List of functional analysis topics

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

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

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

• 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

Non-linear

History