---
title: Functional Analysis
description: Functional analysis studies vector spaces endowed with limit-related structures, focusing on infinite-dimensional spaces and operators acting on them. Learners will understand Banach and Hilbert spaces, linear operators, and their applications to differential equations and quantum mechanics.
category: mathematics
subcategory: real-analysis
difficulty: beginner, intermediate, advanced
url: /subject/functional-analysis
---

# Functional Analysis

Functional analysis studies vector spaces endowed with limit-related structures, focusing on infinite-dimensional spaces and operators acting on them. Learners will understand Banach and Hilbert spaces, linear operators, and their applications to differential equations and quantum mechanics.

## Available Resources

2 Books • 2 Courses • 3 Websites

## Courses

### 1. Geometric Anatomy of Theoretical Physics (Lectures on Functional Analysis)

Learn functional analysis through a geometric lens with Frederic Schuller's lectures! Explore key concepts in theoretical physics.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic

**Tags:** mathematical-physics, differential-geometry, topology, lie-groups, fibre-bundles, lecture-series

### 2. Differential Analysis (MIT 18.155)

**Author:** Richard Melrose

First semester of graduate differential analysis: fundamental solutions for elliptic, hyperbolic and parabolic operators, the method of characteristics, Lebesgue integration, distributions, homogeneous distributions, the Fourier transform and asymptotic methods. Provides 18 lecture note files and problem sets with solutions, preparing readers for modern PDE theory.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/

**Tags:** distribution-theory, partial-differential-equations, fourier-transform, lebesgue-integration, fundamental-solutions

## Websites

### 1. Terence Tao's Analysis Notes

**Author:** Terence Tao

Graduate lecture notes from Terence Tao's UCLA course on harmonic and Fourier analysis, covering maximal inequalities, Calderon-Zygmund and Littlewood-Paley theory, Sobolev spaces and pseudodifferential operators, with homework and a take-home final for practice.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.math.ucla.edu/~tao/247a.1.06f/

**Tags:** harmonic-analysis, fourier-analysis, calderon-zygmund-theory, littlewood-paley-theory, sobolev-spaces

### 2. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

### 3. terrytao.wordpress.com

Terry Tao's blog is a personal mathematics resource with accessible expositions, lecture notes, and problem discussions spanning measure theory and other analysis topics, alongside teaching materials and occasional research updates.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://terrytao.wordpress.com

**Tags:** websites, mathematics-statistics, real-analysis

## Books

### 1. Introductory Functional Analysis with Applications

**Author:** Erwin Kreyszig

Applied-oriented introduction to functional analysis assuming only linear algebra and basic calculus. Covers metric, normed, Banach, and Hilbert spaces, the fundamental theorems, spectral theory, and unbounded operators, with applications to approximation theory and quantum mechanics and many worked problems.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/8126511915?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, real-analysis

### 2. A Course in Functional Analysis

**Author:** John B. Conway

Graduate Texts in Mathematics 96, a standard first-year graduate text covering Hilbert and Banach spaces, locally convex spaces, weak topologies, Banach and C*-algebras, and spectral theory for normal and unbounded operators. Assumes measure theory and real analysis background.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0387972455?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, real-analysis

---

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