Elements of Functional Analysis pp 187-212 | Cite as

# Spectra

Chapter

## Abstract

We fix here a Banach space

*E*over K ℝ or ℂ, and we wish to study the (noncommutative) Banach algebra*L(E)*of continuous linear maps from*E*to*E*, the product operation being composition. We use the same notation ∥·∥ for the norm on*E*and the associated norm on*L(E)*, and we denote by*I*the identity map on*E*. Thus,*I*is the unity of the algebra*L(E)*. An element*T*∈*L(E)*is called**invertible**if it has an inverse in*L(E)*; that is, if there exists a continuous linear map*S*such that*TS = ST = I*. Because composition is associative,*T*has an inverse in*L(E)*if and only if it has a right inverse (an element*U*such that*TU = I*) and a left inverse (an element V such that*VT = I*) in*L(E)*. Clearly, if*T*is invertible, it is bijective and its inverse in*L(E)*is unique and equals the inverse map*T*^{-1}Thus, for*T*∈*L(E)*, the following properties are equivalent:-
*T*is invertible. -
*T*is bijective and*T*^{-1}is continuous. -
ker

*T*= {0}, im*T*=*E*, and*T*^{-1}is continuous.

### Keywords

Radon Toll Prool## Preview

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## Copyright information

© Springer Science+Business Media New York 1999