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Topic: Open mapping theorem

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	Open mapping theorem - Wikipedia, the free encyclopedia
		In mathematics, there are two theorems with the name "open mapping theorem".
		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.
		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.
	en.wikipedia.org /wiki/Open_mapping_theorem (241 words)

	PlanetMath: open mapping theorem (Site not responding. Last check: 2007-11-01)
		There are two important theorems having this name.
		Every surjective continuous linear mapping between two banach spaces is an open mapping.
		This is version 10 of open mapping theorem, born on 2002-12-07, modified 2004-02-23.
	planetmath.org /encyclopedia/OpenMappingTheorem.html (106 words)

	Riemann mapping theorem - Wikipedia, the free encyclopedia
		The theorem was stated (under the assumption that the boundary of U is piecewise smooth) by Bernhard Riemann in 1851 in his PhD thesis.
		The Riemann mapping theorem is the easiest way to prove that any two simply connected domains in the plane are homeomorphic.
		The Riemann mapping theorem can be generalized to the context of Riemann surfaces: If U is a simply-connected open subset of a Riemann surface, then U is biholomorphic to one of the following: the Riemann sphere, the complex plane or the open unit disk.
	en.wikipedia.org /wiki/Riemann_mapping_theorem (1040 words)

	PlanetMath: Baire category theorem (Site not responding. Last check: 2007-11-01)
		The Baire category theorem is often stated as “no non-empty complete metric space is of first category”, or, trivially, as “a non-empty, complete metric space is of second category”.
		In functional analysis, this important property of complete metric spaces forms the basis for the proofs of the important principles of Banach spaces: the open mapping theorem and the closed graph theorem.
		This is version 10 of Baire category theorem, born on 2002-06-04, modified 2004-09-29.
	planetmath.org /encyclopedia/BaireCategoryTheorem.html (428 words)

	Encyclopedia :: encyclopedia : Functional analysis (Site not responding. Last check: 2007-11-01)
		One of the open problems in functional analysis is to prove that every operator on a Hilbert space has a proper subspace which is invariant.
		One spectral theorem (there are more of them) 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.
	www.hallencyclopedia.com /Functional_analysis (712 words)

	Baire category theorem - Wikipedia, the free encyclopedia
		The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.
		Relation to AC The proofs of BCT1 and BCT2 require some form of the axiom of choice; and in fact the statement that every complete pseudometric space is a Baire space is logically equivalent to a weaker version of the axiom of choice called the axiom of dependent choice.
		BCT1 is used to prove the open mapping theorem, the closed graph theorem and the uniform boundedness principle.
	en.wikipedia.org /wiki/Baire_category_theorem (384 words)

	Axiom of choice - Wikipedia, the free encyclopedia
		A third possibility is to prove theorems using neither the axiom of choice nor its negation, a tactic preferred in constructive mathematics.
		Important theorems equivalent to AC There are a remarkable number of important statements that, assuming the axioms of ZF but neither AC nor ¬AC, are equivalent to the axiom of choice.
		The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.
	en.wikipedia.org /wiki/Axiom_of_choice (3539 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		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.
		Verify that g is a continuous linear bijection from Q onto Y; hence it is an isomorphism of topologies; hence it is an open map.
		Hence the composition f is an open map.
	www.math.vanderbilt.edu /~schectex/ccc/addenda/openmap.html (318 words)

	Baire category theorem
		In mathematics, the Baire category theorem is an important tool in the study of complete spaces, such as Banach spaces and Hilbert spaces, that arise in topology and functional analysis.
		The proof of the Baire category theorem uses the axiom of choice; in fact, the Baire category theorem is logically equivalent to a weaker version of the axiom of choice called the axiom of dependent choice[?].
		The Baire category theorem is used in the proof of the open mapping theorem.
	www.findword.org /ba/baire-category-theorem.html (452 words)

	PlanetMath: (Site not responding. Last check: 2007-11-01)
		open (in classification of topological properties according to behaviour under mapping) owned by rspuzio
		open and closed intervals have the same cardinality owned by mps
		open unit disk (in unit disc) owned by drini
	planetmath.org /encyclopedia/O (929 words)

	University of Manitoba: Mathematics -- Abstract (Site not responding. Last check: 2007-11-01)
		The Open Mapping Theorem says that a bounded linear transformation from one Banach space onto another must be an open mapping, while the Tietze Extension Theorem says that a bounded continuous function can always be extended from a closed subset of a normal space to the entire space.
		The two theorems sound as though they are unrelated, but the central argument in the standard proofs of these two theorems are in fact quite similar.
		Distinguishing this common point, we first state an approximation lemma and then present a proof of the Tietze Extension Theorem which is based on the lemma.
	www.umanitoba.ca /faculties/science/mathematics/new/seminars/html/Feb231233192004.html (109 words)

	Mathematics 2KF Complex Analysis 2000: Status of week 12 (Site not responding. Last check: 2007-11-01)
		Then I proved a corollary, stating the the maximum is attained on the boundary if the analytic function in the bounded region D is continuous on the closure of D. Remark the differences when we compare with max.
		Statement of the Open Mapping Theorem: there is a misprint in the text book: D should be a region (and then f(D) is open and connected).
		I shall start by showing that if f_n converges to f uniformly on compact subsets of an open set D and if f_n is analytic, then so is f.
	www.math.ku.dk /~henrikp/mat2kf2000/status12.html (279 words)

	Math 240 Home Page (Driver, 03-04)
		Students are assumed to have taken at the very least a two-quarter sequence in undergraduate real analysis covering in a rigorous manner the theory of limits, continuity and the like in Euclidean spaces and general metric spaces.
		The theorems of Heine-Borel (compactness in Euclidean spaces), Bolzano-Weierstrass (existence of convergent subsequences), the theory of uniform convergence, Riemann integration theory should have been covered.
		Differentiation of measures on R^n and the fundamental theorem of calculus.
	math.ucsd.edu /~driver/240A-C-03-04 (609 words)

	European School on Complex Analysis
		This course is organized by the Universities of Coimbra and Aveiro with the same goals as the ones organized under the programme Socrates, and is open to all young mathematicians interested in Complex Analysis and its applications.
		One the advantages of this approach is that the Nevanlinna functions appear as elements of a transfer matrix and the Padé approximants are the resolvents of a finite matrix approximation to a infinite Jacobi matrix.
		Riemann Mapping Theorem is one of the fundamental results of single variable complex analysis.
	www.mat.uc.pt /~ajplb/7.htm (616 words)

	Diary for Math 507:01, spring 2004
		This mapping is an algebra homomorphism with kernel 0, and maps z to T. The resulting functional calculus has many uses.
		A map in L(H) is compact iff it is the limit of maps with finite dimensional range.
		The collection of continuous linear maps from a normed vector space to a Banach space is a Banach space with the indicated norm (some trouble showing that it is complete!).
	www.math.rutgers.edu /~greenfie/mill_courses/math507/diary.html (4537 words)

	Springer Online Reference Works
		The conditions of the open-mapping theorem are satisfied, for example, by every non-zero continuous linear functional defined on a real (complex) Banach space
		The closed-graph theorem can be considered alongside with the open-mapping theorem.
		A recent comprehensive study of the closed-graph theorem can be found in [a1].
	eom.springer.de /o/o068300.htm (139 words)

	Vasco Brattka's Papers (Site not responding. Last check: 2007-11-01)
		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)

	Graduate Course Descriptions
		These notions will be explained by examining simple concrete examples of dynamical systems such as translations and automorphisms of tori, expanding maps of the interval, topological Markov chains, etc. Fundamental theorems of ergodic theory such as the Poincare recurrence theorem, the Von Neumann mean ergodic theorem and the Birkhoff ergodic theorem will be presented.
		This course is intended to introduce students to a recent technique in Algebraic Geometry based on application of the moment map and toric degenerations.
		One of the simplest examples of the moment map is the logarithm map that takes a point of the complex torus C*^n to the point in R
	www.math.toronto.edu /graduate/courses/descriptions.html (4381 words)

	Learn more about Functional analysis in the online encyclopedia. (Site not responding. Last check: 2007-11-01)
		Since finite-dimensional Hilbert spaces are fully understood in linear algebra, and since morphisms of Hilbert spaces can always be divided into morphisms of spaces with Aleph Null dimensionality, functional analysis of Hilbert spaces mostly deals with the unique Hilbert space of dimensionality Aleph Null, and its morphisms.
		The spectral theorem gives an integral formula for normal operators on a Hilbert space.
		Site Map 2 3 4 5 6 7
	www.onlineencyclopedia.org /f/fu/functional_analysis.html (579 words)

	Syllabus Graduate Courses, Math, TIFR (Site not responding. Last check: 2007-11-01)
		Proper maps; quotient space construction; examples of spheres, real and complex projective spaces, Grassmannians; normal and Hausdorff spaces; paracompact spaces; topological groups and continuous actions; classical groups*.
		Topological vector spaces; Banach spaces; Hilbert spaces; Hahn Banach theorem; open mapping theorem; uniform boundedness principle; bounded linear transformation; linear functionals and dual spaces.
		Basic properties of holomorphic functions; open mapping theorem; maximum modulus theorem; zeros of holomorphic functions, Weierstrass factorisation theorem Riemann mapping theorem; meromorphic functions; essential singularities; Picard's theorem.
	www.math.tifr.res.in /academic/gs/syllabus.html (407 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)

	University of Windsor, Windsor, Ontario Canada
		Engel's Theorem, Lie's Theorem, criterion for semi simplicity, root space decomposition, universal enveloping algebra, PBW basis, representation theory, finite dimensional modules, Harish-Chandra's Theorem.
		Aspects of measure theory and probability, convergence theorems for integrations and expectations, moments and inequalities, construction of Lebesgue-Stieltjes measure, Riemann-Stieltjes integral, comparison of Riemann and Lebesgue integrals, introduction to complex variable, contour integration, characteristics functions, elementary theorems on linear and matrix algebra, generalized and conditional inverses, distributions of quadratic forms.
		The main objective is to present the essentials of large sample theory of statistics with a view toward its application to a variety of problems that generally crop up in other areas.
	www.uwindsor.ca /units/mathstat/mainpages.nsf/printerFriendlyView/C45BEB8413F6DE5B85256D2D0058941D (1231 words)

	Springer Online Reference Works
		The closed-graph theorem has various generalizations; for example: a linear mapping with closed graph from a separable barrelled space into a perfectly-complete space is continuous.
		Closely related theorems are the open-mapping theorem and Banach's homeomorphism theorem.
		also Open-mapping theorem (also for the Banach homeomorphism theorem).
	eom.springer.de /c/c022540.htm (104 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		Do you know the Baire Category theorem, and its application to open mapping theorem of Banach spaces.
		The point is to study the equation as a map from C to P2(C).) What are the universal covering spaces for Riemann surfaces?
		What is the Galois Group of F(p^n) over F(p) (generated by the Frobenius map.) What is the characteristic polynomial of the Frobenius map viewed as a F(p)-vector space map.
	www.math.princeton.edu /graduate/generals/ng_ting_fai (339 words)

	Atlas: An Open Mapping Theorem for Basis Separating Maps by Lawrence Narici (Site not responding. Last check: 2007-11-01)
		As a consequence of the open mapping theorem, a continuous linear bijection H: X --> Y between Banach spaces X and Y must be a linear homeomorphism.
		The support map plays a crucial role in the development of the main results.
		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 # cafv-27.
	atlas-conferences.com /c/a/f/v/27.htm (203 words)

	1.4.3 Functional Analysis -- Dr Belton -- 16 MT (Site not responding. Last check: 2007-11-01)
		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.
		Many consequences of these theorems will be developed.
		The Banach-Alaoglu theorem [Proof of Tychonov's theorem non-examinable].
	www.maths.ox.ac.uk /current-students/undergraduates/handbooks-synopses/2003/html/sect-c-03/node18.html (159 words)

	PlanetMath 2004-01-12 Snapshot: Index of Contributors
		point and a compact set in a Hausdorff space have disjoint open neighborhoods.
		two disjoint compact sets in a Hausdorff space have disjoint open neighborhoods.
		theorem for the direct sum of finite dimensional vector spaces
	simba.cs.uct.ac.za /~hussein/PlanetMath-snapshot_2004-01-12/people.html (391 words)

	[No title]
		Locally compact Hausdorff spaces, Uryshon's Lemma, Tietze Extension Theorem, Partitions of Unity, Alexandrov's compactification, Uryshon's metrization theorem.
		Density and approximation theorems including the use of convolution and the Stone Weierstrass theorem.
		The Spectral Theorem for bounded self-adjoint operators on a Hilbert space.
	math.ucsd.edu /~driver/240-01-02 (285 words)

	reals (Site not responding. Last check: 2007-11-01)
		Metric Spaces: Open and closed sets, convergent sequences, functions and continuity, semi-continuity, separable spaces, complete spaces, compact spaces, Baire category theorem.
		Measurable Functions and Integration:Properties, approximation by simple functions, Egoroff's theorem, monotone convergence theorem, convergence in measure, Fatou's lemma, Lebesgue dominated convergence theorem, signed measures, Lebesgue differentiation theorem, Hahn decomposition theorem, Radon-Nikodym theorem, Lebesgue decomposition theorem, Fubini's theorem, Tonelli's theorem, Stone-Weierstras theorem, Ascoli-Arzela theorem.
		Lp-Spaces: Minkowski and Holder inequalities, Riesz representation theorem.
	www.kent.edu /math/Graduate/Resources/reals.cfm (193 words)

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