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We know that if the closed unit ball in this space is compact then X is of finite dimension. If I can prove that a closed ball of radius $r>0$ in X is compact, can it also be affirmed that X is of finite dimension? If so, can you tell me in which book I can find this result? Thanks in advance.

Juan Figo Math
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  • Yes this is Riesz theorem. A normed space is locally compact iff it's of finite dimension. – Jakobian Jun 27 '23 at 01:37
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    You should be able to find this in almost any book about functional analysis. – Jakobian Jun 27 '23 at 01:38
  • In the functional analysis books I read they talk about the closed unit ball but not about a closed ball with radius r where r is not 1. So I wanted to know about a specific book. – Juan Figo Math Jun 27 '23 at 12:13
  • Those are homeomorphic. Multiplication by a non-zero real number is a homeomorphism in any normed linear space. – Jakobian Jun 27 '23 at 13:58

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