Continuity and Discontinuity of Seminorms on Infinite-Dimensional Vector Spaces. II

Continuity and Discontinuity of Seminorms on Infinite-Dimensional Vector Spaces

Let $S$ be a seminorm on an infinite-dimensional real or complex vector space $X$. Our purpose in this note is to study the continuity and discontinuity properties of $S$ with respect to certain norm-topologies on $X$.

Weak-star point of continuity property and Schauder bases

We characterize the weak-star point of continuity property for subspaces of dual spaces with separable predual and we deduce that the weak-star point of continuity property is determined by subspaces with a Schauder basis in the natural setting of dual spaces of separable Banach spaces. As a consequence of the above characterization we get that a dual space satisfies the Radon-Nikodym property if, and only if, every seminormalized topologically weak-star null tree has a boundedly complete branch, which improves some results in \cite{DF} obtained for the separable case. Also, as a consequence o…

Fixed point theorem for discontinuous mappings on PN spaces

We present a study on strong t-continuity and measure of discontinuity on PN spaces. As an application, we prove a fixed point theorem for a self mapping on PN spaces by means of measure of discontinuity.

Continuity of bilinear maps on direct sums of topological vector spaces

We prove a criterion for continuity of bilinear maps on countable direct sums of topological vector spaces. As a first application, we get a new proof for the fact (due to Hirai et al. 2001) that the map taking a pair of test functions on R^n to their convolution is continuous. The criterion also allows an open problem by K.-H. Neeb to be solved: If E is a locally convex space, regard the tensor algebra T(E) as the locally convex direct sum of the projective tensor powers T^j(E) of E. We show that T(E) is a topological algebra if and only if every sequence of continuous seminorms on E has an u…

拓扑向量空间中的微分

Differentiation in Topological Vector Spaces

本文探讨拓扑向量空间中微分的定义与性质，基于F-半范数构建扩展的ε-δ语言，定义单值映射的连续性及广义Gâteaux和Frechet导数，并应用于非normed空间的优化问题。

Continuity of LF-algebra representations associated to representations of Lie groups

Let G be a Lie group and E be a locally convex topological G-module. If E is sequentially complete, then E and its space of smooth vectors are modules for the algebra D(G) of compactly supported smooth functions on G. However, the module multiplication need not be continuous. The pathology can be ruled out if E is (or embeds into) a projective limit of Banach G-modules. Moreover, in this case the space of analytic vectors is a module for the algebra A(G) of superdecaying analytic functions introduced by Gimperlein, Kroetz and Schlichtkrull. We prove that the space of analytic vectors is a topo…