ABSTRACT
So far, our study of linear algebra has focused mostly on finite-dimensional vector spaces and the linear maps between such spaces. In this chapter, we want to give the reader a small taste of the ideas that are needed to treat linear algebra in the infinite-dimensional setting. Infinite-dimensional vector spaces arise frequently in analysis, where one studies vector spaces of functions. Not surprisingly, to understand such spaces in detail, one needs not just the algebraic concepts of linear algebra but also some tools from analysis and topology. In particular, the notions of the length of a vector, the distance between two vectors, and the convergence of a sequence of vectors are key ingredients in the study of infinite-dimensional spaces.
