This is a list of
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
topics.
See also:
Glossary of functional analysis.
Hilbert space
Functional analysis, classic results
Operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operato ...
Banach space examples
*
Lp space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourba ...
*
Hardy space
In complex analysis, the Hardy spaces (or Hardy classes) H^p are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In real anal ...
*
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
*
Tsirelson space In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an ℓ ''p'' space nor a ''c''0 space can be embedded. The Tsirelson space is reflexive.
It was introduced by B. ...
*
ba space
Real and complex
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
s
Topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s
Amenability
*
Amenable group
*
Von Neumann conjecture
Wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the n ...
s
Quantum theory
''See also
list of mathematical topics in quantum theory''
{{columns-list, colwidth=20em,
*
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, whic ...
*
Observable
In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum ...
*
Operator (physics)
An operator is a function over a space of physical states onto another space of states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Because of this, they a ...
*
Quantum state
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
**
Pure state
**
Fock state,
Fock space
**
Density state
**
Coherent state
*
Heisenberg picture
In physics, the Heisenberg picture or Heisenberg representation is a Dynamical pictures, formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which observables incorporate a dependency on time, but the quantum state, st ...
*
Density matrix
In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while th ...
*
Quantum logic
In the mathematical study of logic and the physical analysis of quantum foundations, quantum logic is a set of rules for manipulation of propositions inspired by the structure of quantum theory. The formal system takes as its starting p ...
*
Quantum operation
*
Wightman axioms
Probability
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
*
Free probability
Free probability is a mathematics, mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of statistical independence, independence, and it is connecte ...
*
Bernstein's theorem
Non-linear
*
Fixed-point theorems in infinite-dimensional spaces
History
*
Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...
(1892–1945)
*
Hugo Steinhaus (1887–1972)
*
John von Neumann
John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
(1903-1957)
*
Alain Connes
Alain Connes (; born 1 April 1947) is a French mathematician, known for his contributions to the study of operator algebras and noncommutative geometry. He was a professor at the , , Ohio State University and Vanderbilt University. He was awar ...
(born 1947)
Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis
*
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...