Let $X$ be a locally convex topological vector space, and suppose $C$ is a balanced, open, convex nbhd of $0$. I want to show that the Minkowski functional $\mu_C(x)=\inf(\lbrace t>0 \mid t^{-1}x \in C \rbrace)$ is a semi-norm, that is $\mu_C(tx)=\vert t \vert p_C(x)$ for all $x \in X$ and $t \in \Phi$ being either the complex or real numbers. I suppose I can use the fact that $C$ can be blown up to include any $x$ in $X$, but then it might no longer be balanced?
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Using definitions, homothety preserves the property of $C $ to be balanced. – Jean Marie Mar 21 '16 at 08:43
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So for any scaling $tC$ of $C$ by an arbitrary scalar $t \in \Phi$ as above, $tC$ is still balanced and convex? – Bartuc Mar 21 '16 at 12:13
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Does this answer your question? Norm induced by convex, open, symmetric, bounded set in $\Bbb R^n$. – ViktorStein Jun 06 '20 at 14:28