absolutely convex subset



A subset BB of a vector space VV over \mathbb{R} or \mathbb{C} is absolutely convex if λx+μyB\lambda x + \mu y \in B whenever x,yBx, y \in B and |λ|+|μ|1{|\lambda|} + {|\mu|} \le 1.


Absolutely convex subsets are closely related to semi-norms. Given a vector space VV and an absolutely convex subset BVB \subseteq V, we define V BV_B to be the linear span of BB in VV. Then let μ B:V B\mu_B \colon V_B \to \mathbb{R} be the Minkowski functional of BB. That is, μ B\mu_B is given by:

μ B(v)=inf{t>0:tvB} \mu_B(v) = \inf\{ t \gt 0 : t v \in B\}

Then μ B\mu_B is a semi-norm on V BV_B and E BV B/kerμ BE_B \coloneqq V_B/\ker \mu_B is a normed space.

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