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Let $X$ be a Normed Vector Space. For any $x\in X$ and $r>0$, let $W:=\{y∈X:∥y−x∥≤r\}$. Prove: $W$ is closed and if $\dim(X)<\infty$ $W$ is compact.

I have no problems show that it is closed, but do not know how to show it is compact. Any suggestions?

Michael Hardy
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Peter G
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1 Answers1

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Recall that, for finite dimensional spaces, compact sets are the bounded and closed ones. Therefore, after proving that $ W $ is closed, it is enough to check that it is bounded.

ACV
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