Functional Analysis

Year 4 · 5 sections · 14 concepts · 13 practice questions

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What's covered

free1 · Normed and Banach spaces2 concepts

Definition 1.1 · Normed and Banach spaces
Theorem 1.2 · Finite vs infinite dimensions (Riesz's lemma)

2 · Bounded operators and duals3 concepts

Theorem 2.1 · Bounded operators and the operator norm
Theorem 2.2 · Neumann series and invertibility
Definition 2.3 · Dual spaces

3 · Hilbert spaces4 concepts

Definition 3.1 · Hilbert spaces
Theorem 3.2 · Orthogonal projection onto closed subspaces
Theorem 3.3 · Riesz representation theorem
Theorem 3.4 · Orthonormal bases and Parseval

4 · The big theorems3 concepts

Theorem 4.1 · Hahn–Banach theorem
Theorem 4.2 · Uniform boundedness principle (Banach–Steinhaus)
Theorem 4.3 · Open mapping and closed graph theorems

5 · Compact operators and the spectral theorem2 concepts

Definition 5.1 · Compact operators and the spectrum
Theorem 5.2 · Spectral theorem for compact self-adjoint operators

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Theorem 1.2 · Finite vs infinite dimensions (Riesz's lemma)

Infinite dimensions are different: the closed unit ball is never compact. (In $\ell^2$ , the basis vectors $e_n$ are all $\sqrt2$ apart.) So compactness arguments need replacing, e.g. with compact operators or weak topologies.

Proof

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