Open mapping theorem (functional analysis)
In functional analysis, the 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 continuous linear operator between Banach spaces is surjective then it is an open map. More precisely, (Rudin 1973, Theorem 2.11):
Open Mapping Theorem. If X and Y are Banach spaces and A : X → Y is a surjective continuous linear operator, then A is an open map (i.e. if U is an open set in X, then A(U) is open in Y).
The proof uses the Baire category theorem, and completeness of both X and Y 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 X and Y are taken to be Fréchet spaces.

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Open mapping theorem (functional analysis) - Wikipedia, the free ...
In functional analysis, the open mapping theorem, also known as the Banach– Schauder theorem (named after Stefan Banach and Juliusz Schauder), is a ...
en.wikipedia.org/wiki/Open_mapping_theorem_(functional_analysis)
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Open mapping theorem - Wikipedia, the free encyclopedia
Open mapping theorem may refer to: Open mapping theorem (functional analysis ) or Banach–Schauder theorem states that a surjective continuous linear ...
en.wikipedia.org/wiki/Open_mapping_theorem
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Functional Analysis Notes
Functional Analysis. Alexander C. R. Belton ... 2.5 The Open-Mapping Theorem . ..... As is usual practise in functional analysis, we shall frequently blur the.
www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf
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Lectures in functional analysis - University of Michigan
These notes are for a one-semester graduate course in Functional Analysis, ... Banach theorem, open mapping theorem, closed graph theorem, principle of uni- .
www-personal.umich.edu/~romanv/teaching/2010-11/602/functional-analysis.pdf
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open mapping theorem | What's new
Theorem 1 (Closed graph theorem (functional analysis)) Let {X, Y} ... (Conversely , one can deduce the open mapping theorem from the closed graph theorem by ...
terrytao.wordpress.com/tag/open-mapping-theorem/
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Closed graph and open mapping theorems for normed cones
graph and an open mapping type theorem for quasi-normed cones. ... on general topology and functional analysis have recently been obtained in order to ...
www.ias.ac.in/mathsci/vol118/may2008/PM3069.PDF
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Open Mapping and Closed Graph Theorems
Functional Analysis Notes. 8 The open mapping theorem and the closed graph theorem. The open mapping theorem. Recall that an open subset U ⊂ X of a ...
www-users.york.ac.uk/~fr606/fa_notes/8_open_mapping_and_closed_graph_theorems.pdf
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functional analysis - Does the open mapping theorem imply the ...
May 19, 2012 ... On the other hand, the three classical consequences of the Baire category theorem in basic functional analysis — the open mapping theorem, ...
math.stackexchange.com/questions/146910/
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fa.functional analysis - Counterexample for the Open Mapping ...
Nov 3, 2010 ... I would like to ask a counterexample for the classical theorem in functional analysis: the open mapping theorem in the case that \$Y\$ is Banach, ...
mathoverflow.net/questions/44716/counterexample-for-the-open-mapping-theorem
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Functional Analysis - University of…
Functional analysis 2012-13, MWF 1:25, Vincent 02 In reverse chronological order: ... 11 Unbounded operators on Hilbert spaces 10 Fourier transforms, tempered ...
www.math.umn.edu/~garrett/m/fun/
Search results for "Open mapping theorem (functional analysis)"
Open mapping theorem (functional analysis) in science
[PDF]Lectures in functional analysis - University of Michigan
These notes are for a one-semester graduate course in Functional Analysis, ... Banach theorem, open mapping theorem, closed graph theorem, principle of uni- .
Functional analysis - Wikipedia, the free encyclopedia
An important part of functional analysis is the extension of the theory of measure, ... The open mapping theorem and closed graph theorem. ... edition, 1980; Vogt, D., Meise, R.: Introduction to Functional Analysis, Oxford University Press, 1997.
[PDF]Functional analysis lecture notes T.B. Ward - University of East Anglia
32. 4. Uniform boundedness. 32. 5. An application of uniform boundedness to Fourier series. 34. 6. Open mapping theorem. 36. 7. Hahn–Banach theorem. 38.
[PDF]Functional Analysis Notes
2.5 The Open-Mapping Theorem . .... undergraduates at the University of Oxford. .... As is usual practise in functional analysis, we shall frequently blur the.
[PDF]Functional Analysis Notes Fall 2004 Prof. Sylvia Serfaty - New York ...
Mar 14, 2006 ... These are notes from a one-semester graduate course in Functional Analysis given by Prof. ... ences, New York University, in the Fall of 2004. Thanks to ... 2.4 The Open Mapping Theorem and Closed Graph Theorem . . . . 18.
Download or subscribe to free content from Functional Analysis by University of ... Theorem, Chapter 5, printed slides 102-108, Proof of Open Mapping Theorem.
[PDF]Functional Analysis Lecture Notes
Robert J. Zimmer, Essential Results of Functional Analysis, University of ...... The Open Mapping Theorem and the Uniform Boundedness Principle join the Hahn ...
Functional Analysis - University of Minnesota
Nov 15, 2013 ... Functional Analysis ... Functional analysis 2012-13, MWF 1:25, Vincent 02 .... Open Mapping Theorem, Hahn-Banach Theorem, in the simple ...
MA3FA1 Functional Analysis I - Module descriptions- University of ...
May 9, 2013 ... Co-requisites: MA3TLA Topology and Linear Analysis and MA3MTI ... of Functional Analysis: the Hahn-Banach Theorem, the Open Mapping ...
Functional Analysis - Wikibooks, open books for an open world
Functional Analysis can mean different things, depending on who you ask. ... In both parts, we give principal results e.g., the closed graph theorem, resulting in ...
Books on the term Open mapping theorem (functional analysis)
Exercises in Functional Analysis
C. Costara, Dumitru Popa, 2003
This book of exercises in Functional Analysis contains almost 450 exercises (all with complete solutions), providing supplementary examples, counter-examples and applications for the basic notions usually presented in an introductory course ...
Elements of Functional Analysis (Graduate Texts in Mathematics)
1999
This book presents the fundamental function spaces and their duals, explores operator theory and finally develops the theory of distributions up to significant applications such as Sobolev spaces and Dirichlet problems. Includes an assortment of well formulated exercises, with answers and hints collected at the end of the book.
Elementary Functional Analysis
Barbara D. MacCluer, 2008
The theorems of the title of this section are closely related; we will prove the open mapping theorem first, using the Baire category theorem, and then derive the closed graph theorem from it. An open map is one for which the image of every ...
Spectral Theory of Operators on Hilbert Spaces
Carlos S. Kubrusly, 2012
From the reviews:“Kubrusly’s book … situates the spectral theorem in a very interesting broader context. … Kubrusly intends the book for a one-semester graduate course and … he closes each of his five (pretty beefy) chapters with sections titled ‘Additional Propositions.’ … I think this is a nice idea: it will make for a very solid learning experie...
Convex Functional Analysis
Andrew J. Kurdila, Michael Zabarankin, 2005
2.1.3 The Open Mapping Theorem We studied several alternative characterizations of continuity of functions in the previous chapter. Perhaps the most abstract definition is cast in terms of inverse images of open sets. A function f : X — > Y is ...
Introductory Functional Analysis: With Applications to Boundary Value Problems and Finite Elements (Texts in Applied...
1997
Providing an introduction to functional analysis, this text treats in detail its application to boundary-value problems and finite elements, and is distinguished by the fact that abstract concepts are motivated and illustrated wherever possible. It is intended for use by senior undergraduates and graduates in mathematics, the physical sciences and ...
Beginning Functional Analysis
Karen Saxe, 2002
Theorems. from. Functional. Analysis. In this section we present the open mapping theorem, the Banach-Steinhaus theorem, and the Hahn-Banach theorem. We also discuss their history and applications. The first was proved by Banach, the ...
Deconvolution of Images and Spectra: Second Edition (Dover Books on Engineering)
2012
A technique of signal and image processing, deconvolution is applied to difficult-to-read data. This text conveys a clear understanding of the field, assuming only a working knowledge of calculus. Self-contained chapters emphasize applications rather than theory, offering a practical and easy-to-understand approach. 1997 edition.
Introductory Functional Analysis: With Applications to ...
B.D. Reddy, 1998
In Theorem 2 it was seen that a continuous operator T may be characterized by the property that the inverse image of an open set in ... The answer is provided by one of the important theorems in functional analysis, the open mapping theorem.
Techniques of Constructive Analysis (Universitext)
2006
This book is an introduction to constructive mathematics with an emphasis on techniques and results obtained in the last twenty years. The text covers fundamental theory of the real line and metric spaces, focusing on locatedness in normed spaces and with associated results about operators and their adjoints on a Hilbert space. The first appendix g...

Blog posts on the term
Open mapping theorem (functional analysis)
Closed Graph Theorem implies Open Mapping Theorem | Mathematics@CUHK
Today Wongting raised the question in the real analysis prelims at UW about proving that the Closed Graph Theorem (CGT) implies the Open Mapping Theorem (OMT). This is a standard exercise in the first course of functional analysis. Let us recall the three basic theorems. We assume \$latex X,Y\$ are Banach spaces. Recall that a…
cuhkmath.wordpress.com/2011/09/06/closed-graph-theorem-implies-open-mapping-theorem/
Functional Analysis Course, IIT Kharagpur Mathematics Video Tutorials, P.D. Srivastava
Functional Analysis Video Lectures, IIT Kharagpur Online Course, free tutorials and lecture notes, free download, Educational Lecture Videos
freevideolectures.com/Course/3145/Functional-Analysis
The closed graph theorem in various categories | What's new
The closed graph theorem in various categories 20 November, 2012 in expository, math. AG, math.
terrytao.wordpress.com/2012/11/20/the-closed-graph-theorem-in-various-categories/
The Open Mapping Theorem & Consequences | Sizes of Sets
The Open Mapping Theorem & Consequences March 17, 2012 in Complex Analysis Viewed as geometric mappings, non-constant holomorphic functions map connected open sets in the complex plane to open sets in the complex plane. This statement is known as The Open Mapping Theorem.
dakecht.wordpress.com/2012/03/17/open-mapping-theorem-all-the-way/
Advanced Analysis, Notes 9: Banach spaces (the three big theorems) | Noncommutative Analysis
Until now we had not yet seen a theorem about Banach spaces --- the Hahn--Banach theorems did not require the space to be complete. In this post we learn the three big theorems about operators on Banach spaces: the principle of uniform boundedness, the open mapping theorem, and the closed graph theorem. It is common that…