Some generalizations of Schauder’s theorem in locally convex spaces
Volker Wróbel · Pacific Journal of Mathematics · 1975
Let E 19 E 21 E z , Eι be four locally convex Hausdorff spaces (l.c.s.); denote by -£^(2?*,E k ) the set of all continuous linear operators from E t into E k with the topology of uniform convergence on bounded subsets of E t .Given two linear operators fe£f(E u E 2 ) and ge£r(E t ,EJ 9 consider the generalized adjoint operator Horn (/, g): ^b{E 2J E z -> & t (E l9 E 4 ) defined by u -> Horn (/, g)u = g ° u of.This paper deals with transformation properties of Horn (/, g) and their interactions with those of / and g.This purpose may be illustrated by a result due to K. Vala which generalizes Schauder's wellknown theorem concerning precompact operators and their adjoints on normed spaces: Let all spaces under consideration be normed, let / and g both be nonzero.Then Horn (/, g) is a precompact operator if and only if / and g are precompact operators.In the present paper bounded and precompact operators on l.c.s. are investigated.