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| | == Louis Vuitton Scarf Winter Online == |
| | footer = The graph of the [[cubic function|cubic]] function {{nowrap|1=''f''(''x'') = ''x''<sup>3</sup> − 9''x''}} on the interval <nowiki>[-4,4]</nowiki> is closed because the function is [[continuous function|continuous]]. The graph of the [[Heaviside function]] from <nowiki>[-2,2]</nowiki> is not closed, because the function is not continuous.
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| | image1 =cubicpoly.png
| | If you the contractor,[http://www.zetroz.com/images/content/info.asp?tid=231 Louis Vuitton Scarf Winter Online], This is ideal, right? Repeat business with no cost of the sale. Well, He shows up to give me the estimate,[http://www.zetroz.com/images/content/info.asp?tid=606 Cheap Louis Vuitton Scarf Tag Outlet], Let call him Gino, And Gino has it already written by the time I can secure my infant and get outside to talk to him. It a standard seal job with an additional piece of beefing up a section of the driveway that was sunken over the years. <br><br>This type of rivalry had not been seen in the NBA since the days of Chamberlain and Russell. This rivalry was one of heart,[http://www.zetroz.com/images/content/info.asp?tid=452 Louis Vuitton Rose Scarf], skill, Determination,[http://www.zetroz.com/images/content/info.asp?tid=295 Louis Vuitton Scarf Genuine On Sale], And beauty. These two players brought the NBA to the lime light once again with this rivalry. There are a good number of trusted and reputed companies which present discount electronics products at your doorsteps in the quickest possible moment through reliable postal products. The most beautiful feature of Chinese language wholesale electronics is you can get even a particular piece for wholesale price just in case of almost all electronics goods. The many items sold within wholesale electronics can be purchased with one 12 months warranty and cost free repairs or replacement in the damaged product is additionally available. |
| | alt1 = A cubic function
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| | image2 = Dirac distribution CDF.svg
| | == Louis Vuitton Scarf Mens On Sale == |
| | alt2 = The Heaviside function
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| In [[mathematics]], the '''closed graph theorem''' is a basic result which characterizes [[continuous function]]s in terms of their [[graph of a function|graph]]s. There are several versions of the theorem.
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| == The closed graph theorem ==
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| In mathematics, there are several results known as the "closed graph theorem".
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| For any function {{nowrap|''T'' : ''X'' → ''Y''}}, we define the ''graph'' of ''T'' to be the set
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| :<math>\lbrace (x,y) \in X\times Y \mid Tx=y\rbrace.</math>
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| In point-set topology, the closed graph theorem states the following:
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| If ''X'' is a [[topological space]] and ''Y'' is a [[Compact space|compact]] [[Hausdorff space]], then the graph of ''T'' is closed if and only if ''T'' is [[continuous function (topology)|continuous]].<ref>{{harvtxt|Munkres|2000}}, p. 171</ref>
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| The rest of the section concerns functional analysis, where the closed graph theorem states the following:
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| If ''X'' and ''Y'' are Banach spaces, and {{nowrap|''T'' : ''X'' → ''Y''}} is a [[linear operator]], then ''T'' is continuous if and only if its graph is [[closed set|closed]] in {{nowrap|''X'' × ''Y''}} (with the [[product topology]]).
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| In the latter case we say that ''T'' is a [[closed linear operator|closed operator]]. Note that the operator is required to be everywhere-defined, i.e., the [[domain (mathematics)|domain]] ''D(T)'' of ''T'' is ''X''. This condition is necessary, as there exist closed linear operators that are unbounded (not continuous); a prototypical example is provided by the derivative operator on ''C([0,1])'' (whose domain is a strict subset of ''C([0,1])'').
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| The usual proof of the closed graph theorem employs the [[open mapping theorem (functional analysis)|open mapping theorem]]. In fact, the closed graph theorem, the open mapping theorem and the [[bounded inverse theorem]] are all [[logically equivalent|equivalent]]. This equivalence also serves to demonstrate the necessity of ''X'' and ''Y'' being Banach; one can construct linear maps that have unbounded inverses in this setting, for example, by using either continuous functions with compact support or by using sequences with finitely many non-zero terms along with the supremum norm.
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| The closed graph theorem can be reformulated as follows. If {{nowrap|''T'' : ''X'' → ''Y''}} is a linear operator between Banach spaces, then the following are equivalent:
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| # For every sequence {''x''<sub>''n''</sub>} in ''X'', if the sequence {''x''<sub>''n''</sub>} converges in ''X'' to some element ''x'', then the sequence {''T''(''x''<sub>''n''</sub>)} in ''Y'' also converges, and its limit is ''T''(''x'').
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| # For every sequence {''x''<sub>''n''</sub>} in ''X'', if the sequence {''x''<sub>''n''</sub>} converges in ''X'' to some element ''x'' and the sequence {''T''(''x''<sub>''n''</sub>)} in ''Y'' converges to some element ''y'', then {{nowrap|''y'' {{=}} ''T''(''x'')}}.
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| == Generalization == | |
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| The closed graph theorem can be generalized to more abstract [[topological vector spaces]] in the following ways:
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| :A linear operator from a [[barrelled space]] ''X'' to a [[Fréchet space]] ''Y'' is [[continuous linear operator|continuous]] if and only if its graph is closed in the space ''X''×''Y'' equipped with the product topology.
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| and there is a version that does not require ''Y'' to be locally convex:<ref>{{harvtxt|Trèves|1995}}, p. 173</ref>
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| :If the graph of a linear map between two [[F-space]]s is closed then the map is continuous.
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| and an even more general version of the closed graph theorem is<ref>Treves p. 169</ref>
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| :Suppose that ''X'' and ''Y'' are two topological vector spaces (they need not be Hausdorff nor locally convex) with the following property: If ''G'' is any closed subspace of <math>X \times Y</math> and ''u'' is any continuous map of ''G'' onto ''X'', then ''u'' is an open mapping. Under this condition, if <math>f : X \to Y</math> is a linear map whose graph is closed then ''f'' is continuous.
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| === The Borel Graph Theorem === | |
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| The Borel graph theorem, proved by L. Schwartz,t shows that the closed graph theorem is valid for linear maps defined on and valued in most spaces encountered in Analysis.<ref>Treves p. 549</ref> Recall that a topological space is called a [[Polish space]] if it is a separable complete metrix space and that a [[Souslin space]] is the continuous image of a Polish space. The weak dual of a separable Frechet space and the strong dual of a separable Frechet-Montel space are Souslin spaces. Also, the space of distributions and all [[Lp-space]]s over open subsets of Euclidean space as well as many other spaces that occur in analysis are Souslin spaces. The Borel graph theorem sates:<ref>Treves p. 549</ref>
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| :Let ''X'' and ''Y'' be locally convex Hausdorff spaces and let <math>u : X \to Y</math> be linear. If ''X'' is the inductive limit of an arbitrary family of Banach spaces, if ''Y'' is a Souslin space, and if the graph of ''u'' is a Borel set in <math>X \times Y</math>, then ''u'' is continuous.
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| An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces. A topological space ''X'' is called a <math>K_{\sigma \delta}</math> if it is the countable intersection of countable unions of compact sets. A Hausdorff topological spae ''Y'' is called '''K-analytic''' if it is the continuous image of a <math>K_{\sigma \delta}</math> space (that is, if there is a <math>K_{\sigma \delta}</math> space ''X'' and a continuous map of ''X'' onto ''Y''). Every compact set is K-analytic so that there are non-separable K-analyitic spaces. Also, every Polish, Souslin, and reflexive Frechet space is K-analytic as is the weak dual of a Frechet space. The generalized theorems states:<ref>Treves pp. 557-558</ref>
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| :Let ''X'' and ''Y'' be locally convex Hausdorff spaces and let <math>u : X \to Y</math> be linear. If ''X'' is the inductive limit of an arbitrary family of Banach spaces, if ''Y'' is a K-analytic space, and if the graph of ''u'' is closed in <math>X \times Y</math>, then ''u'' is continuous.
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| ==See also== | |
| * [[Discontinuous linear map]]
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| * [[Webbed space]]
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| ==Notes==
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| {{reflist}}
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| == References ==
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| * {{citation | last = Munkres | first = James | authorlink = James Munkres | title=Topology | edition=2nd | publisher=[[Upper Saddle River: Prentice Hall]] | year=2000 | isbn=0-13-181629-2 | pages=163–172}}
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| * {{citation | last = Trèves | first = François | authorlink = François Trèves | title=Topological Vector Spaces, Distributions and Kernels | publisher=[[Academic Press, Inc.]] | year=1995 | isbn=0-486-45352-9 | pages=169–173, 549–557}}
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| * {{citation | last = Folland | first = Gerald B. | authorlink = Gerald Folland | title=Real Analysis: Modern Techniques and Their Applications | edition=1st | publisher=[[John Wiley & Sons]] | year = 1984 | isbn=978-0-471-80958-6 }}
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| * {{citation| first=Walter|last=Rudin|authorlink=Walter Rudin|title=Functional analysis|publisher=Tata MacGraw-Hill|year=1973}}.
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| * {{planetmath reference|id=6472|title=Proof of closed graph theorem }}
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| {{Functional Analysis}}
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| [[Category:Functional analysis]]
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| [[Category:Theorems in functional analysis]]
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