Functional Analysis

Bounded Operators on Hilbert Space

A guided tour through the theory of bounded linear operators on a Hilbert space — from the geometry of the underlying space to norms, adjoints, spectra, and compactness.

Diagram of a bounded operator T mapping the unit ball of a Hilbert space H to a bounded ellipse, contained in a disk of radius equal to the operator norm ||T||.
A bounded operator maps the unit ball to a set that stays inside a disk of radius $\|T\|$.

Let $H$ be a Hilbert space over $\mathbb{C}$ (or $\mathbb{R}$). The set $B(H)$ of bounded linear operators $T : H \to H$ is itself a rich mathematical object: a Banach space under the operator norm, a $C^*$-algebra under composition and adjoint, and the natural home for spectral theory. This site walks through the core ideas, in order, with definitions, theorems, and worked examples at each stage.

How to read this Each page builds on the last. If you already know the basics of Hilbert spaces, feel free to jump straight to Bounded Linear Operators.

Contents

01

Hilbert Spaces

Inner products, norms, completeness, orthogonality, and orthonormal bases.

 02

Bounded Linear Operators

Definition of boundedness, equivalence with continuity, and $B(H)$.

 03

The Operator Norm

Equivalent formulas for $\|T\|$, submultiplicativity, and completeness of $B(H)$.

 04

The Adjoint

Existence and uniqueness of $T^*$, its properties, and the $C^*$-identity.

 05

Special Classes

Self-adjoint, unitary, normal, and projection operators.

 06

Spectrum

Resolvent set, spectrum, spectral radius, and spectra of special operators.

 07

Compact Operators

Compact operators as limits of finite-rank operators, and their spectral theory.

 08

Worked Examples

Multiplication, shift, and integral operators, computed explicitly.