Advertisement

Properties of Linear and Nonlinear Operators

Part of the Birkhäuser Advanced Texts / Basler Lehrbücher book series (BAT)

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

Keywords

Hilbert Space Banach Space Linear Operator Compact Operator Nonlinear Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Birkhäuser Verlag AG 2007

Personalised recommendations