# List of functional analysis topics

This is a list of functional analysis topics.
See also: Glossary of functional analysis.

## Hilbert space

* Bra–ket notation
* Definite bilinear form
* Direct integral
* Euclidean space
* Fundamental theorem of Hilbert spaces
* Gram–Schmidt process
* Hellinger–Toeplitz theorem
* Hilbert space
* Inner product space
* Legendre polynomials
* Matrices
* Mercer's theorem
* Min-max theorem
* Normal vector
* Orthonormal basis
* Orthogonal complement
* Orthogonalization
* Parallelogram law
** Normal matrix, normal operator
** Orthogonal matrix
** Unitary matrix
* Semi-Hilbert space
** Diagonal matrix
** Eigenvector, eigenvalue, eigenfunction
** Hermitian operator self-adjoint operator, Hermitian adjoint
** Hilbert matrix
** Shift operator
** Symmetric matrix
* Parseval's identity
* Rayleigh quotient
* Reproducing kernel Hilbert space
* Riesz representation theorem
* Rigged Hilbert space
* Spectral theorem, Spectral theory
* Trace class

## Functional analysis, classic results

* Normed vector space
* Unit ball
* Banach space
* Hahn–Banach theorem
* Dual space
* Predual
* Weak topology
* Reflexive space
* Polynomially reflexive space
* Baire category theorem
* Open mapping theorem (functional analysis)
* Closed graph theorem
* Uniform boundedness principle
* Arzelà–Ascoli theorem
* Banach–Alaoglu theorem
* Measure of non-compactness
* Banach–Mazur theorem

## Operator theory

* Bounded linear operator
** Continuous linear extension
** Compact operator
** Approximation property
** Invariant subspace
* Spectral theory
** Spectrum of an operator
** Essential spectrum
** Spectral density
* Topologies on the set of operators on a Hilbert space
** norm topology
** ultrastrong topology
** strong operator topology
** weak operator topology
** weak-star operator topology
** ultraweak topology
* Singular value (or S-number)
* Fredholm operator
* Fuglede's theorem
* Compression (functional analysis)
* Friedrichs extension
* Stone's theorem on one-parameter unitary groups
* Stone–von Neumann theorem
* Functional calculus
** Continuous functional calculus
** Borel functional calculus
* Hilbert–Pólya conjecture

## Banach space examples

* Lp space
* Hardy space
* Sobolev space
* Tsirelson space
* ba space

## Real and complex algebras

* Uniform norm
* Matrix norm
* Spectral radius
* Normed division algebra
* Stone–Weierstrass theorem
* Banach algebra
* *-algebra
* B*-algebra
* C*-algebra
** Universal C*-algebra
** Spectrum of a C*-algebra
* Positive element
* Positive linear functional
* operator algebra
** nest algebra
** reflexive operator algebra
** Calkin algebra
* Gelfand representation
* Gelfand–Naimark theorem
* Gelfand–Naimark–Segal construction
* Von Neumann algebra
** Abelian von Neumann algebra
* von Neumann double commutant theorem
** Commutant, bicommutant
* Topological ring
* Noncommutative geometry
* Disk algebra
* Colombeau algebra

## Topological vector spaces

* Barrelled space
* Bornological space
* Bourbaki–Alaoglu theorem
* Dual pair
* F-space
* Fréchet space
* Krein–Milman theorem
* Locally convex topological vector space
* Mackey topology
* Mackey–Arens theorem
* Montel space
* Polar set
* Polar topology
* Seminorm

## Amenability

* Amenable group
* Von Neumann conjecture

## Wavelets

* Basis function
* Daubechies wavelet
* Haar wavelet
* Morlet wavelet
* Mexican hat wavelet
* Complex Mexican hat wavelet
* Hermitian wavelet
* Discrete wavelet transform
* Continuous wavelet
* Continuous wavelet transform

## Quantum theory

See also list of mathematical topics in quantum theory
* Mathematical formulation of quantum mechanics
* Observable
* Operator (physics)
* Quantum state
** Pure state
** Fock state, Fock space
** Density state
** Coherent state
* Heisenberg picture
* Density matrix
* Quantum logic
* Quantum operation
* Wightman axioms

## Probability

* Free probability
* Bernstein's theorem

## Non-linear

* Fixed-point theorems in infinite-dimensional spaces

## History

* Stefan Banach (1892–1945)
* Hugo Steinhaus (1887–1972)
* John von Neumann (1903-1957)
* Alain Connes (born 1947)
* Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis
* Earliest Known Uses of Some of the Words of Mathematics: Matrices and Linear Algebra