Normed Spaces

Abstract

Following a review of vector spaces and Hausdorff’s maximality principle, this chapter introduces normed spaces: the combination of a vector space with a natural distance function that is translation invariant and scaling homogeneous. The most important examples are $\mathbb {C}^N$, the sequence spaces $\ell ^p$, and the function spaces $L^p(A)$. With respect to a norm, addition and scalar multiplication are continuous, and all balls have the same sha pe. Every normed space can be completed to a Banach space. Finally, series are defined, and various tests for their convergence are proved.