Topological Vector Spaces

The topological vector space V (over a topological field K) is locally convex iff each element of some neighborhood base at O, is convex. A subset B of V is bounded iff for every neighborhood U of O and some positive t. When some open subset of V is bounded, V is locally bounded. A subset S of V is circled iff and for every complex number z such that ||z| ≤ 1 and every x in S. For a circled bounded neighborhood of U of O, the associated Minkowski functional M _U is: $$ M_U :V \mathrel\backepsilon {\text{x}} \mapsto \inf \{ \alpha :\alpha \geqslant 0,{\text{x}} \in \alpha U\} $$. A map p : V ∋ x ↦ p(x) ∈ [0,∞) is a quasinorm iff for some k in [1,∞): a) p(x + y) ≤ k (p(x) + p(y)); b) p|t|x) When k = 1, p is a seminorm. A seminorm p is a norm iff {p(x) = 0} ⇔ {x = O}. A subset R of V is radial or absorbing iff for every finite subset F of V and some real number r(F), λR:⊃ F if |λ| ≥ r(F).