Extensions of best approximation and coincidence theorems

LetXbe a Hausdorff compact space,Ea topological vector space on whichE*separates points,F:X→2Ean upper semicontinuous multifunction with compact acyclic values, andg:X→Ea continuous function such thatg(X)is convex andg−1(y)is acyclic for eachy∈g(X). Then either (1) there exists anx0∈Xsuch thatgx0∈Fx0or (2) there exist an(x0,z0)on the graph ofFand a continuous seminormponEsuch that0<p(gx0−z0)≤p(y−z0) for all y∈g(X). A generalization of this result and its application to coincidence theorems are obtained. Our aim in this paper is to unify and improve almost fifty known theorems of others.

Some remarks about Mackey convergence

In this paper, we examine Mackey convergence with respect toK-convergence and bornological (Hausdorff locally convex) spaces. In particular, we prove that: Mackey convergence and local completeness imply propertyK; there are spaces havingK- convergent sequences that are not Mackey convergent; there exists a space satisfying the Mackey convergence condition, is barrelled, but is not bornological; and if a space satisfies the biackey convergence condition and every sequentially continuous seminorm is continuous, then the space is bornological.

A variant of a fixed point theorem of Browder-Fan and Reich

LetSbe a convex, weakly compact subset of a locally convex Hausdorff space(E,τ)andf:S→Ebe a continuous multifunction from its weak topologyωtoτ. letρbe a continuous seminorm on(E,τ)and for subsetsA,BofEletp(A,B)=inf{p(x−y):x ϵ A, y ϵ B}. In this paper, sufficient conditions are developed for the existence of anx ϵ Ssatisfyingp(x,fx)=p(fx,S). The result is then used to prove several fixed point theorems.

The spectral radius in a locally convex algebra

In (1), Allan introduced the concept of the spectrum of an element of a locally convex algebra, and developed a spectral theory for pseudo-complete algebras. In a commutative Banach algebra the spectral radius is a continuous seminorm, and so it is natural to investigate continuity properties of the spectral radius in various classes of locally convex algebra. Continuity is too strong a condition to be expected in any general case, and an interesting property to investigate appears to be lower semi-continuity. We shall show easily that in a commutative pseudo-complete locally m-convex algebra (in the sense of (4)) the spectral radius is lower semi-continuous. We shall then exhibit a commutative complete metrizable algebra in which lower semi-continuity fails to hold.

(LF)-spaces with absolute bases

Suppose E is a locally convex space over a field K which can be the real line or the complex plane. Then a basis for E is a sequence (xk) of elements of E such that, if x ∈ E, x can be expressed uniquely aswhere ξk ∈K for each k. If this representation converges absolutely, i.e. iffor every continuous seminorm p on E, then (xk) is called an absolute basis for E. If the mappings x → ξk from E into K are continuous for each k, then (xk) is a Schauder basis for E. The purpose of this paper is to prove some results for (LF)-spaces with bases and to use them to extend some theorems due to Pietsch. We recall that an (F)-space is a complete metrizable locally convex space and an (LF)-space the inductive limit of a strictly increasing sequence of (F)-spaces (En, τn) such that τn+1|En = τn for all n.