Topological Vector Spaces

Theorem 1.4. Let X be a tvs and let F be a local base at 0. Then (i) U, V ∈ F ⇒ there exists W ∈ F such that W ⊆ U ∩ V . (ii) If U ∈ F , there exists V ∈ F such that V + V ⊆ U . (iii) If U ∈ F , there exists V ∈ F such that αV ⊆ U for all α ∈ K such that |α| ≤ 1. (iv) Any U ∈ J is absorbing, i.e. if x ∈ X, there exists δ > 0 such that ax ∈ U for all a such that |a| ≤ δ. Conversely, let X be a linear space and let F be a non-empty family of subsets of X which satisfy (i)–(iv), define a topology J by :