In mathematics, a family of sets is of finite character if for each , belongs to if and only if every finite subset of belongs to . That is,
A family of sets of finite character enjoys the following properties:
Let be a vector space, and let be the family of linearly independent subsets of . Then is a family of finite character (because a subset is linearly dependent if and only if has a finite subset which is linearly dependent). Therefore, in every vector space, there exists a maximal family of linearly independent elements. As a maximal family is a vector basis, every vector space has a (possibly infinite) vector basis.
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