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