List of functional analysis topics

Hilbert space

Definite bilinear form

Euclidean space

Inner product space

Hilbert space

Parallelogram law

Orthonormal basis

Orthogonalization

Orthogonal complement

Gram–Schmidt process

Legendre polynomials

Matrices

Orthogonal matrix

Unitary matrix

Normal matrix, normal operator

Symmetric matrix

Hermitian operator self-adjoint operator, Hermitian adjoint

Eigenvector, eigenvalue, eigenfunction

Diagonal matrix

Shift operator

Hilbert matrix

Normal vector

Parseval's identity

Riesz representation theorem

Bra–ket notation

Spectral theorem, Spectral theory

Rayleigh quotient

Mercer's theorem

Reproducing kernel Hilbert space

Trace class

Min-max theorem

Rigged Hilbert space

Hellinger–Toeplitz theorem

Direct integral

Semi-Hilbert space

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

Hermitian hat 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 (1903-1959)

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