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Let $X$ be a real or complex topological vector space and let $A\subseteq X$ be absorbing, balanced, and convex. Then the Minkowski functional $\mu_A$ is a seminorm. Can we add some conditions that make $\mu_A$ a norm? This post gives such a condition, but it assumes $X$ is already a normed space. I wonder whether we can formulate more general conditions in which $X$ is only a TVS.

WillG
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    $\mu_A$ is a norm if and only if $A$ contains no vector subspace other than ${0}$. See https://www.wikiwand.com/en/Minkowski_functional – Kavi Rama Murthy Aug 22 '23 at 04:42
  • @geetha290krm If $X$ is a complex vector space and $A$ contains only a real nontrivial subspace, does the theorem apply? – WillG Aug 23 '23 at 11:19
  • Actually I just realized the statement on Wikipedia assumes $A$ is balanced. Thus, if $A$ contains a real subspace, it contains a complex one. – WillG Aug 23 '23 at 11:39

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