Editing List of functional analysis topics

{{Short description|None}}
This is a '''list of [[functional analysis]] topics'''.

See also: [[Glossary of functional analysis]].

== Hilbert space ==

{{columns-list|colwidth=20em|
* [[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]]
* [[Matrix (mathematics)|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==
{{columns-list|colwidth=20em|
*[[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]]==
{{columns-list|colwidth=20em|
*[[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 [[associative algebra|algebra]]s==
{{columns-list|colwidth=20em|
*[[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 space]]s==
{{columns-list|colwidth=20em|
*[[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]]

==[[Wavelet]]s==
{{columns-list|colwidth=20em|
*[[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]]''
{{columns-list|colwidth=20em|
*[[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 on monotone functions|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)
*[http://www.economics.soton.ac.uk/staff/aldrich/Calculus%20and%20Analysis%20Earliest%20Uses.htm Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis]
*[http://www.economics.soton.ac.uk/staff/aldrich/matrices.htm Earliest Known Uses of Some of the Words of Mathematics: Matrices and Linear Algebra]

[[Category:Functional analysis|*]]
[[Category:Mathematics-related lists|Functional analysis]]