Range preserving maps between the spaces of continuous functions with values in a locally convex space

Abstract

Let C ( X , E ) C(X,E) be the linear space of all continuous functions on a compact Hausdorff space X X with values in a locally convex space E E . We characterize maps T : C ( X , E ) → C ( Y , E ) T:C(X,E)\to C(Y,E) which satisfy R a n ( T F − T G ) ⊂ R a n ( F − G ) \mathrm {Ran}(TF-TG)\subset \mathrm {Ran}(F-G) for all F , G ∈ C ( X , E ) F,G\in C(X,E) .