Branches of Mathematics
Functional Analysis
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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.
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 Cross-Tradition Connections
Sources
1. Wikipedia: Functional Analysis
Wikimedia FoundationLead sectionQuote, Lead section
the study of vector spaces endowed with some kind of limit-related structure
1. Wikipedia: Functional Analysis
Wikimedia FoundationNormed vector spaces sectionQuote, 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.
1. Wikipedia: Functional Analysis
Wikimedia FoundationHilbert spaces section, invariant subspace problemQuote, 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.
Stone-Weierstrass Theorem (Wikipedia)
WikipediaOpen problems sectionQuote, Open 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.
Stone-Weierstrass Theorem (Wikipedia)
WikipediaHistory sectionQuote, History section
Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert (after whom they are named)
Stone-Weierstrass Theorem (Wikipedia)
Wikipedialead paragraphQuote, 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.
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