Every vector space V has a finite set S which both spans V and is linearly independent in V.

Question

Quizplus · Accepted Answer

This statement is false. While every finite-dimensional vector space does have a finite spanning set that is also linearly independent (by the finite basis theorem), this is not necessarily true for infinite-dimensional vector spaces. In fact, there are many examples of infinite-dimensional vector spaces which do not have a finite spanning set that is also linearly independent, such as the space of all polynomials or the space of all continuous functions.

Every Vector Space V Has a Finite Set S Which