BASES FOR INFINITE DIMENSIONAL VECTOR SPACES MATH 513 LINEAR ALGEBRA SUPPLEMENT

We have proven that every finitely generated vector space has a basis. But what about vector spaces that are not finitely generated, such as the space of all continuous real valued functions on the interval [0, 1]? Does such a vector space have a basis? By definition, a basis for a vector space V is a linearly independent set which generates V. But we must be careful what we mean by linear combinations from an infinite set of vectors. The definition of a vector space gives us a rule for adding two vectors, but not for adding together infinitely many vectors. By successive additions, such as (v1 + v2) + v3, it makes sense to add any finite set of vectors, but in general, there is no way to ascribe meaning to an infinite sum of vectors in a vector space. Therefore, when we say that a vector space V is generated by or spanned by an infinite set of vectors {v1, v2,...}, we mean that each vector v in V is a finite linear combination λi1vi1 + · · · + λinvin of the vi’s. Likewise, an infinite set of vectors {v1, v2,...} is said to be linearly independent if the only finite linear combinatio