Branches of Mathematic

Functional Analysis

Analysis

Functional analysis is the branch of mathematical analysis that studies vector spaces of functions equipped with limit-related structure, such as a norm or an inner product, together with the linear maps between them that respect that structure. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Central Question
Which infinite-dimensional vector spaces of functions carry enough limit structure, a norm or an inner product complete under it, to let the tools of linear algebra and calculus be applied to them together. 1
Key Debate
Whether every bounded linear operator on a Hilbert space has a proper invariant subspace, a question about the operators the field's own foundational spaces support that remains an open problem in functional analysis despite the field's century of development. 1
Classification
Pure or Applied
Pure Mathematics 1
Connections

Associated With

Includes

Source Allen Shields (Wikipedia)
Source Andrey Kolmogorov (Wikipedia)
Source Andrey Tychonoff (Wikipedia)
Source Arthur Milgram (Wikipedia)
Source Aryeh Dvoretzky (Wikipedia)
Source Banach-Alaoglu Theorem (Wikipedia)
Source Béla Bollobás (Wikipedia)
Source Bent Fuglede (Mathematics Genealogy Project)
Source Edgar Lorch (Wikipedia)
Source Wikipedia: Eduard Helly
Source Einar Hille (Wikipedia)
Source Erhard Schmidt (Wikipedia)
Source Frank Smithies (Wikipedia)
Source Fréchet Derivative (Wikipedia)
Source Frigyes Riesz (Wikipedia)
Source Wolfram MathWorld: Fuglede's Conjecture
Source Gian-Carlo Rota (Wikipedia)
Source MacTutor biography of Gottfried Köthe, St Andrews
Source Griffith C. Evans (Wikipedia)
Source Gustave Choquet, Biography (MacTutor History of Mathematics)
Source Wikipedia: Hans Hahn (mathematician)
Source Heinz Bauer (Wikipedia)
Source Wikipedia: Hugo Steinhaus
Source Invariant Subspace Problem (Wikipedia)
Source Irving Segal (Wikipedia)
Source Wikipedia: Isaac Namioka
Source Jean Dieudonné (Wikipedia)
Source Wikipedia: Juliusz Schauder
Source Krein-Milman theorem, Wikipedia
Source Laurent Schwartz, Biography (MacTutor History of Mathematics)
Source Leonid Kantorovich (Wikipedia)
Source Schauder fixed point theorem, Wikipedia
Source MacTutor biography of Mark Krein, St Andrews
Source Marshall H. Stone (Wikipedia)
Source Mercer's theorem, Wikipedia
Metric Space, Concepts
Source Namioka's theorem - Wikipedia
Source Nelson Dunford (Wikipedia)
Source MacTutor biography of Otto Toeplitz, St Andrews
Source Paul Halmos (Wikipedia)
Source Paul Lévy, Biography (MacTutor History of Mathematics)
Source Rellich-Kondrachov theorem, Wikipedia
Source Roger Godement (Wikipedia)
Source Wikipedia: Shizuo Kakutani
Source Spectral Theory of Compact Operators (Wikipedia)
Source Stefan Banach (Wikipedia)

Showing 1 to 50 of 53 in Includes, page 1 of 2.

Sources
1. Wikipedia: Functional Analysis
Wikimedia Foundation
  • Lead section
    the study of vector spaces endowed with some kind of limit-related structure
  • Normed vector spaces section
    These spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics, machine learning, partial differential equations, and Fourier analysis.
  • Hilbert spaces section, invariant subspace problem
    One of the open problems in functional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace.
  • Hilbert spaces
    One of the open problems in functional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace.
  • In Group: Functional and Operator Analysis, lead paragraph
    Functional analysis is a branch of mathematical analysis
View the Source
Schauder fixed point theorem, Wikipedia
Includes: Leray-Schauder Fixed-Point Theorem, Lead
Quote, Includes: Leray-Schauder Fixed-Point Theorem, Lead
The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to locally convex topological vector spaces
View the Source
Fréchet Derivative (Wikipedia)
Includes: Fréchet Derivative, Lead
Quote, Includes: Fréchet Derivative, Lead
the Fréchet derivative is a derivative defined on normed spaces
View the Source
Bent Fuglede (Mathematics Genealogy Project)
Includes: Bent Fuglede, Degree record
Quote, Includes: Bent Fuglede, Degree record
Functional analysis
View the Source
Edgar Lorch (Wikipedia)
Includes: Edgar Lorch, Lead, spectral theory
Quote, Includes: Edgar Lorch, Lead, spectral theory
functional analysis, especially spectral theory
View the Source
Paul Halmos (Wikipedia)
Includes: Paul Halmos, Lead
Quote, Includes: Paul Halmos, Lead
made fundamental advances in the areas of mathematical logic, probability theory, operator theory, ergodic theory, and functional analysis
View the Source
Heinz Bauer (Wikipedia)
Includes: Heinz Bauer, Biography, functional analysis
Quote, Includes: Heinz Bauer, Biography, functional analysis
His research focuses were potential theory, probability theory, and functional analysis.
View the Source
Gian-Carlo Rota (Wikipedia)
Includes: Gian-Carlo Rota, Career
Quote, Includes: Gian-Carlo Rota, Career
Rota began his career as a functional analyst, but switched to become a distinguished combinatorialist.
View the Source
Erhard Schmidt (Wikipedia)
Includes: Erhard Schmidt, Mathematics
Quote, Includes: Erhard Schmidt, Mathematics
he made important contributions to functional analysis
View the Source
Andrey Tychonoff (Wikipedia)
Includes: Andrey Tychonoff, Lead
Quote, Includes: Andrey Tychonoff, Lead
known for important contributions to topology, functional analysis, mathematical physics
View the Source
Nelson Dunford (Wikipedia)
Includes: Nelson Dunford, Lead
Quote, Includes: Nelson Dunford, Lead
known for his work in functional analysis
View the Source
Aryeh Dvoretzky (Wikipedia)
Includes: Aryeh Dvoretzky, Lead, field
Quote, Includes: Aryeh Dvoretzky, Lead, field
He is best known for his work in functional analysis, statistics and probability.
View the Source
Arthur Milgram (Wikipedia)
Includes: Arthur Milgram, Lead, branch
Quote, Includes: Arthur Milgram, Lead, branch
He made contributions in functional analysis, combinatorics, differential geometry, topology, partial differential equations, and Galois theory.
View the Source
Griffith C. Evans (Wikipedia)
Includes: Griffith C. Evans, Biography, early research
Quote, Includes: Griffith C. Evans, Biography, early research
functional analysis
View the Source
Marshall H. Stone (Wikipedia)
Wikimedia FoundationIncludes: Marshall H. Stone, Lead sentence
Quote, Includes: Marshall H. Stone, Lead sentence
functional analysis
View the Source
Vito Volterra (Wikipedia)
Includes: Vito Volterra, Lead, functional analysis
Quote, Includes: Vito Volterra, Lead, functional analysis
being one of the founders of functional analysis
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Leonid Kantorovich (Wikipedia)
Includes: Leonid Kantorovich, Mathematics
Quote, Includes: Leonid Kantorovich, Mathematics
Kantorovich had important results in functional analysis, approximation theory, and operator theory
View the Source
Béla Bollobás (Wikipedia)
Includes: Béla Bollobás, Lead
Quote, Includes: Béla Bollobás, Lead
including functional analysis, combinatorics, graph theory, and percolation theory
View the Source
Timothy Gowers (Wikipedia)
Includes: Timothy Gowers, Lead
Quote, Includes: Timothy Gowers, Lead
for research connecting the fields of functional analysis and combinatorics
View the Source
Jean Dieudonné (Wikipedia)
Includes: Jean Dieudonné, Lead
Quote, Includes: Jean Dieudonné, Lead
notable for research in abstract algebra, algebraic geometry, and functional analysis
View the Source
Allen Shields (Wikipedia)
Includes: Allen Shields, Lead, functional analysis
Quote, Includes: Allen Shields, Lead, functional analysis
who worked on measure theory, complex analysis, functional analysis and operator theory
View the Source
Frank Smithies (Wikipedia)
Includes: Frank Smithies, Lead
Quote, Includes: Frank Smithies, Lead
who worked on integral equations, functional analysis, and the history of mathematics
View the Source
Irving Segal (Wikipedia)
Includes: Irving Segal, Lead
Quote, Includes: Irving Segal, Lead
in functional and harmonic analysis
View the Source
Roger Godement (Wikipedia)
Includes: Roger Godement, Lead, field of work
Quote, Includes: Roger Godement, Lead, field of work
known for his work in functional analysis as well as his expository books
View the Source
Einar Hille (Wikipedia)
Includes: Einar Hille, Career, functional analysis
Quote, Includes: Einar Hille, Career, functional analysis
Later in his career his interests turned more towards functional analysis
View the Source
Invariant Subspace Problem (Wikipedia)
WikipediaIncludes: Invariant Subspace Problem, Lead sentence
Quote, Includes: Invariant Subspace Problem, Lead sentence
In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking
View the Source
Spectral Theory of Compact Operators (Wikipedia)
Includes: Spectral Theory of Compact Operators, Lead sentence
Quote, Includes: Spectral Theory of Compact Operators, Lead sentence
In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets.
View the Source
Banach-Alaoglu Theorem (Wikipedia)
Wikimedia FoundationIncludes: Banach-Alaoglu Theorem, Lead sentenceView the Source
Krein-Milman theorem, Wikipedia
Includes: Krein-Milman Theorem, Lead sentenceView the Source
Mercer's theorem, Wikipedia
Includes: Mercer's Theorem, Lead sentence
Quote, Includes: Mercer's Theorem, Lead sentence
In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function o
View the Source
Namioka's theorem - Wikipedia
Includes: Namioka's Theorem, Lead sentence
Quote, Includes: Namioka's Theorem, Lead sentence
In functional analysis, Namioka's theorem is a result concerning the relationship between separate continuity and joint continuity
View the Source
Frigyes Riesz (Wikipedia)
Includes: Frigyes Riesz, Lead paragraph [in-branch]
Quote, Includes: Frigyes Riesz, Lead paragraph [in-branch]
functional analysis
View the Source
Stefan Banach (Wikipedia)
Includes: Stefan Banach, Lead paragraph [in-branch]
Quote, Includes: Stefan Banach, Lead paragraph [in-branch]
functional analysis
View the Source
Wikipedia: Eduard Helly
Includes: Eduard Helly, Lead
Quote, Includes: Eduard Helly, Lead
Helly wrote important contributions on functional analysis
View the Source
Wikipedia: Shizuo Kakutani
Includes: Shizuo Kakutani, Lead
Quote, Includes: Shizuo Kakutani, Lead
Kakutani received two awards of the Japan Academy, the Imperial Prize and the Academy Prize in 1982, for his scholarly achievements in general and his work on functional analysis in particular.
View the Source
Wikipedia: Juliusz Schauder
Includes: Juliusz Schauder, Lead
Quote, Includes: Juliusz Schauder, Lead
Juliusz Paweł Schauder ([ˈjulʲjuʂ ˈpavɛw ˈʂau̯dɛr]; 21 September 1899, October 1943) was a Polish mathematician known for his work in functional analysis, partial differential equations and mathematical physics.
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Wikipedia: Hugo Steinhaus
Includes: Hugo Steinhaus, Lead
Quote, Includes: Hugo Steinhaus, Lead
He is credited with "discovering" mathematician Stefan Banach, with whom he gave a notable contribution to functional analysis through the Banach-Steinhaus theorem.
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Wikipedia: Hans Hahn (mathematician)
Includes: Hans Hahn, Lead
Quote, Includes: Hans Hahn, Lead
Hans Hahn (; German: [haːn]; 27 September 1879-24 July 1934) was an Austrian mathematician and philosopher who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory.
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Wikipedia: Isaac Namioka
Includes: Isaac Namioka, Lead
Quote, Includes: Isaac Namioka, Lead
Isaac Namioka (April 25, 1928, September 25, 2019) was a Japanese-American mathematician who worked in general topology and functional analysis.
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Andrey Kolmogorov (Wikipedia)
Includes: Andrey Kolmogorov, Lead paragraph
Quote, Includes: Andrey Kolmogorov, Lead paragraph
He also gave fundamental contributions to the mathematics of topology, intuitionistic logic, turbulence, classical mechanics, functional analysis, algorithmic information theory and computational complexity.
View the Source
Wolfram MathWorld: Fuglede's Conjecture
Wolfram MathWorldIncludes: Fuglede's Conjecture, Subject classifications
Quote, Includes: Fuglede's Conjecture, Subject classifications
Calculus and Analysis Functional Analysis
View the Source
Rellich-Kondrachov theorem, Wikipedia
Includes: Rellich-Kondrachov Theorem, Lead section
Quote, Includes: Rellich-Kondrachov Theorem, Lead section
theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Austrian-German mathematician
View the Source
MacTutor biography of Gottfried Köthe, St Andrews
Includes: Gottfried Köthe, Biography
Quote, Includes: Gottfried Köthe, Biography
who worked in abstract algebra, functional analysis and topological vector spaces
View the Source
MacTutor biography of Otto Toeplitz, St Andrews
Includes: Otto Toeplitz, Biography
Quote, Includes: Otto Toeplitz, Biography
he developed a general theory of infinite dimensional spaces
View the Source
MacTutor biography of Mark Krein, St Andrews
Includes: Mark Krein, Summary
Quote, Includes: Mark Krein, Summary
Mark G Krein was a Soviet mathematician who worked in functional analysis, operator theory, classical analysis and representation theory.
View the Source
Gustave Choquet, Biography (MacTutor History of Mathematics)
Includes: Gustave Choquet, Summary
Quote, Includes: Gustave Choquet, Summary
Gustave Choquet was a French mathematician who worked in functional analysis, potential theory, topology and measure theory.
View the Source
Laurent Schwartz, Biography (MacTutor History of Mathematics)
Includes: Laurent Schwartz, Summary
Quote, Includes: Laurent Schwartz, Summary
Laurent Schwartz was a French mathematician who is best known for his work in the theory of distributions.
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Paul Lévy, Biography (MacTutor History of Mathematics)
Includes: Paul Lévy, Summary
Quote, Includes: Paul Lévy, Summary
Paul Lévy was a French mathematician who, after starting out as an expert on functional analysis, made important advances in probability theory.
View the Source
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