Functional analysis: Difference between revisions

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'''Functional analysis''' is a branch of [[mathematical analysis]], the core of which is formed by the study of [[vector space]]s endowed with some kind of limit-related structure (e.g. [[Inner product space#Definition|inner product]], [[Norm (mathematics)#Definition|norm]], [[Topological space#Definition|topology]], etc.) and the [[linear transformation|linear function]]s defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of [[function space|spaces of functions]] and the formulation of properties of transformations of functions such as the [[Fourier transform]] as transformations defining [[continuous function|continuous]], [[unitary operator|unitary]] etc. operators between function spaces. This point of view turned out to be particularly useful for the study of [[differential equations|differential]] and [[integral equations]].

The usage of the word ''[[functional (mathematics)|functional]]'' as a noun goes back to the [[calculus of variations]], implying a [[Higher-order function|function whose argument is a function]]. The term was first used in [[Jacques Hadamard|Hadamard]]'s 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist [[Vito Volterra]].<ref>{{Cite web | last=Lawvere|first=F. William|title=Volterra's functionals and covariant cohesion of space| url=http://www.acsu.buffalo.edu/~wlawvere/Volterra.pdf |archive-url=https://web.archive.org/web/20030407030553/http://www.acsu.buffalo.edu/~wlawvere/Volterra.pdf |archive-date=2003-04-07|url-status=live| website=acsu.buffalo.edu | publisher=Proceedings of the May 1997 Meeting in Perugia}}</ref><ref>{{Cite book| url=http://dx.doi.org/10.1142/5685|title=History of Mathematical Sciences|date=October 2004| page=195| publisher=WORLD SCIENTIFIC| doi=10.1142/5685|isbn=978-93-86279-16-3|last1=Saraiva|first1=Luís}}</ref> The theory of nonlinear functionals was continued by students of Hadamard, in particular [[Maurice René Fréchet|Fréchet]] and [[Paul Lévy (mathematician)|Lévy]]. Hadamard also founded the modern school of linear functional analysis further developed by [[Frigyes Riesz|Riesz]] and the [[Lwów School of Mathematics|group]] of [[Poland|Polish]] mathematicians around [[Stefan Banach]].

In modern introductory texts on functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular [[Dimension (vector space)|infinite-dimensional spaces]].<ref>{{Cite book| last1=Bowers|first1=Adam|title=An introductory course in functional analysis|last2=Kalton|first2=Nigel J.| publisher=[[Springer Science & Business Media]]|year=2014|pages=1}}</ref><ref>{{Cite book| last=Kadets| first=Vladimir| title=A Course in Functional Analysis and Measure Theory|publisher=[[Springer Publishing|Springer]] | year=2018|pages=xvi|trans-title=КУРС ФУНКЦИОНАЛЬНОГО АНАЛИЗА}}</ref> In contrast, [[linear algebra]] deals mostly with finite-dimensional spaces, and does not use topology. An important part of functional analysis is the extension of the theory of [[measure (mathematics)|measure]], [[integral|integration]], and [[probability]] to infinite dimensional spaces, also known as '''infinite dimensional analysis'''.

==Normed vector spaces==

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===Hilbert spaces===

[[Hilbert space]]s can be completely classified: there is a unique Hilbert space [[up to]] [[isomorphism]] for every [[cardinal number|cardinality]] of the [[orthonormal basis]].<ref>{{Cite book| last=Riesz|first=Frigyes| url=https://www.worldcat.org/oclc/21228994|title=Functional analysis|date=1990|publisher=Dover Publications| others = Béla Szőkefalvi-Nagy, Leo F. Boron|isbn=0-486-66289-6|edition=Dover |___location=New York|oclc=21228994| pages = 195–199}}</ref> Finite-dimensional Hilbert spaces are fully understood in [[linear algebra]], and infinite-dimensional [[Separable space|separable]] Hilbert spaces are isomorphic to [[Sequence space#ℓp spaces|<math>\ell^{\,2}(\aleph_0)\,</math>]]. Separability being important for applications, functional analysis of Hilbert spaces consequently mostly deals with this space. 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]]. Many special cases of this [[invariant subspace problem]] have already been proven.

===Banach spaces===

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Examples of Banach spaces are [[Lp space|<math>L^p</math>-spaces]] for any real number {{nowrap|<math>p\geq1</math>.}} Given also a measure <math>\mu</math> on set {{nowrap|<math>X</math>,}} then {{nowrap|<math>L^p(X)</math>,}} sometimes also denoted <math>L^p(X,\mu)</math> or {{nowrap|<math>L^p(\mu)</math>,}} has as its vectors equivalence classes <math>[\,f\,]</math> of [[Lebesgue-measurable function|measurable function]]s whose [[absolute value]]'s <math>p</math>-th power has finite integral; that is, functions <math>f</math> for which one has

:<math display="block">\int_{X}\left|f(x)\right|^p\,d\mu(x) < +\infty.</math>

If <math>\mu</math> is the [[counting measure]], then the integral may be replaced by a sum. That is, we require

:<math display="block">\sum_{x\in X}\left|f(x)\right|^p<+\infty .</math>

Then it is not necessary to deal with equivalence classes, and the space is denoted {{nowrap|<math>\ell^p(X)</math>,}} written more simply <math>\ell^p</math> in the case when <math>X</math> is the set of non-negative [[integer]]s.

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The theorem was first published in 1927 by [[Stefan Banach]] and [[Hugo Steinhaus]] but it was also proven independently by [[Hans Hahn (mathematician)|Hans Hahn]].

<blockquote>'''{{math theorem | name = Theorem (Uniform Boundedness Principle).''' | math_statement = Let <math>X</math> be a [[Banach space]] and <math>Y</math> be a [[normed vector space]]. Suppose that <math>F</math> is a collection of continuous linear operators from <math>X</math> to <math>Y</math>. If for all <math>x</math> in <math>X</math> one has

:<math display="block">\sup\nolimits_{T \in F} \|T(x)\|_Y < \infty, </math>

:<math>\sup\nolimits_{T \in F} \|T(x)\|_Y < \infty, </math>

then

:<math display="block">\sup\nolimits_{T \in F} \|T\|_{B(X,Y)} < \infty.</math></blockquote>}}

:<math>\sup\nolimits_{T \in F} \|T\|_{B(X,Y)} < \infty.</math></blockquote>

===Spectral theorem===

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There are many theorems known as the [[spectral theorem]], but one in particular has many applications in functional analysis.

<blockquote>'''{{math theorem | name = Spectral theorem.'''<ref>{{Cite book|last=Hall|first=Brian C.|url={{google books |plainurl=y |id=bYJDAAAAQBAJ|page=147}}|title=Quantum Theory for Mathematicians|date=2013-06-19|publisher=[[Springer Science & Business Media]]|isbn=978-1-4614-7116-5|page=147|language=en}}</ref>
|math_statement = Let <math>A</math> be a bounded self-adjoint operator on a Hilbert space <math>H</math>. Then there is a [[measure space]] <math>(X,\Sigma,\mu)</math> and a real-valued [[ess sup|essentially bounded]] measurable function <math>f</math> on <math>X</math> and a unitary operator <math>U:H\to L^2_\mu(X)</math> such that

:<math display="block"> U^* T U = A \;</math>

:<math> U^* T U = A \;</math>

where ''T'' is the [[multiplication operator]]:

<math display="block"> [T \varphi](x) = f(x) \varphi(x). </math>

:and <math> [\|T \varphi](x)| = \|f(x) \varphi(x). |_\;infty</math>.}}

and <math>\|T\| = \|f\|_\infty</math>.</blockquote>

This is the beginning of the vast research area of functional analysis called [[operator theory]]; see also the [[spectral measure#Spectral measure|spectral measure]].

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The [[Hahn–Banach theorem]] is a central tool in functional analysis. It allows the extension of [[Bounded operator|bounded linear functionals]] defined on a subspace of some [[vector space]] to the whole space, and it also shows that there are "enough" [[continuous function (topology)|continuous]] linear functionals defined on every [[normed vector space]] to make the study of the [[dual space]] "interesting".

<blockquote>'''{{math theorem | name = Hahn–Banach theorem:'''<ref name="rudin">{{Cite book | last=Rudin | first=Walter | url={{google books |plainurl=y |id=Sh_vAAAAMAAJ}} | title=Functional Analysis | date=1991 | publisher=McGraw-Hill | isbn=978-0-07-054236-5 | language=en}}</ref>
| math_statement = If <math>p:V\to\mathbb{R}</math> is a [[sublinear function]], and <math>\varphi:U\to\mathbb{R}</math> is a [[linear functional]] on a [[linear subspace]] <math>U\subseteq V</math> which is [[dominate (mathematics)|dominated]] by <math>p</math> on <math>U</math>; that is,

:<math display="block">\varphi(x) \leq p(x)\qquad\forall x \in U</math>

then there exists a linear extension <math>\psi:V\to\mathbb{R}</math> of <math>\varphi</math> to the whole space <math>V</math> which is [[dominate (mathematics)|dominated]] by <math>p</math> on <math>V</math>; that is, there exists a linear functional <math>\psi</math> such that

:<math>\varphi(x) \leq p(x)\qquad\forall x \in U</math>

<math display="block">\begin{align}

:<math>\psi(x) &= \varphi(x)\qquad &\forall x\in U,</math> \\

then there exists a linear extension <math>\psi:V\to\mathbb{R}</math> of <math>\varphi</math> to the whole space <math>V</math> which is [[dominate (mathematics)|dominated]] by <math>p</math> on <math>V</math>; that is, there exists a linear functional <math>\psi</math> such that

:<math>\psi(x) &\le p(x)\qquad &\forall x\in V.</math></blockquote>

\end{align}</math>}}

:<math>\psi(x)=\varphi(x)\qquad\forall x\in U,</math>

:<math>\psi(x) \le p(x)\qquad\forall x\in V.</math></blockquote>

===Open mapping theorem===

{{main|Open mapping theorem (functional analysis)}}

The [[open mapping theorem (functional analysis)|open mapping theorem]], also known as the Banach–Schauder theorem (named after [[Stefan Banach]] and [[Juliusz Schauder]]), is a fundamental result which states that if a [[Bounded linear operator|continuous linear operator]] between [[Banach space]]s is [[surjective]] then it is an [[open map]]. More precisely,:<ref name=rudin/>

:{{math theorem | name = '''Open mapping theorem.''' | math_statement = If <math>X</math> and <math>Y</math> are Banach spaces and <math>A:X\to Y</math> is a surjective continuous linear operator, then <math>A</math> is an open map (that is, if <math>U</math> is an [[open set]] in <math>X</math>, then <math>A(U)</math> is open in <math>Y</math>).}}

: '''Open mapping theorem.''' If <math>X</math> and <math>Y</math> are Banach spaces and <math>A:X\to Y</math> is a surjective continuous linear operator, then <math>A</math> is an open map (that is, if <math>U</math> is an [[open set]] in <math>X</math>, then <math>A(U)</math> is open in <math>Y</math>).

The proof uses the [[Baire category theorem]], and completeness of both <math>X</math> and <math>Y</math> is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a [[normed space]], but is true if <math>X</math> and <math>Y</math> are taken to be [[Fréchet space]]s.

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{{main|Closed graph theorem}}

The closed graph theorem states the following:

If <math>X</math> is a [[topological space]] and <math>Y</math> is a [[Compact space|compact]] [[Hausdorff space]], then the graph of a linear map <math>T</math> from <math>X</math> to <math>Y</math> is closed if and only if <math>T</math> is [[continuous function (topology)|continuous]].<ref>{{Cite book | last=Munkres | first=James R. | url={{google books |plainurl=y |id=XjoZAQAAIAAJ}} | title=Topology | date=2000 | publisher=Prentice Hall, Incorporated | isbn=978-0-13-181629-9 | language=en | page= 171}}</ref>

===Other topics===

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==References==

{{Reflist}}

==Further reading==

* Aliprantis, C.D., Border, K.C.: ''Infinite Dimensional Analysis: A Hitchhiker's Guide'', 3rd ed., Springer 2007, {{ISBN|978-3-540-32696-0}}. Online {{doi|10.1007/3-540-29587-9}} (by subscription)