2 - Preliminaries

Summary

We begin with a brief summary of important facts from functional analysis – some with proofs, some without. Throughout, X, Y, and so on, are normed linear spaces over ℝ. If there is no danger of confusion, we will use ∥ · ∥ to denote the norm in any given normed space; if two or more spaces enter into the discussion, we will use ∥ · ∥_X, and so forth, to further identify the norm in question.

Continuous Linear Operators

Given a linear map T : X → Y, recall that the following are equivalent:

1. (i) T is continuous at 0 ∈ X.

2. (ii) T is continuous.

3. (iii) T is uniformly continuous.

4. (iv) T is Lipschitz; that is, there exists a constant C < ∞ such that ∥ T x – T y ∥_Y ≤ C∥ x – y ∥_X for all x, y ∈ X.

5. (v) T is bounded; that is, there exists a constant C < ∞ such that ∥T x∥_Y ≤ C∥x∥_X for all x ∈ X.

If a linear map T : X → Y is bounded, then there is, in fact, a smallest constant C satisfying ∥T x∥_Y ≤ C∥x∥_X for all x ∈ X. Indeed, the constant

called the norm of T, works; that is, it satisfies ∥T x∥_Y ≤ ∥T ∥ ∥x∥_X and it's the smallest constant to do so. Further, it's not hard to see that (2.1) actually defines a norm on the space B(X, Y) of all bounded, continuous, linear maps T : X → Y.