locally convex space

locally convex space

[′lō·kə·lē ′kän‚veks ‚spās]
(mathematics)
A Hausdorff topological vector space E such that every neighborhood of any point x belonging to E contains a convex neighborhood of x.
References in periodicals archive ?
For a Tychonoff space X, we denote by L(X), V(X), F(X), and A(X) the free locally convex space, the free topological vector space, the free topological group, and the free abelian topological group over X, respectively.
(Park [23]) Let k [greater than or equal to] 1 and, for each h [member of] [Z.sub.k], let [Y.sub.h] be a nonempty compact convex subset of a locally convex space [E.sub.h], and [V.sub.h] [member of] V([Y.sub.h], [Y.sub.h+1]).
(1) For a given locally convex space E, a convex function f : E [right arrow] R U (+[infinity]} is called a proper convex function if f (x) < [infinity] for some x [member of] E.
(b) Let (X, [tau]) be a metrizable locally convex space. Then, there is a fuzzy norm (N, [conjunction]) on X such that [[tau].sub.N] = [tau].
Et., Generalized difference sequence spaces defined by a modulus function in a locally convex space, Soochow J.
In this paper r stands for the set of real numbers, K will denote the field of real or complex numbers (we will call them scalars), X a Hausdorff normal topological space and E a quasi-complete locally convex space space over K with topology generated by an increasing family of semi-norms [[parallel]*[parallel].sub.p], p [member of] P; E' will denote the topological dual of E.
Let Y be a (real) separated locally convex space; and K, some (convex) cone of it ([alpha]K + [beta]K [[subset].bar] K for each [alpha], [beta] [greater than or equal to] 0).
It then provides generalizations of the classical result of the Orlicz-Pettis theorem to delta multiplier convergent series with values in a locally convex space. Other topics covered include generalizations of the Hanh-Schur theorem to delta multiplier convergent series, double series, and automatic continuity of matrix mappings between sequence spaces.
Altin, Generalized difference sequence spaces defined by Orlicz function in a locally convex space, J.
Altin and M.Et, Generalized difference sequence spaces defined by a modulus function in a locally convex space, Soochow J.Math., 31 (2) (2005), 233-243.
The existence of such a resolution in a locally convex space E has shown to be equivalent to the existence of a so-called G-base of absolutely convex neighborhoods of the origin in the strong dual [E',sub.[beta] of E.
If a locally convex space X admits an operator without closed invariant subspaces then L(X) is strongly generated by two elements.
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