Projections in Certain Continuous Function Spaces C(H) and Subspaces of C(H) Isomorphic with C(H)

Abstract

If A and B are sets then A — B = {x| x £A, x ∉ B}. This notation is also used if A and B are linear spaces. If X and Y are Banach spaces an embedding of X into Y is a continuous linear mapping u of X onto a closed subspace of F which is 1 — 1. In this case X is said to be embedded in Y. If |ux| = |x| for every x ∈ X (| … | stands for norm), then u embeds Xisometrically into Y. If u is onto then X and Y are isomorphic and if, in addition, |ux| = |x| for every x ∈ X, then X and Y are isometric. Then an embedding u has a continuous inverse u-1 (4, p. 36) defined on uX and this fact is used below without further reference. The conjugate space of X is denoted by X′. Unless otherwise noted, all topological spaces considered are Hausdorff spaces.