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	~~'''~~Functional Analysis''' is a branch of [[mathematics]] that studies [[vector spaces]] endowed with some kind of limit-related structure (e.g., inner product, norm, topology) and the [[linear functions]] defined on these spaces and respecting these structures in a suitable sense. It has its historical roots in the study of functional spaces, particularly transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. This field of mathematics is related to many other areas of mathematics and science, particularly [[calculus of variations]], [[complex analysis]], and [[quantum mechanics]].		{{DISPLAYTITLE:Functional Analysis}}

	== ~~Definition and Scope~~ ==		== Overview ==
			[[File:Drum_vibration_mode12.gif\|thumb\|right\|Vibration modes of a drum, an example of functional analysis in physics.]]
			'''Functional analysis''' is a branch of [[mathematics]] concerned with the study of [[vector spaces]] and operators acting upon them. It is a fundamental area of [[mathematical analysis]] with applications in various fields such as [[quantum mechanics]], [[signal processing]], and [[differential equations]].

	Functional analysis ~~focuses on~~ the study of [[~~space (mathematics)\|spaces~~]] ~~of functions~~ and ~~their mapping properties~~. The ~~primary objects~~ of ~~interest are the~~ [[Banach ~~space~~]]s and [[~~Hilbert space~~]]~~s. Banach spaces are vector spaces equipped with a norm that turns them into a complete metric space. Hilbert spaces, a subset of Banach spaces, are endowed with an inner product that allows~~ for ~~a geometric interpretation of vector space concepts~~.		== Historical Background ==
			Functional analysis emerged in the early 20th century as a distinct field of study. It was developed to address problems in [[integral equations]] and [[differential equations]]. The work of mathematicians such as [[David Hilbert]], [[Stefan Banach]], and [[John von Neumann]] laid the foundation for modern functional analysis.

	== Key Concepts ==		== Key Concepts ==

	=== ~~Normed Spaces and Banach~~ Spaces ===		=== Vector Spaces ===
			A [[vector space]] is a collection of objects called vectors, which can be added together and multiplied by scalars. Functional analysis often deals with infinite-dimensional vector spaces, such as [[Hilbert spaces]] and [[Banach spaces]].

	A ~~'''normed space'''~~ is a ~~vector space V over the field ℝ or ℂ, together with a norm~~ function that assigns a non-negative ~~scalar~~ to each vector. A ~~normed space~~ is ~~complete (i.e., all Cauchy sequences~~ in ~~the space converge) is called~~ a ~~'''Banach~~ space~~'''~~.		=== Norms and Metrics ===
			A [[norm]] is a function that assigns a non-negative length or size to each vector in a vector space. A [[metric]] is a function that defines a distance between each pair of vectors in a space. These concepts are crucial for understanding the structure of vector spaces in functional analysis.

	=== ~~Inner Product and Hilbert Spaces~~ ===		=== Operators ===
			Operators are mappings between vector spaces that preserve the vector space structure. In functional analysis, operators are often linear, meaning they satisfy the properties of additivity and homogeneity. Important classes of operators include [[bounded operators]], [[compact operators]], and [[self-adjoint operators]].

	An '''inner product space''' is a vector space with an additional structure called an inner product. This inner product allows for the definition of angles and lengths in the vector space. A complete inner product space is known as a '''Hilbert space''', which is central to the study of functional analysis due to its geometric properties.		=== Spectral Theory ===
			[[Spectral theory]] studies the spectrum of operators, which generalizes the concept of [[eigenvalues]] and [[eigenvectors]] from finite-dimensional spaces to infinite-dimensional spaces. This theory is essential in understanding the behavior of operators in functional analysis.
	=== ~~Linear Operators and Bounded Operators~~ ===

	~~In functional analysis~~, ~~a '''linear operator''' is a mapping between two vector spaces that preserves~~ the ~~operations~~ of ~~vector addition~~ and ~~scalar multiplication~~. ~~A linear operator is said to be '''bounded''' if there~~ is ~~a constant C such that for all vectors v~~ in ~~its domain,~~ the ~~norm~~ of ~~the image of v is less than or equal to C times the norm of v. The study of bounded~~ operators ~~on Hilbert spaces is a key aspect of~~ functional analysis.

	== Applications ==		== Applications ==
			Functional analysis has numerous applications in both pure and applied mathematics. In [[quantum mechanics]], it provides the framework for the formulation of [[quantum states]] and [[observables]]. In [[signal processing]], it is used to analyze and manipulate signals. Functional analysis also plays a critical role in solving [[partial differential equations]] and in the study of [[dynamical systems]].

	~~Functional analysis has numerous applications across mathematics and physics. In~~ [[~~quantum mechanics~~]]~~, for example, the states of a quantum system are represented by vectors in a Hilbert space, and physical observables are represented by linear operators on these spaces. In~~ [[~~partial differential equations~~]]~~, functional analysis techniques are used to prove the existence and uniqueness of solutions under given conditions.~~		== Related Pages ==
			* [[Linear algebra]]
	~~== See Also ==~~		* [[Measure theory]]

	* [[Topology]]		* [[Topology]]
	* [[Measure (mathematics)\|Measure Theory]]
	* [[Spectral theory]]
	* [[Operator theory]]		* [[Operator theory]]
			* [[Fourier analysis]]

	~~== Categories ==~~		[[Category:Mathematical analysis]]

	~~[[Category:Functional Analysis]]~~
	~~[[Category:Mathematics]]~~
	[[Category:Mathematical ~~Analysis~~]]

	~~{{math-stub}}~~

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'''Functional Analysis''' is a branch of [[mathematics]] that studies [[vector spaces]] endowed with some kind of limit-related structure (e.g., inner product, norm, topology) and the [[linear functions]] defined on these spaces and respecting these structures in a suitable sense. It has its historical roots in the study of functional spaces, particularly transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. This field of mathematics is related to many other areas of mathematics and science, particularly [[calculus of variations]], [[complex analysis]], and [[quantum mechanics]].

== Definition and Scope ==

Functional analysis focuses on the study of [[space (mathematics)|spaces]] of functions and their mapping properties. The primary objects of interest are the [[Banach space]]s and [[Hilbert space]]s. Banach spaces are vector spaces equipped with a norm that turns them into a complete metric space. Hilbert spaces, a subset of Banach spaces, are endowed with an inner product that allows for a geometric interpretation of vector space concepts.

== Key Concepts ==

=== Normed Spaces and Banach Spaces ===

A '''normed space''' is a vector space V over the field ℝ or ℂ, together with a norm function that assigns a non-negative scalar to each vector. A normed space is complete (i.e., all Cauchy sequences in the space converge) is called a '''Banach space'''.

=== Inner Product and Hilbert Spaces ===

An '''inner product space''' is a vector space with an additional structure called an inner product. This inner product allows for the definition of angles and lengths in the vector space. A complete inner product space is known as a '''Hilbert space''', which is central to the study of functional analysis due to its geometric properties.

=== Linear Operators and Bounded Operators ===

In functional analysis, a '''linear operator''' is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. A linear operator is said to be '''bounded''' if there is a constant C such that for all vectors v in its domain, the norm of the image of v is less than or equal to C times the norm of v. The study of bounded operators on Hilbert spaces is a key aspect of functional analysis.

== Applications ==

Functional analysis has numerous applications across mathematics and physics. In [[quantum mechanics]], for example, the states of a quantum system are represented by vectors in a Hilbert space, and physical observables are represented by linear operators on these spaces. In [[partial differential equations]], functional analysis techniques are used to prove the existence and uniqueness of solutions under given conditions.

== See Also ==

* [[Topology]]
* [[Measure (mathematics)|Measure Theory]]
* [[Spectral theory]]
* [[Operator theory]]

== Categories ==

[[Category:Functional Analysis]]
[[Category:Mathematics]]
[[Category:Mathematical Analysis]]

{{math-stub}}

Functional analysis - Revision history

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