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The following statement is stated in Nicolae Dinculeanu, Integration on Locally Compact Spaces without proof:

Suppose that $T$ is a locally compact Hausdorff space and $E$ is a Banach space. Given a compact set $K\subseteq T$ and a continuous function $f:K\rightarrow E$, $f$ can be extended to a continuous function $f_{1}:T\rightarrow E$ with compact support.

If $E$ is the real line, then it is classic, and the proof can be found in Gerald B. Folland, Real Analysis, but now $E$ is a Banach space, the situation is much more complicated.

@pre-kidney has provided a link addressing the issue. But I do not think that answers my question because $T$ is locally compact.

Perhaps someone can point me a standard reference for that statement.

user284331
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1 Answers1

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If in addition it is assumed that $T$ is Hausdorff, then this is proven by Bill Johnson in this mathoverflow thread. Further discussion and generalizations appear there as well.

pre-kidney
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  • That post discusses about a closed set $K$ instead of a compact set, I wonder if the compactness in my question makes the answer easier? And I imposed also the condition that such continuous extension $f_{1}$ must be compactly supported. – user284331 Apr 16 '17 at 07:41
  • If you read the answer posted by Bill Johnson, his approach is very simple and general and appears to carry over to these variations. – pre-kidney Apr 16 '17 at 19:50