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	PlanetMath: Hahn-Banach theorem
		The Hahn-Banach theorem is a foundational result in functional analysis.
		We first consider an abstract version of this theorem, and then give the more classical result as a corollary.
		This is version 7 of Hahn-Banach theorem, born on 2002-08-01, modified 2003-04-13.
	planetmath.org /encyclopedia/HahnBanachTheorem.html (184 words)

	Amazon.com: "Hahn-Banach Theorem": Key Phrase page (Site not responding. Last check: 2007-10-08)
		However, the Hahn-Banach Theorem establishes properties for normed linear spaces in general and its proof depends...
		The Hahn-Banach Theorem 77 Radon-Nikodym Theorem can be interpreted as an identification (isometrically isomorphic) of L' (I p I) with {rl E M(X...
		By the Hahn-Banach Theorem, F has a continuous extension to ft. The Riesz representation theorem implies there is a vector i E H with...
	www.amazon.com /phrase/Hahn_Banach-Theorem (338 words)

	Hahn–Banach theorem - Wikipedia, the free encyclopedia
		In mathematics, the Hahn–Banach theorem is a central tool in functional analysis.
		It allows one to extend linear operators defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space interesting.
		It is named for Hans Hahn and Stefan Banach who proved this theorem independently in the 1920s.
	en.wikipedia.org /wiki/Hahn-Banach_theorem (409 words)

	Learn more about Functional analysis in the online encyclopedia. (Site not responding. Last check: 2007-10-08)
		In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear functionals.
		The notion of derivative is extended to arbitrary functions between Banach spaces; it turns out that the derivative of a function at a certain point is really a continuous linear map.
		The Hahn-Banach theorem is about extending functionals from a subspace to the full space, in a norm-preserving fashion.
	www.onlineencyclopedia.org /f/fu/functional_analysis.html (579 words)

	Hahn-Banach Theorem (Site not responding. Last check: 2007-10-08)
		AMCA: Pretopologies and geometric Hahn-Banach theorem by Josep Rubio-Massegu...
		AMCA: The Hahn-Banach theorem and maximal monotonicity by Stephen Simons...
		Five Theorems in the theory of Riesz spaces...
	www.scienceoxygen.com /math/630.html (116 words)

	Continuous linear extension - Wikipedia, the free encyclopedia
		In functional analysis, it is often convenient to define something on a normed vector space by defining it on a dense set and extending it to the whole space.
		This theorem is sometimes called the BLT theorem, where BLT stands for bounded linear transformation.
		If V is not dense in X, then the Hahn-Banach theorem may sometimes be used to show that an extension exists.
	en.wikipedia.org /wiki/Continuous_linear_extension (385 words)

	Search Results for Banach
		Banach's father had never given his son much support, but now once he left school he quite openly told Banach that he was now on his own.
		There is the Hahn-Banach theorem on the extension of continuous linear functionals, the Banach-Steinhaus theorem on bounded families of mappings, the Banach-Alaoglu theorem, the Banach fixed point theorem and the Banach-Tarski paradoxical decomposition of a ball.
		Banach was by this time on the staff at Lvov and the school rapidly grew in importance.
	www-history.mcs.st-and.ac.uk /Search/historysearch.cgi?SUGGESTION=Banach&CONTEXT=1 (4583 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)

	[No title]
		I have some confusion relating the geometric form and the extension form of the Hahn Banach theorem.
		The extension form basically says that given any bounded linear functional in a subspace M of a vector space, X then it is possible to extend the functional to the entire space without blowing up the norm.
		The version in Consequences of the Axiom of Choice (287) is the following: Let V be a separable normed linear space, and p (from V to R) be a subadditive and positively homogeneous map.
	www.math.niu.edu /~rusin/known-math/99/hahn-banach2 (1089 words)

	Functional Analysis
		Emphasis on common Banach spaces of k-times continuously differentiable functions.
		[ Hahn-Banach theorems ] [pdf] These are the general versions for (real or complex} locally convex topological vectorspaces.
		[ A 'good spectral theorem'] [ pdf ] This detailed outline sets up basic ideas about von Neumann algebras, direct integrals of Hilbert spaces, recovering more traditional (less illuminating) things like `resolutions of the identity' as artifacts.
	www.math.umn.edu /~garrett/m/fun (545 words)

	1.4.3 Functional Analysis -- Dr Belton -- 16 MT (Site not responding. Last check: 2007-10-08)
		The first involves the fundamental theorems of linear operators on Banach spaces, the open mapping theorem, the closed graph theorem and the principle of uniform boundedness, together with the Hahn-Banach theorem in its full glory.
		The second part of the course is an introduction to the theory of Banach algebras.
		Normed algebras and Banach algebras; subalgebra and ideals; quotient algebras.
	www.maths.ox.ac.uk /current-students/undergraduates/handbooks-synopses/2003/html/sect-c-03/node18.html (159 words)

	Reversal via Hahn-Banach
		Corollary 4.5 The extended Hahn-Banach theorem, EHB, is provable in
		Corollary 4.7 The Hahn-Banach theorem, HB, is provable in
		This result may be compared to our Theorem 4.4.
	www.math.psu.edu /simpson/papers/sep-l2h/node4.html (381 words)

	Hilbert's Program
		Gödel had found the same result already independently: the second incompleteness theorem, asserting that the system of Principia does not prove the formalization of the claim that the system of Principia is consistent (provided it is).
		Detlefsen presents several lines of defense, one of which is similar to the one just described: arguing that a version of the ω-rule is finitarily acceptable, although not capable of formalization (however, see Ignjatovic 1994).
		Reverse mathematics seeks to give a precise answer to this question by investigating which theorems of classical mathematics are provable in weak subsystems of analysis which are reducible to finitary mathematics (in the sense discussed in the preceding paragraph).
	plato.stanford.edu /entries/hilbert-program (7534 words)

	The Hahn-Banach Theorem for Real Vector Spaces - Bauer (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Abstract: The Hahn-Banach Theorem is one of the most fundamental results in functional analysis.
		We present a fully formal proof of two versions of the theorem, one for general linear spaces and another for normed spaces.
		6 Hahn-Banach theorem - Nowak, Trybulec - 1993
	citeseer.ist.psu.edu /330681.html (365 words)

	Formal Topology
		The beginning of formal (or point-free topology) can be traced back to Whitehead (1915: revue de Metaphysique, ``Theorie relationnelle de l'espace''), and a survey of the state of the art in 1940 was written by Menger.
		There is another older algebraic tradition: Dedekind and Weber (1882) analysed a purely algebraic presentation of Riemann surfaces where the notion of points is derived (valuation on a field; I have yet to understand the connections with Stone's representation theorem).
		The published constructions of the generic site model of a geometric theory are not optimal, since they use logic in the construction of the model.
	www.cs.chalmers.se /~coquand/formal.html (661 words)

	question.html
		Among the various definitions which are equivalent (in set-theory with the axiom of choice) to super-reflexivity, there is one which is: "not to satisfy the finite tree property".
		This theorem is provable in ZF In set theory with the axiom of choice, it can be proved that uniformly smooth spaces do not satisfy the finite tree property: the classical proof (see [Bea85]) relies on a proof by contradiction, and the use of envelopes of Banach spaces (quotients of ultraproducts of Banach spaces).
		Is there a computable mapping f such that for every finite dimensional normed space E, if t is a witness of the fact that E does not satisfy the finite tree property, then f(t) is a witness of the fact that the continuous dual E' does not satisfy the finite tree property.
	www2.univ-reunion.fr /~mar/question.html (1466 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		define linear operators, their spectrum (with matrices serving as examples) and their spectral radius, derive spectral radius theorem, understand the definitions of self-adjoint, isometric, unitary and normal operators on Hilbert spaces and their spectra, be able to apply these ideas to matrices, understand integral operators as examples of linear operators.
		Linear operators, invertibility, spectrum, spectral radius, spectral radius theorem, self-adjoint, isometric, unitary and normal operators on Hilbert spaces and their spectra, applications to matrices, integral operators.
		Models, diagrams, preservation theorems, Löwenheim-Skølem theorems, partial isomorphisms, elimination of quantifiers, countable categoricity, types, key examples which will be explored and used for illustration include: the random graph; the rationals as a partially ordered set; the reals as an ordered field.
	www.ma.man.ac.uk /ug/OldFile_s4sem2.htm (4551 words)

	The Hahn Banach Theorem
		This is pretty complicated, and not used very often, so if this isn't scaffolding for a larger theorem, you might want to skip ahead.
		This is the Hahn (biography) Banach (biography) theorem.
		This may close the gap, but it is still possible to select e, and extend f to all of s.
	www.mathreference.com /top-ban,hahn.html (1109 words)

	A Direct Proof of the Localic Hahn-Banach Theorem (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Abstract: Using the notion of entailment relation [Sco74] we give a direct proof of the localic version of Hahn-Banach's theorem.
		0.2: A constructive topological proof of van der Waerden's theorem - Coquand (1993)
		13 A globalization of the hahn-banach theorem (context) - Mulvey, Pelletier - 1991
	citeseer.ist.psu.edu /coquand99direct.html (255 words)

	Proceedings of the American Mathematical Society
		We prove that all integral polynomials over a Banach space are extendible.
		C. Gupta, Malgrange theorem for nuclearly entire functions of bounded type on a Banach space, Ph.D.Thesis, I.M.P.A., Rio de Janeiro and University of Rochester (1966).
		J. Jaramillo, A. Prieto, and I. Zalduendo, The bidual of the space of polynomials on a Banach space, Math.
	www.ams.org /proc/1999-127-01/S0002-9939-99-04485-8/home.html (507 words)

	Atlas: Pretopologies and geometric Hahn-Banach theorem by Josep Rubio-Massegu (Site not responding. Last check: 2007-10-08)
		The geometric Hahn-Banach theorem is studied for different pretopologies.
		However, the theorem does not hold in general for pretopologies which are finer than the algebraic one.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caey-78.
	atlas-conferences.com /cgi-bin/abstract/caey-78 (183 words)

	Fall Semester 2004, Math 362A (Site not responding. Last check: 2007-10-08)
		Syllabus: The course will start off with a discussion of the basic principles of functional analysis such as the Hahn-Banach theorem, the open mapping theorem, the closed graph theorem and the uniform boundedness principle.
		These are fundamental theorems which are used in various areas of mathematics such as applied analysis, representation theory, operator theory, operator algebras and noncommutative geometry.
		If time permits, we will present the spectral theorem for normal operators on Hilbert space, which leads to the functional calculus, an important tool in operator theory and operator algebras.
	math.vanderbilt.edu /~bisch/362A_fall04.html (336 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Calendar Description: Hilbert and Banach spaces; applications to Fourier series and numerical analysis.
		Objectives: This course is an introduction to functional analysis and is designed for the advanced undergraduate or beginning graduate mathematics student as well as physics and engineering majors who are finding increasing applications of functional analysis in their disciplines.
		Usually students in the course are evaluated on the basis of assignments, one in-class term test and a final examination.
	www.mathstat.uoguelph.ca /courses/descriptions/63422.html (195 words)

	MT5815 (Site not responding. Last check: 2007-10-08)
		The module will familiarise students with the basic notions of functional analysis, that is analysis on (typically infinite dimensional) normed spaces and Hilbert space.
		understand the statement of the spectral theorem and be able to apply it to simple situations.
		Construction of new Banach spaces from old ones including product spaces and quotient spaces.
	www-maths.mcs.st-andrews.ac.uk /ug/hon5/MT5815.shtml (201 words)

	sem11_7 (Site not responding. Last check: 2007-10-08)
		ABSTRACT: The well-known Hahn-Banach theorem can be stated in the following way.
		If T is an operator of rank 1 from X to Y and Z is a Banach space containing X as a proper subspace, then there exists a norm-preseving extension of T to Z.
		The talk is devoted to generalizations of this theorem for operators of higher rank.
	arts-sciences.cua.edu /math/MIO/seminars/sem11_7.htm (93 words)

	Simon Eveson: Graduate Course in Analysis (Site not responding. Last check: 2007-10-08)
		This course is in four parts, run jointly between York and Leeds, and is aimed at first year graduate students, although of course anyone else is also welcome to attend.
		Weeks 1-3 in York Simon Eveson A brief introduction to Banach spaces, their duals, and L
		Literature: Douglas Banach Algebra Techniques in Operator Theory, Nikolski Treatise on the Shift Operator.
	www-users.york.ac.uk /~spe1/analysis/course.html (237 words)

	Atlas: The Hahn-Banach theorem with infinite values by Pere Rubio-Diaz (Site not responding. Last check: 2007-10-08)
		We obtain an extension of the Hahn-Banach theorem that permits to work with pretopologies in a vector space E in which the neighbourhoods need not be absorbent in E. In the present case the seminorms must be replaced by sublinear functionals which might take the values +\infty or -\infty.
		We present three further generalizations of the Hahn-Banach theorem for real vector spaces.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caey-77.
	atlas-conferences.com /cgi-bin/abstract/caey-77 (150 words)

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