1

The author wrote the following assertion which makes me panic for a while:

Let $T$ be a locally compact Hausdorff space and $K\subseteq T$ a compact set, and $E$ a Banach space. Suppose that $g:K\rightarrow E$ is continuous, then $g$ can be extended to a continuous function $f:T\rightarrow E$ with compact support such that $\|f\|\leq 2\|g\|$, where $\|\cdot\|$ denotes the uniform norm.

Note that $T$ may not be normal, so the classical Tietze Extension Theorem is not applicable. Plus, the target space is a Banach space, not $\mathbb{R}$, and so classical Tietze Extension Theorem fails again.

Actually I have asked the similar question before, but I do not completely understand the solution though.

But here the situation is slightly different than the old question, because I need only $\|f\|\leq 2\|g\|$, I do not require the same boundedness extension, so I suspect that one can do it in a more constructively way.

Any idea?

user284331
  • 55,591
  • 1
    The one-point compactification of $T$ will be a compact Hausdorff space and therefore normal. So the Tietze extension theorem would do the job if $E$ were $\mathbb R$ (or a finite-dimensional Banach space, since you could deal with one coordinate at a time). I don't immediately see how to handle infinite-dimensional Banach spaces; maybe sleep will help. – Andreas Blass Jun 25 '20 at 02:35

0 Answers0