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 Functional Analysis
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Source Banach-Alaoglu Theorem (Wikipedia)
Source Frigyes Riesz (Wikipedia)
Source Invariant Subspace Problem (Wikipedia)
Source Krein-Milman theorem, Wikipedia
Source Mercer's theorem, Wikipedia
Source Namioka's theorem - Wikipedia
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.
View the Source Stone-Weierstrass Theorem (Wikipedia)
WikipediaOpen problems section
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.
History section
Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert (after whom they are named)
lead paragraph
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function.
View the Source 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
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