Hilbert space

• 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
• Positive-definite
• 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
• 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-Polya 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

• Locally convex topological vector space
• F-space
• Barrelled space
• Frechet space
• Seminorm
• Dual pair
• Polar topology
• Polar set
• Mackey-Arens theorem
• Mackey topology
• Bourbaki-Alaoglu theorem
• Bornological space
• Montel space

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

• 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
• Alain Connes (born 1947)