Banach Spaces

Abstract

Banach spaces form a special class of normed vector spaces. Compared with other normed spaces, Banach spaces have the advantage that it is easier to check that a sequence of vectors in the space is convergent. We give the formal definition of a Banach space in Section 3.1. As example of a Banach space we consider the set of continuous functions on a closed and bounded interval. Other important Banach spaces, the sequence spaces $\ell^p(\mathbb{N})$, are introduced in Section 3.2. In Section 3.3 we continue the analysis of bounded linear operators initiated in Section 2.4.