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Topic: Closed graph theorem

	Closed graph theorem - Wikipedia, the free encyclopedia
		In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph.
		The restriction on the domain is needed due to the existence of closed unbounded linear operators.
		The usual proof of the closed graph theorem employs the open mapping theorem.
	en.wikipedia.org /wiki/Closed_graph_theorem (272 words)

	Graph of a function - Wikipedia, the free encyclopedia
		In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc. Graphing on a Cartesian plane is sometimes referred to as curve sketching.
		For general functions, the graphic representation cannot be applied and the formal definition of the graph of a function suits the need of mathematical statements, e.g., the closed graph theorem in functional analysis.
		The concept of the graph of a function is generalised to the graph of a relation.
	en.wikipedia.org /wiki/Graph_of_a_function (268 words)

	Closed graph theorem: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-06)
		Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions....
		In topology and related branches of mathematics, a closed set is a set whose complement is open setopen....
		The banach-alaoglu theorem (also known as alaoglus theorem) states that the closed ball (mathematics)unit ball of the dual space of a normed vector...
	www.absoluteastronomy.com /encyclopedia/c/cl/closed_graph_theorem.htm (905 words)

	Function (mathematics) - Wikipedia, the free encyclopedia
		The graph of a function f is the set of all ordered pairs (x, f(x)), for all x in the domain X.
		If X and Y are subsets of R, the real numbers, then this definition coincides with the familiar sense of "graph" as a picture or plot of the function, with the ordered pairs being the Cartesian coordinates of points.
		Sometimes a function can be modified, often by replacing the domain with a subset of the domain, and making corresponding changes in the codomain and graph, so that the modified function has an inverse that is a function.
	en.wikipedia.org /wiki/Function_(mathematics) (3304 words)

	An Introduction to Banach Space Theory
		The Eberlein-Smulian theorem is obtained in this section, as is the result due to Krein and Smulian that the closed convex hull of a weakly compact subset of a Banach space is itself weakly compact.
		Schauder's theorem relating the compactness of a bounded linear operator to that of its adjoint is presented, as is the characterization of operator compactness in terms of the bounded-weak*-to-norm continuity of the adjoint.
		Gantmacher's theorem is obtained, as well as the equivalence of the weak compactness of a bounded linear operator to the weak*-to-weak continuity of its adjoint.
	www.math.lsa.umich.edu /~meggin/ibst.html (2875 words)

	PlanetMath: proof of closed graph theorem
		We remark that these projections are continuous, by definition of the product of Banach spaces.
		"proof of closed graph theorem" is owned by Koro.
		This is version 2 of proof of closed graph theorem, born on 2004-11-12, modified 2004-11-12.
	planetmath.org /encyclopedia/ProofOfClosedGraphTheorem.html (128 words)

	Learn more about Functional analysis in the online encyclopedia. (Site not responding. Last check: 2007-11-06)
		The spectral theorem gives an integral formula for normal operators on a Hilbert space.
		The Hahn-Banach theorem is about extending functionals from a subspace to the full space, in a norm-preserving fashion.
		The open mapping theorem and closed graph theorem.
	www.onlineencyclopedia.org /f/fu/functional_analysis.html (579 words)

	E. I. Smirnov (Site not responding. Last check: 2007-11-06)
		The closed graph theorem is one of the most important results of functional analysis.
		The development of the descriptive theory of sets, functional analysis, theory of structures and topological methods in measure theory led to the appearance of constructively complex objects (analytical and projective sets, pseudotopology, the L. Schwartz spaces of test functions and distributions, M. De Wilde spaces etc.), which found various fruitful applications.
		Morever A. Grothendieck's problem about the construction of the classes of locally convex spaces possessing the closed graph theorem and broad properties of permanence found positive solution in the works of L. Schwartz, W. Slowikowski, D. Raikov, M. De Wilde, P.P. Zabreiko, E.I. Smirnov and many others ([6], [4], [11], [12]).
	www.utm.edu /staff/jschomme/topology/c/a/a/j/132.htm (459 words)

	AMCA: Hausdorff Spectra and the Closed Graph Theorem by E. I. Smirnov (Site not responding. Last check: 2007-11-06)
		J the closed graph theorem is true for linear operators acting from X to Y. The class J is closed relative to the operations of forming products and projective and inductive limits of countably many spaces.
		Morever A. Grothendieck's problem about the construction of the classes of locally convex spaces possessing the closed graph theorem and broad properties of permanence found positive solution in the works of L. Schwartz, W. Sowikowski, D. Raikov, M.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/a/j/32.htm (389 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		To close the blowout preventers of a well to control the entry of water, gas, oil, or other formation fluids into the wellbore during a drilling operation.
		The event that occurs when two planets or other celestial bodies are nearest to each other as they orbit about the sun or other primary.
		] A plug used to close openings of various components of a round of ammunition, that is, primer or nose of an unfused projectile.
	www.accessscience.com /Dictionary/C/C30/DictC30.html (2679 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Since f is continuous, its graph is closed.
		If K is closed and X is an F-space, then Q is an F-space, when equipped with the quotient F-norm given by q
		To prove p is open, refer to 15.31.f: The p-saturation of an open set G is equal to G+K. That's the union of the open sets G+k (for k in K), so it is open.
	www.math.vanderbilt.edu /~schectex/ccc/addenda/openmap.html (318 words)

	Open mapping theorem - (Site not responding. Last check: 2007-11-06)
		In functional analysis, the open mapping theorem, also known as the Banach-Schauder theorem, is a fundamental result which states: if A : X → Y is a surjective continuous linear operator between Banach spaces X and Y, then A is an open map (i.e.
		→ y it follows that y = 0, then A is continuous (Closed graph theorem).
		In complex analysis, the open mapping theorem states that if U is a connected open subset of the complex plane C and f : U → C is a non-constant holomorphic function, then f is an open map (i.e.
	psychcentral.com /psypsych/Open_mapping_theorem (287 words)

	Continuity and All That
		The usefulness of this theorem can hardly be overstated - but the simplest, in our case, is that in Euclidian space, if a set X is closed and bounded, then we can claim it is compact.
		Theorem: (Compact Range of Continuous Functions) X is a compact metric space, Y is a metric space and let
		There is indeed a closed graph, but it is not u.s.c.
	cepa.newschool.edu /het/essays/math/contin.htm (1663 words)

	Closed Graph Theorem (Site not responding. Last check: 2007-11-06)
		LCGMiner: Levelwise Closed Graph Pattern Mining from Large Databases...
		Closed graph theorem article - Closed graph theorem mathematics functional analy...
		AMCA: Hausdorff Spectra and the Closed Graph Theorem by E. Smirnov...
	www.scienceoxygen.com /math/637.html (118 words)

	PlanetMath: closed graph theorem
		is continuous if and only if its graph is a closed
		Cross-references: product topology, subset, closed, continuous, Banach spaces, between, linear mapping
		This is version 4 of closed graph theorem, born on 2002-12-09, modified 2003-07-27.
	planetmath.org /encyclopedia/ClosedGraphTheorem.html (60 words)

	[No title]
		Indeed, a theorem of Kolmogorov shows that a topological vector space is normable if and only if it is Hausdorff, locally convex, and locally bounded.
		This is a corollary of the Closed Graph Theorem (which is true for F-norms, not just for norms).
		Indeed, Wright proved (in a paper published in 1977) this variant of the Closed Graph Theorem: if we replace conventional ZF + AC with the alternative ZF + DC + BP, then any linear map from an F-space into a Hausdorff topological vector space is continuous.
	www.math.niu.edu /~rusin/known-math/94/func_space (1281 words)

	S. Poljak (Site not responding. Last check: 2007-11-06)
		The result is obtained by the study of coloring iterated adjoints of a digraph by iterated antichains of a poset.
		In this paper, we determine which closed neighborhoods $B$ of zero in a real locally convex space $E$ of dimension at least 3 have the property that for every 2 dimensional subspace $F$ there is a continuous linear projection $P$ onto $F$ with $P(B)\subseteq B$.
		Abstract:Filling a possible gap in the literature, we give a complete and readable proof of this trace theorem, which also shows that the imbedding constant is uniformly bounded for $T \downarrow 0$.
	www.maths.tcd.ie /EMIS/journals/CMUC/cmuc9102/abs/smazat.htm (1402 words)

	Vasco Brattka's Papers (Site not responding. Last check: 2007-11-06)
		We investigate the computable content of certain theorems which are sometimes called the ``principles'' of the theory of Banach spaces.
		Among these the main theorems are the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem.
		From the computational point of view these theorems are interesting, since their classical proofs rely more or less on the Baire Category Theorem and therefore they count as ``non-constructive''.
	www.informatik.fernuni-hagen.de /thi1/vasco.brattka/publications/banach.html (292 words)

	Diary for Math 507:01, spring 2004
		We prepared for Runge's Theorem by deriving some simple consequences of H-B, such as existence of enough continuous linear functionals to separate points in a normed vector space.
		We worked on the H-B theorem with an upper bound that is sublinear, a bit weaker than a seminorm.
		I began by discussing the statement of Runge's Theorem, and got the existence of an almost paradoxical sequence of pointwise convergent polynomials.
	www.math.rutgers.edu /~greenfie/mill_courses/math507/diary.html (4537 words)

	ipedia.com: Functional analysis Article (Site not responding. Last check: 2007-11-06)
		One spectral theorem (there are more of them) gives an integral formula for normal operators on a Hilbert space.
		Many very important theorems require the Hahn-Banach theorem which itself requires Zorn's lemma in case of an infinite dimensional space.
		To better understand the long-term evolution of mathematics, one should have a look at how the acceptance of the axiom of choice, being equivalent to Zorn's lemma, has changed during time.
	www.ipedia.com /functional_analysis.html (647 words)

	Department of Mathematics, University of Strathclyde
		Closed Operators: Graphs and closed operators; Closed Graph Theorem and Bounded Inverse Theorem.
		Inverse of an injective closed operator is closed.
		Statement of the Spectral Theorem on infinite-dimensional spaces.
	www.maths.strath.ac.uk /ungrad/classes/921.htm (267 words)

	math lessons - Hellinger-Toeplitz theorem
		In functional analysis, a branch of mathematics, the Hellinger-Toeplitz theorem states that an everywhere defined symmetric operator on a Hilbert space is bounded.
		It is named for Ernst David Hellinger and Otto Toeplitz.
		The Hellinger-Toeplitz theorem leads to some technical difficulties in the mathematical formulation of quantum mechanics.
	www.mathdaily.com /lessons/Hellinger-Toeplitz_theorem (212 words)

	Function Mathematics (Site not responding. Last check: 2007-11-06)
		The graph of a function f is the set of all ordered pairs(x, f(x)), for all x in the domain X.
		There are theorems formulated or proved most easily in terms of the graph, such as the closed graph theorem.
		In the context of category theory, a function no longer represents a rule for taking an input to an output, but instead represents a relationship between its domain and its codomain.
	www.gay-village.wikiverse.org /function-mathematics (2942 words)

	Graduate Course Descriptions
		Isomorphism theorems, group actions, Jordan-Hölder theorem, Sylow theorems, direct and semidirect products, finitely generated abelian groups, simple groups, symmetric groups, linear groups, nilpotent and solvable groups, generators and relations.
		The fact that e^{\pi\sqrt{163}} is so close to an integer has a beautiful explanation that involves class field theory and theory of elliptic curves with complex multiplication.
		In particular, it involves the concept of Hilbert class field, the maximal unramified abelian expansion whose Galois group is the class group.
	www.math.toronto.edu /graduate/courses/descriptions.html (4148 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Complex Analysis (mostly Sarnak) State and prove the Riemann Mapping Theorem (I used the standard normal family proof in Ahlfors).
		He didn't want much detail, but apparently someone claimed to have solved Goldbach last week, and he wanted me to understand why he was skeptical.
		A little discussion of how it's actually the discrete version of the gamma function, which I didn't know (neither did Nelson, it seemed, from his reaction).
	www.math.princeton.edu /graduate/generals/vanderkam_jeffrey (372 words)

	Open Mapping Theorem (Site not responding. Last check: 2007-11-06)
		AMCA: An Open Mapping Theorem for Basis Separating Maps by Lawrence Narici...
		Open Mappings of Probability Measures and the Skorokhod Representation Theorem...
		Open mapping theorem article - Open mapping theorem mathematics theorems functio...
	www.scienceoxygen.com /math/467.html (98 words)

	[No title]
		We will spend another part of the course discussing the theory of operators on Hilbert space, including the spectral theorem and the rudiments of C*-algebra theory; and a third discussing distribution theory.
		I hope to show how the seemingly abstract techniques of functional analysis can be brought to bear on concrete problems such as those arising from linear partial differential equations.
		I will assume some familiarity with this material (not total mastery!) Detailed lecture notes and outlines from the course given by Professor Arnold (Spring 1997) are available on the Penn State MathNet.
	www.math.psu.edu /roe/503/home.html (436 words)

	msg Linear_algebra In linear algebra linear algebra a scalar scalar lambda...
		In functional analysis functional analysis, the spectrum spectrum of a bounded linear operator "A" on a Banach space Banach space is the set of scalar ν such that νI-"A" does not have a bounded two-sided inverse.
		Note that by the closed graph theorem closed graph theorem, if a bounded operator has an inverse, the inverse is necessarily bounded.
		If the underlying Banach space is finite dimensional, then the spectrum of "A" is the same of the set of eigenvalues of "A".
	www.biodatabase.de /Eigenvalue (465 words)

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