K-1 - Banach Spaces and Topology (I)

Abstract

Banach spaces were defined by S. Banach and others. The novel idea of Banach is to combine point–set topological ideas with the linear theory to obtain powerful theorems such as Banach–Steinhaus Theorem, Open-Mapping Theorem, and Closed Graph Theorem. Both general topology and the theory of Banach spaces continue to benefit from the cross-fertilization of analysis and topology, some of which are highlighted in this chapter. According to the definition of norm (║║) and norm space, a Banach space is a complete normed space. If Y is a linear subspace of X, where (X, ║║is a normed space, then Y is normed with the restriction of ║║to Y. If X is a Banach space and if Y is a closed linear subspace of X, then Y is a Banach space. A linear map between normed spaces is continuous if it is continuous at a point, and in this case, the map is a Lipschitz map. A scalar-valued liner map ƒ on a normed space is called a linear functional on X. The linear functional ƒ is continuous if and only if (iff) the null space ƒ−1 (0) is closed.