Locally Convex Topological Vector Spaces

Abstract

1. Completeness 1.1 Definition. Let X be a set. A collection F of subsets of X is called a filter if the following are satisfied: (1) F = ∅ and ∅ / ∈ F ; (2) if A ∈ F and B ∈ F then A ∩ B ∈ F ; (3) if A ∈ F and A ⊂ C ⊂ X then C ∈ F ; 1.2 Definition. Let X be an set. A collection F of subsets of X is called a filter base if the following are satisfied: (1) F = ∅ and ∅ / ∈ F ; (2) if A ∈ F and B ∈ F then there exists C ∈ F such that C ⊂ A ∩ B; Note that every filter base A is contained in a unique filter F with the property that each element of F contains an element of A. In fact, F is the set of all subsets of X which contain an element of A.