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 QuestionWhich 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 DebateWhether 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 Connections
Associated With
Includes
Source Andrey Kolmogorov (Wikipedia)
Source Banach-Alaoglu Theorem (Wikipedia)
Source Bent Fuglede (Mathematics Genealogy Project)
Source Wikipedia: Eduard Helly
Source Fréchet Derivative (Wikipedia)
Source Frigyes Riesz (Wikipedia)
Source Wolfram MathWorld: Fuglede's Conjecture
Source MacTutor biography of Gottfried Köthe, St Andrews
Source Gustave Choquet, Biography (MacTutor History of Mathematics)
Source Wikipedia: Hans Hahn (mathematician)
Source Wikipedia: Hugo Steinhaus
Source Invariant Subspace Problem (Wikipedia)
Source Wikipedia: Isaac Namioka
Source Wikipedia: Juliusz Schauder
Source Krein-Milman theorem, Wikipedia
Source Laurent Schwartz, Biography (MacTutor History of Mathematics)
Source Schauder fixed point theorem, Wikipedia
Source MacTutor biography of Mark Krein, St Andrews
Source Mercer's theorem, Wikipedia
Source Namioka's theorem - Wikipedia
Source MacTutor biography of Otto Toeplitz, St Andrews
Source Paul Lévy, Biography (MacTutor History of Mathematics)
Source Rellich-Kondrachov theorem, Wikipedia
Source Wikipedia: Shizuo Kakutani
Source Spectral Theory of Compact Operators (Wikipedia)
Source Stefan Banach (Wikipedia)
Sources
1. Wikipedia: Functional Analysis
Wikimedia FoundationLead 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.
View the Source Schauder fixed point theorem, Wikipedia
Includes: Leray-Schauder Fixed-Point Theorem, LeadQuote, 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
Fréchet Derivative (Wikipedia)
Includes: Fréchet Derivative, LeadQuote, Includes: Fréchet Derivative, Lead
the Fréchet derivative is a derivative defined on normed spaces
Bent Fuglede (Mathematics Genealogy Project)
Includes: Bent Fuglede, Degree recordQuote, Includes: Bent Fuglede, Degree record
Functional analysis
Invariant Subspace Problem (Wikipedia)
WikipediaIncludes: Invariant Subspace Problem, Lead sentenceQuote, 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
Spectral Theory of Compact Operators (Wikipedia)
Includes: Spectral Theory of Compact Operators, Lead sentenceQuote, 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.
Banach-Alaoglu Theorem (Wikipedia)
Wikimedia FoundationIncludes: Banach-Alaoglu Theorem, Lead sentenceView the Source Krein-Milman theorem, Wikipedia
Mercer's theorem, Wikipedia
Includes: Mercer's Theorem, Lead sentenceQuote, Includes: Mercer's Theorem, Lead sentence
In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function o
Namioka's theorem - Wikipedia
Includes: Namioka's Theorem, Lead sentenceQuote, Includes: Namioka's Theorem, Lead sentence
In functional analysis, Namioka's theorem is a result concerning the relationship between separate continuity and joint continuity
Frigyes Riesz (Wikipedia)
Includes: Frigyes Riesz, Lead paragraph [in-branch]Quote, Includes: Frigyes Riesz, Lead paragraph [in-branch]
functional analysis
Stefan Banach (Wikipedia)
Includes: Stefan Banach, Lead paragraph [in-branch]Quote, Includes: Stefan Banach, Lead paragraph [in-branch]
functional analysis
Wikipedia: Eduard Helly
Includes: Eduard Helly, LeadQuote, Includes: Eduard Helly, Lead
Helly wrote important contributions on functional analysis
Wikipedia: Shizuo Kakutani
Includes: Shizuo Kakutani, LeadQuote, 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.
Wikipedia: Juliusz Schauder
Includes: Juliusz Schauder, LeadQuote, 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.
Wikipedia: Hugo Steinhaus
Includes: Hugo Steinhaus, LeadQuote, 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.
Wikipedia: Hans Hahn (mathematician)
Includes: Hans Hahn, LeadQuote, 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.
Wikipedia: Isaac Namioka
Includes: Isaac Namioka, LeadQuote, 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.
Andrey Kolmogorov (Wikipedia)
Includes: Andrey Kolmogorov, Lead paragraphQuote, 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.
Wolfram MathWorld: Fuglede's Conjecture
Wolfram MathWorldIncludes: Fuglede's Conjecture, Subject classificationsQuote, Includes: Fuglede's Conjecture, Subject classifications
Calculus and Analysis Functional Analysis
Rellich-Kondrachov theorem, Wikipedia
Includes: Rellich-Kondrachov Theorem, Lead sectionQuote, Includes: Rellich-Kondrachov Theorem, Lead section
theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Austrian-German mathematician
MacTutor biography of Gottfried Köthe, St Andrews
Includes: Gottfried Köthe, BiographyQuote, Includes: Gottfried Köthe, Biography
who worked in abstract algebra, functional analysis and topological vector spaces
MacTutor biography of Otto Toeplitz, St Andrews
Includes: Otto Toeplitz, BiographyQuote, Includes: Otto Toeplitz, Biography
he developed a general theory of infinite dimensional spaces
MacTutor biography of Mark Krein, St Andrews
Includes: Mark Krein, SummaryQuote, Includes: Mark Krein, Summary
Mark G Krein was a Soviet mathematician who worked in functional analysis, operator theory, classical analysis and representation theory.
Gustave Choquet, Biography (MacTutor History of Mathematics)
Includes: Gustave Choquet, SummaryQuote, Includes: Gustave Choquet, Summary
Gustave Choquet was a French mathematician who worked in functional analysis, potential theory, topology and measure theory.
Laurent Schwartz, Biography (MacTutor History of Mathematics)
Includes: Laurent Schwartz, SummaryQuote, Includes: Laurent Schwartz, Summary
Laurent Schwartz was a French mathematician who is best known for his work in the theory of distributions.
Paul Lévy, Biography (MacTutor History of Mathematics)
Includes: Paul Lévy, SummaryQuote, 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.
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