It is well known that if a normed space is finite-dimensional, then all norms are equivalent. Does the reverse implication hold as well? Thank you for your help!
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Hint: If $X$ is infinite dimensional there exists a discontinuous linear functional $f$ on it. Define $\|x\|'=\|x\|+|f(x)|$ to get a new norm which is not equivalent to the original norm.
Kavi Rama Murthy
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Can you explain why $f $ exists? – infinity Feb 26 '20 at 12:38
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@infinity Let $H$ be a Hamel basis and ${h_n)$ be distinct point in it. Without loss of generality assume that $|h_n|=1$ for all $n$. Let $f(h_n)=n$, ($n=1,2...$) and $f(h)=0$ for all other $h \in H$. Extend $f$ by linearity to all of $X$. Then $f$ is linear but not continuous. – Kavi Rama Murthy Feb 26 '20 at 23:15