A representation theorem for isometries of C(X,E)

Ka-Sing Lau

Vol. 60 (1975), No. 1, 229–233

DOI: 10.2140/pjm.1975.60.229

Abstract

Let X, Y be compact Hausdorff spaces and let E, F be Banach spaces such that their duals are strictly convex. We show that a linear map T : C(X,E) → C(Y,F) is an isometric isomorphism if and only if there exists a homeomorphism ϕ : Y → X and a continuous map λ from Y to the set of isometric isomorphisms from E onto F (with the strong topology) such that Tf(y) = λ(y) ⋅ f(ϕ(y)) for all y ∈ Y , f ∈ C(X,E).

Mathematical Subject Classification 2000

Primary: 46E40

Secondary: 47B37

Milestones

Received: 22 August 1974

Published: 1 September 1975