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