Compact sets in locally convex spaces

RESTRICTED p-CENTERS FOR SETS IN REAL LOCALLY CONVEX SPACES

Let X ,Z be a pair of real linear spaces put in duality by a separating bilinear form 〈, 〉, and endowed with compatible locally convex topologies respectively. A subset F of X is said to be p-bounded if sup p(x) : x ∈ F < ∞. We denote the collection of all nonempty p-bounded subsets of X by p(X ). Given x ∈ X , F ∈ p(X ), and V ⊆ X , write rp(F ; x) = sup p(y − x) : y ∈ F , radp(F ;V ) = inf rp...

Approximate Convexity and Submonotonicity in Locally Convex Spaces

Locally Compact, Ω1-compact Spaces

This paper is centered on an extremely general problem: Problem. Is it consistent (perhaps modulo large cardinals) that a locally compact space X must be the union of countably many ω-bounded subspaces if every closed discrete subspace of X is countable [in other words, if X is ω1-compact]? A space is ω-bounded if every countable subset has compact closure. This is a strengthening of countable ...

Some results on functionally convex sets in real Banach spaces

‎We use of two notions functionally convex (briefly‎, ‎F--convex) and functionally closed (briefly‎, ‎F--closed) in functional analysis and obtain more results‎. ‎We show that if $lbrace A_{alpha} rbrace _{alpha in I}$ is a family $F$--convex subsets with non empty intersection of a Banach space $X$‎, ‎then $bigcup_{alphain I}A_{alpha}$ is F--convex‎. ‎Moreover‎, ‎we introduce new definition o...

Locally Compact Path Spaces

It is shown that the space X [0,1], of continuous maps [0, 1] → X with the compact-open topology, is not locally compact for any space X having a nonconstant path of closed points. For a T1-space X, it follows that X [0,1] is locally compact if and only if X is locally compact and totally path-disconnected. AMS Classification: 54C35, 54E45, 55P35, 18B30, 18D15