Editing Functional analysis (section)

==Major and foundational results==

There are four major theorems which are sometimes called the four pillars of functional analysis: 

* the [[Hahn–Banach theorem]]
* the [[Open mapping theorem (functional analysis)|open mapping theorem]]
* the [[Closed graph theorem (functional analysis)|closed graph theorem]]
* the [[uniform boundedness principle]], also known as the [[Banach–Steinhaus theorem]]. 

Important results of functional analysis include:

===Uniform boundedness principle===
{{main|Banach-Steinhaus theorem}}
The [[uniform boundedness principle]] or [[Banach–Steinhaus theorem]] is one of the fundamental results in functional analysis. Together with the [[Hahn–Banach theorem]] and the [[open mapping theorem (functional analysis)|open mapping theorem]], it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of [[continuous linear operator]]s (and thus bounded operators) whose domain is a [[Banach space]], pointwise boundedness is equivalent to uniform boundedness in operator norm.

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]].

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

===Spectral theorem===
{{main|Spectral theorem}}
There are many theorems known as the [[spectral theorem]], but one in particular has many applications in functional analysis.

{{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>
where ''T'' is the [[multiplication operator]]:
<math display="block"> [T \varphi](x) = f(x) \varphi(x). </math>
and <math>\|T\| = \|f\|_\infty</math>.}}

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

There is also an analogous spectral theorem for bounded [[normal operator]]s on Hilbert spaces. The only difference in the conclusion is that now <math>f</math> may be complex-valued.

===Hahn–Banach theorem===
{{main|Hahn–Banach theorem}}
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".

{{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 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 dominated by <math>p</math> on <math>V</math>; that is, there exists a linear functional <math>\psi</math> such that
<math display="block">\begin{align}
\psi(x) &= \varphi(x) &\forall x\in U, \\
\psi(x) &\le p(x) &\forall x\in V.
\end{align}</math>}}

===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>).}}

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.

===Closed graph theorem===
{{main|Closed graph theorem}}
{{math theorem | name = Closed graph theorem | math_statement = 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===
{{main|List of functional analysis topics}}