List of functional analysis topics

This is a list of functional analysis topics, by Wikipedia page.

Contents

1 Hilbert space 2 Functional analysis, classic results 3 Operator theory 4 Banach space examples 5 Real and complex algebras 6 Topological vector spaces 7 Amenability 8 Wavelets 9 Quantum theory 10 Probability 11 Non-linear 12 History

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
• Closed graph theorem
• Uniform boundedness principle
• Arzelà-Ascoli theorem
• Banach-Alaoglu theorem
• Measure of non-compactness

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
• Star algebra
• B-star-algebra
• C-star-algebra
  • Universal C-star-algebra
  • Spectrum of a C-star-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

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

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)