Extending pointwise bounded equicontinuous collections of functions

Abstract

We prove that for a subspace $A$ of a space $X$, the following statements are equivalent: (1) for any Fréchet space $Y$, every pointwise bounded equicontinuous subset of $C(A, Y)$ can be extended to a pointwise bounded equicontinuous subset of $C(X, Y)$; (2) every pointwise bounded equicontinuous subset of $C(A)$ can be extended to a pointwise bounded equicontinuous subset of $C(X)$; (3) for any Fréchet space $Y$, every function $f\in C(A, Y)$ can be extended to a function $g\in C(X, Y)$. This theorem and other results obtained in this paper generalize several known theorems due to Flood, Frantz and Heath-Lutzer-Zenor, etc.