usual choice is to take to be the class of locally convex spaces and the class of continuous linear maps; the inductive limit UL then always exists [1, § 16 C]. If is again the continuous linear mappings but contains only normed spaces, the corresponding inductive limit UN may not always exist. However, if in addition we require that contains just contractions (norm-decreasing linear mappings), then an inductive limit Uc will exist if every uij is a contraction [2]. We shall give a condition under which these limits coincide (as far as possible), and consider the corresponding condition for projective limits.

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.

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