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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.2012-10-15
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    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.2012-10-15

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