# Compactifications

• Ethan Akin
Chapter
Part of the The University Series in Mathematics book series (USMA)

## Abstract

For a real Banach space (written 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)
For example B(R) is the closed interval [−1, 1]. For a bounded linear operator T: E 1E 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)
Of course by linearity $$\left| {T\left( x \right)} \right| \leqslant \left\| T \right\|\left| x \right|$$ for all x ∈ 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.

## Authors and Affiliations

• Ethan Akin
• 1
1. 1.The City CollegeNew YorkUSA