Functional Analysis

Prerequisite path Preparation & skip guidance · Measure & Integration · Linear Algebra

Studying Functional Analysis

Finite-dimensional linear algebra changes character when completeness, topology and boundedness become part of the problem. Melrose develops Banach and Hilbert spaces alongside the main operator theorems and compact spectral theory. Yosida takes operator semigroups toward evolution equations. The semigroup route answers a more specialized question and should follow, rather than stand in for, the initial normed-space and bounded-operator work.

What to work on

Prove completeness or exhibit an incomplete normed space.
Compute norms of elementary bounded operators and functionals.
Choose between uniform boundedness, open mapping, closed graph and Hahn-Banach arguments.
Use orthogonal projections, adjoints and compact-operator spectral theory.

A starting text

Functional Analysis: Lecture Notes for 18.102 — Richard Melrose. Chapter 1, sections 2-6 and 8-12 for Banach spaces and the main operator theorems. Then chapter 3, sections 8-11 for projections and adjoints and 14-18 for compactness, weak convergence and the compact self-adjoint spectral theorem.

Choosing what to read

Test continuity and boundedness on explicit operators, keeping their domains visible. Compare a complete sequence space with an incomplete normed subspace, and identify which proof fails without completeness. When applying an open-mapping or uniform-boundedness theorem, write the hypotheses before the conclusion. In Hilbert space, calculate a projection and its orthogonality relation. Save unbounded generators and evolution problems for the point where dense domains, convergence and operator composition can be handled precisely.

Functional Analysis: Lecture Notes for 18.102

Richard Melrose

Establish completeness, duality and compact-operator arguments before specialization.

How it reads: Course-length development of Banach and Hilbert spaces with operator theorems.
What it expects: Real analysis, measure theory and linear algebra.
What to keep in mind: The compact self-adjoint spectral theorem is not a spectral theorem for every bounded or unbounded operator.
Practice: Exercises provided; public hints and selected worked solutions verified within chapter 5. Coverage of every problem is not claimed. Chapter 5 Problems and solutions, starting at printed page 133; Problems - Chapter 1, Hints for some problems, and the subsequent Solutions to problems sections.

Complete public PDF.

Edition, reading range and source

Chapter 1, sections 2-6 and 8-12 for Banach spaces and the main operator theorems. Then chapter 3, sections 8-11 for projections and adjoints and 14-18 for compactness, weak convergence and the compact self-adjoint spectral theorem.

Source · Checked 2026-10-16

Lectures on Semi-Group Theory and its Application to Cauchy’s Problem in Partial Differential Equations

Kôsaku Yosida

Study how operator families encode time evolution after the main Banach-space tools.

How it reads: Specialized operator-semigroup route to the Cauchy problem.
What it expects: Functional analysis and differential equations.
What to keep in mind: Historical notation may need translation; this is an application branch rather than a first functional-analysis course.
Practice: Public worked solutions were not verified; reconstruct the semigroup property in a concrete evolution example as an editorial exercise.

Complete TIFR archive PDF; individual-use access only, with no redistribution.

Edition, reading range and source

Use the semigroup and Cauchy-problem developments indicated by the archive's contents; exact reading sections were not newly checked.

Source · Checked 2026-10-11

Practice

Functional-analysis problems, hints and selected solutions. Chapter 5 Problems and solutions, starting at printed page 133; Problems - Chapter 1, Hints for some problems, and the subsequent Solutions to problems sections.. Exercises provided; public hints and selected worked solutions verified within chapter 5. Coverage of every problem is not claimed.

Try these before moving on

Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.

On C[0,1] with the supremum norm compute the operator norm of L(f)=integral from 0 to 1 of t f(t) dt, proving both an upper bound and attainment.
A hint
Bound the absolute integral using the supremum norm. A constant function can attain the bound because the weight is nonnegative.
Work through the solution
Take $C[0,1]$ over either $\mathbb R$ or $\mathbb C$ . The integral defines a linear functional, and for every $f$ in this space,

$|L(f)|\le\int_0^1 t|f(t)|\,dt\le\|f\|_\infty\int_0^1t\,dt=\tfrac12\|f\|_\infty.$

Thus $L$ is bounded and $\|L\|\le1/2$ . For the constant function $f(t)=1$ , $\|f\|_\infty=1$ and $L(f)=1/2$ , proving $\|L\|\ge1/2$ . Consequently $\|L\|=1/2$ , and the supremum in the operator norm definition is attained. An upper bound alone would not determine the norm.

If this was the sticking point:
- Real Analysis — Review the supremum norm and integral inequalities.
- Linear Algebra — Review linear maps and how linear functionals fit that definition.

Already comfortable with this? Skip Banach-space review if completeness and boundedness proofs are routine. Skip the Hilbert-space selection only if you can complete the compact-operator check and distinguish spectrum from point spectrum; finite-dimensional eigenvalue calculations alone do not suffice.

Review the prerequisites · Compare the full subject collection

Where this can lead

Fourier & Harmonic Analysis — Use Lp estimates, Hilbert-space methods and operators in Fourier analysis.
Differential Equations & PDEs — Continue to weak solutions or evolution equations with the required measure and Sobolev preparation.

Reading-source dates

Functional Analysis: Lecture Notes for 18.102 · 2026-10-16

Reading choices revised 2026-10-17.

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Introduction to Functional Analysis (18.102)

PDF

Richard Melrose · Course notes · Intermediate

A course-length development of Banach spaces, Lebesgue integration, Hilbert spaces and applications. The author notes ongoing revisions; useful before more specialized graduate functional analysis.

Preparation: Proof-based real analysis; metric spaces; linear algebra.

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Additional reading

Search the complete catalogue.

Intermediate reading

Linear Functional Analysis

Bryan Patrick Rynne and Martin Alexander Youngson · Textbook · Intermediate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

An entry course in normed-space and operator methods. Suitable before advanced spectral theory or nonlinear analysis.

Preparation: Proof-based analysis and linear algebra.

Official page / access details

Edition and sources

Version: First edition, 2000 · Checked 2026-10-11

Graduate reading

Lectures on Semi-Group Theory and its Application to Cauchy’s Problem in Partial Differential Equations

Kôsaku Yosida · Long lecture notes · Graduate · English · PDF

Develops operator semigroups for evolution equations. Historical notation may differ from current treatments. Individual-use access only; do not redistribute.

Preparation: Functional analysis; differential equations.

Use: Connect abstract operators and PDE.

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Edition and sources

Version: TIFR volume 8; retypeset digital edition (see PDF imprint for publication year) · Checked 2026-10-11

Lectures on Some Fixed Point Theorems of Functional Analysis

F. F. Bonsall · Long lecture notes · Graduate · English · PDF

Develops fixed-point principles in functional-analytic settings. Historical notation may differ from current treatments. Individual-use access only; do not redistribute.

Preparation: Banach spaces; topology.

Use: Connect existence methods and analysis.

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Edition and sources

Version: TIFR volume 26; retypeset digital edition (see PDF imprint for publication year) · Checked 2026-10-11

Lectures on the Ekeland Variational Principle with Applications and Detours

Djairo G. de Figueiredo · Long lecture notes · Graduate · English · PDF

Develops variational techniques for existence and optimization. Historical notation may differ from current treatments. Individual-use access only; do not redistribute.

Preparation: Functional analysis; nonlinear analysis.

Use: Apply variational principles to nonlinear problems.

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Edition and sources

Version: TIFR volume 81; retypeset digital edition (see PDF imprint for publication year) · Checked 2026-10-11

Topics in Linear and Nonlinear Functional Analysis

Gerald Teschl · Book draft · Graduate · English · Full-text access unverified

No complete PDF is currently offered on the linked author page. The page says the text will return 24 months after publication; this is not a promise of current free access.

Treats Banach and Hilbert spaces, spectral theory, degree and nonlinear operators. The author currently says the full PDF is withdrawn while the book is being printed.

Preparation: Real analysis; measure theory; linear algebra.

Use: Plan graduate functional-analysis study; consult current publication status.

Author / publication-status page

Edition and sources

Version: Author page updated July 17, 2026; publication in preparation · Checked 2026-10-11

Mathematical Methods in Quantum Mechanics

Gerald Teschl · Textbook · Graduate · English · PDF

The AMS permits one stored copy for personal use; do not redistribute.

Self-adjoint operators, spectral theory and scattering are developed alongside Schrödinger operators.

Preparation: Real analysis; linear algebra; differential equations.

Use: Connect operator theory with mathematical physics.

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Edition and sources

Version: Second edition, 2014; authorized author PDF · Checked 2026-10-11

S. Kesavan · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Develops the functional-analytic toolkit with differential-equation and variational applications. Practice duality and weak-convergence arguments before applications.

Preparation: Real analysis; measure theory; normed spaces.

Official page / access details

Edition and sources

Version: Second edition, 2023 · Checked 2026-10-11

Functional Analysis: Fundamentals and Applications

Michel Willem · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Relates Lebesgue, dual and Sobolev spaces to capacity and elliptic problems. A functional route into variational PDE tools.

Preparation: Real analysis and integration.

Official page / access details

Edition and sources

Version: Second edition, 2022 · Checked 2026-10-11

Functional Analysis

Erdoğan S. Şuhubi · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A long treatment of infinite-dimensional spaces and operators. Use as a supporting reference when a shorter course leaves gaps.

Preparation: Real analysis; metric spaces; linear algebra.

Official page / access details

Edition and sources

Version: 2003 · Checked 2026-10-11

Lectures on Optimisation: Theory and Algorithms

Jean Céa · Long lecture notes · Graduate · English · PDF

Complete institutional PDF, authorized for individual use only. Copyright remains with TIFR; retypesetting may retain errors. This is not a redistribution licence.

Connects variational analysis and optimization with algorithms. This is an applied branch of functional analysis, not a general Banach-space textbook.

Preparation: Functional analysis; variational methods.

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Edition and sources

Version: TIFR Lectures on Mathematics 53; authorized retypeset historical edition — original date on title page · Checked 2026-10-11

Function Spaces

Mikko Salo · Lecture notes · Graduate · English · PDF

The complete 67-page manuscript opened at users.jyu.fi.

A self-contained course segment on interpolation, fractional Sobolev regularity, and Besov and Triebel–Lizorkin spaces.

Preparation: Measure theory; Fourier transform; basic Banach spaces.

Use: Bridge standard Sobolev spaces to refined regularity tools.

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Edition and sources

Version: Fall 2008 course notes · Checked 2026-10-13

Functional Analysis and Operator Algebras: An Introduction

John M. Erdman · Textbook · Graduate · English · PDF

The complete 230-page text opened at web.pdx.edu.

Exercises connect Banach and Hilbert spaces, bounded operators and spectral questions with operator algebras.

Preparation: Real analysis, linear algebra and metric-space topology.

Use: Study functional analysis through problems before progressing to operator algebras.

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Edition and sources

Version: October 4, 2015 · Checked 2026-10-13

Research reading

Lectures on Numerical Methods for Non-Linear Variational Problems

Roland Glowinski · Long lecture notes · Research · English · PDF

Complete institutional PDF, authorized for individual use only. Copyright remains with TIFR; retypesetting may retain errors. This is not a redistribution licence.

Develops variational formulations and computational methods for nonlinear problems. An applied specialization requiring analytic foundations.

Preparation: Functional analysis; variational PDE; numerical analysis.

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Edition and sources

Version: TIFR Lectures on Mathematics 65; authorized retypeset historical edition — original date on title page · Checked 2026-10-11

Lectures on Analytic Geometry

Peter Scholze · Long lecture notes · Research · English · PDF

Free PDF at the author or authorized archive.

Develops liquid vector spaces and analytic rings. The preface identifies its analytic-space definition as a historical stepping stone.

Preparation: Condensed mathematics; derived algebra; functional analysis.

Use: Study liquid techniques and compare with later analytic stacks.

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Full-text source checked: 2026-11-03

Found via — References: Lectures on Analytic Geometry

Edition and sources

Version: May 2026 revision of the winter 2019/20 course · Checked 2026-10-13