Banach spaces and Hilbert spaces are well-known examples of topological vector space. Is there any topological vector space which is not normal?
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@AnneBauval the discrete topology is incompatible with the vector space structure. It prevents scalar multiplication from being continuous. The cofinite topology has the same problem. – CyclotomicField Sep 15 '22 at 21:36
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Do you mean "which is not a normed vector space"? (Normal is something else.) Note that Hilbert spaces are particular cases of Banach spaces. Examples of t.v.s. which are not normed are locally convex t.v.s.. – Anne Bauval Sep 15 '22 at 21:41
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To expand on what @AnneBauval has said, are you looking for exotic examples of topological vector spaces or topological vector spaces with a certain property? – CyclotomicField Sep 15 '22 at 21:46
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@Anne Bauval I mean normal space not normed vector space. – murad.ozkoc Sep 17 '22 at 03:15
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1Does this answer your question? $\mathbb{R}^\mathbb{R}$ is not normal – Anne Bauval Sep 18 '22 at 22:19