Copies of the class notes are on the internet in PDF format as given below. These notes have not bee classroom tested and may contain typographical errors.

Preface and Introduction.

Chapter I. Preliminaries.

• Section I.1. Sets and Functions.
• Section I.2. Metric and Normed Linear Spaces.
• Section I.3. The Lebesgue Integral.
• Section I.4. Abstract Measure Theory.
• Section I.5. Two Convergence Arguments.
• Section I.6. Equicontinuity.
• Study Guide 1.

Chapter II. Hilbert Spaces.

• Section II.1. The Geometry of Hilbert Space.
• Section II.2. The Reisz Lemma.
• Section II.3. Orthonormal Bases.
• Section II.4. Tensor Products of Hilbert Spaces.
• Section II.5. Ergodic Theory: An Introduction.
• Study Guide 2.

Chapter III. Banach Spaces.

• Section III.1. Definitions and Examples.
• Section III.2. Duals and Double Duals.
• Section III.3. The Hahn-Banach Theorem.
• Section III.4. Operations on Banach Spaces.
• Section III.5. The Baire Category Theorem and Its Consequences.
• Study Guide 3.

Chapter IV. Topological Spaces.

• Section IV.1. General Notions.
• Section IV.2. Nets and Convergence.
• Section IV.3. Compactness.
• Section IV.4. Measure Theory on Campact Sapces.
• Section IV.5. Weak Topologies on Banach Spaces.
• Study Guide 4.

Chapter V. Locally Convex Spaces.

• Section V.1. General Properties.
• Section V.2. Frechet Space.
• Section V.3. Functions of Rapid Decrease and the Tempered Distributions.
• Section V.4. Inductive Limits: Generalized Functions and Weak Solutions of Partial Differential Equations.
• Section V.5. Fixed Point Theorems.
• Section V.6. Applications of Fixed Point Theorems.
• Section V.7. Topologies on Locally Convex Spaces: Duality Theory and the Strong Dual Topology.
• Study Guide 5.

Chapter VI. Bounded Operators.

• Section VI.1. Topologies on Bounded Operators.
• Section VI.2. Adjoints.
• Section VI.3. The Spectrum.
• Section VI.4. Positive Operators and the Polar Decomposition.
• Section VI.5. Compact Operators.
• Section VI.6. The Trace Class and Hilbert-Schmidt Ideals.
• Study Guide 6.

Additional Chapter.

• Chapter VII. The Spectral Theorem.
• Chapter VIII. Unbounded Operators.
• Chapter IX. The Fourier Transform.

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