Does there exist a compact normed linear space?
All I can see is that if there is any it must be finite-dimensional. But does there exist any other than the trivial one i.e. $(0)$?
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Is the real line or the complex line compact? – kimchi lover Feb 16 '20 at 18:12
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No @kimchilover.............. – Jave Feb 16 '20 at 18:13
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1There you are, then: if there is a non-zero element of your space, it contains a copy of the scalar multiples of that element. – kimchi lover Feb 16 '20 at 18:14
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See here: https://math.stackexchange.com/questions/941897/examples-of-compact-sets-that-are-infinite-dimensional-and-not-bounded – Math1000 Feb 16 '20 at 20:36