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Let $X$ be a Banach space and $K$ a compact subset of $X$ and consider for a given $\eta>0$ the closed ball $C(0,\eta)$ centered at $0$ of radius $\eta$.

How can I show that $K+C(0,\eta)=\{x+y: x\in K~\mbox{and }y\in C(0,\eta)~\}$ is not (sequentially) compact (except when $X$ has finite dimension or $K$ is empty)?

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Your title doesn't seem to relate to your question. –  Chris Eagle Oct 15 '12 at 15:30
    
You are asking if the sum of a compact set and a closed ball is compact in a Banach space. The closed balls need not be compact. –  leo Oct 15 '12 at 15:39
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If $K$ is empty, the set is empty, and if it's not empty then take $x_0\in K$: we get $K+C(0,\eta)\supset x_0+C(0,\eta)=C(x_0,\eta)$. If it was compact, so would be the closed unit ball.

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If $K$ is empty, then the sum is empty, hence compact. –  Chris Eagle Oct 15 '12 at 15:29
    
@ChrisEagle Right. I should have note that. –  Davide Giraudo Oct 15 '12 at 15:31
    
...and the closed unit ball is compact iff.. –  leo Oct 15 '12 at 15:32
    
@kevin This says also how to construct a counter example: take $X$ a space where the closed unit ball is not compact and consider $K$ a compact subset of $X$ –  leo Oct 15 '12 at 15:38
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