Properties of Linear and Nonlinear Operators

Abstract

In this section we point out some fundamental properties of linear operators in Banach spaces. The key assertions presented are the Uniform Boundedness Principle, the Banach-Steinhaus Theorem, the Open Mapping Theorem, the Hahn-Banach Theorem, the Separation Theorem, the Eberlain-Smulyan Theorem and the Banach Theorem. We recall that the collection of all continuous linear operators from a normed linear space X into a normed linear space Y is denoted by $\mathcal{L}\left( {X,{\mathbf{ }}Y} \right)$ , and $\mathcal{L}\left( {X,{\mathbf{ }}Y} \right)$ is a normed linear space with the norm

$$ ||A||_{\mathcal{L}\left( {X,{\mathbf{ }}Y} \right)} = \sup \left\{ {||Ax||_Y :||x||{\text{x}} \leqslant 1} \right\} $$

.