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 ) 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
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.