---
title: Sobolev Spaces
description: Sobolev spaces are vector spaces of functions equipped with norms involving both the functions and their weak derivatives. Learners will understand how to apply these spaces to establish the existence and regularity of solutions to partial differential equations.
category: mathematics
subcategory: functional-analysis
difficulty: beginner, intermediate, advanced
url: /subject/sobolev-spaces
---

# Sobolev Spaces

Sobolev spaces are vector spaces of functions equipped with norms involving both the functions and their weak derivatives. Learners will understand how to apply these spaces to establish the existence and regularity of solutions to partial differential equations.

## Available Resources

2 Books • 3 Websites

## Websites

### 1. Terry Tao's Notes on Sobolev Spaces

Terence Tao's blog posts and UCLA graduate lecture notes tagged Sobolev spaces, covering Sobolev embedding, weak derivatives, fractional Sobolev norms and their role in harmonic analysis. Readers learn the estimates and heuristics behind modern nonlinear PDE arguments.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://terrytao.wordpress.com/tag/sobolev-spaces/

**Tags:** sobolev-spaces, harmonic-analysis, partial-differential-equations, lecture-notes, functional-analysis

### 2. nLab: Sobolev Space

nLab wiki entry defining Sobolev spaces as spaces of weakly differentiable functions with integrability conditions, with pointers to the standard literature. A concise definitional reference for readers who already know measure theory and basic functional analysis.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/Sobolev+space

**Tags:** sobolev-spaces, functional-analysis, weak-derivatives, function-spaces

### 3. 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

## Books

### 1. Functional Analysis, Sobolev Spaces and Partial Differential Equations

**Author:** Haim Brezis

Graduate textbook developing functional analysis (Hahn–Banach, weak topologies, Lp spaces, compact operators, spectral theory) and applying it to Sobolev spaces and the variational theory of elliptic, parabolic and hyperbolic PDEs. Includes extensive exercises, many with solutions.

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

**Link:** https://www.amazon.com/dp/0387709134?tag=edmonddante07-20

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

### 2. Sobolev Spaces

**Author:** Robert A. Adams, John J. F. Fournier

Standard reference monograph on Sobolev spaces, covering weak derivatives, embedding and compactness theorems, interpolation, extension operators, fractional-order spaces and Orlicz–Sobolev spaces. Readers gain the function-space toolkit used in elliptic PDE and the calculus of variations.

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

**Link:** https://www.amazon.com/dp/0123958415?tag=edmonddante07-20

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

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