Recurrence in Topological Dynamics pp 101-131 | Cite as

# Compactifications

Chapter

## Abstract

For a real Banach space (written
For example
Of course by linearity \(\left| {T\left( x \right)} \right| \leqslant \left\| T \right\|\left| x \right|\) for all x ∈ .

*B*space hereafter)*E*we let*B*(*E*) denote the unit ball:$$B\left( E \right) = \left\{ {x \in E:\left| x \right| \leqslant 1} \right\}$$

(5.1)

*B*(*) is the closed interval [−1, 1]. For a bounded linear operator***R***T*:*E*_{1}→*E*_{2}between*B*spaces, the operator norm of*T*can be described as:$$\left\| T \right\| = {\sup _{x \in B\left( {{E_1}} \right)}}\left| {T\left( x \right)} \right|$$

(5.2)

*E*_{l}. The set*L*(*E*_{1},*E*_{2}) of all such bounded linear operators is a*B*space with the operator norm, and its unit ball is the set of operators of norm at most 1. Equivalently:$$B\left( {L\left( {{E_1},{E_2}} \right)} \right) = \left\{ {T \in L\left( {{E_1},{E_2}} \right):T\left( {B\left( {{E_1}} \right)} \right) \subset B\left( {{E_2}} \right)} \right\}$$

(5.3)

## Keywords

Closed Subset Open Family Uniform Space Open Filter Invariant Subset
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.

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## Copyright information

© Springer Science+Business Media New York 1997