Normed Spaces

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 shape. Every normed space can be completed to a Banach space. Finally, series are defined, and various tests for their convergence are proved.