Suppose I have a vector space $K$ which consists of real-valued sequences with only finitely many non-zero terms. I would like to show that there doesn't exist a norm on $K$ that would make it become a complete metric space. My technique is to use the Baire Category Theorem. However, I run into problems because of the generality of the theorem.
Would anyone be kind enough to offer me some tips? thank you!
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T. Eskin
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user145416
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2What difficulties are you facing? Baire's theorem is very good for this. Note that any finite-dimensional linear subspace of a normed space is closed and every proper linear subspace of a normed space has empty interior. – Daniel Fischer May 13 '14 at 20:07
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$K$ has a countable basis, and the rest is in this question. – Luiz Cordeiro May 14 '14 at 20:19