Linear Operators

Overview¶

Once we have Banach spaces, the next step is to study maps between them. A bounded linear operator $A : X \to Y$ satisfies $\|Ax\|_Y \leq M\|x\|_X$, and the space $\mathcal{L}(X,Y)$ of all such operators is itself a Banach space.

The deep results of this chapter all rely on the Baire Category Theorem—a topological result about complete metric spaces. From it we derive:

• Banach–Steinhaus (Uniform Boundedness): A pointwise bounded family of operators is uniformly bounded.

• Open Mapping Theorem: A surjective bounded operator between Banach spaces is an open map.

• Closed Graph Theorem: An operator with a closed graph is automatically bounded.

We also study compact operators, which map bounded sets to precompact sets. These operators behave much like matrices in finite dimensions, and their spectral theory is the gateway to Fredholm theory and applications in PDEs.

What You Will Learn¶

• Bounded vs. continuous operators and the operator norm.

• The Baire Category Theorem and its consequences.

• Banach–Steinhaus, Open Mapping, and Closed Graph Theorems.

• Compact operators and their spectral properties.