Finite character

In mathematics, a family F {displaystyle {mathcal {F}}} of sets is of finite character provided it has the following properties: In mathematics, a family F {displaystyle {mathcal {F}}} of sets is of finite character provided it has the following properties: A family F {displaystyle {mathcal {F}}} of sets of finite character enjoys the following properties: Let V be a vector space, and let F be the family of linearly independent subsets of V. Then F is a family of finite character (because a subset X ⊆ V is linearly dependent iff X 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. This article incorporates material from finite character on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.