A simple proof of the closed graph theorem

 
 
 
  • Abstract
  • Keywords
  • References
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  • Abstract


    Assume that A is a closed linear operator defined on all of a Hilbert space H. Then, A is bounded. This classical theorem is proved on the basis of uniform boundedness principle. The proof is easily extended to Banach spaces.


  • Keywords


    Closed Graph Theorem; Closed Linear Operator; Uniform Boundedness Principle; New Short Proof of The Closed Graph Theorem

  • References


      [1] N.Dunford, J. Schwartz, Linear operators, Part I, Interscience, New York, 1958.

      [2] P. Halmos, A Hilbert space problem book, Springer-Verlag, New York, 1974. (problems 52 and 58)

      [3] J. Hennefeld, A non-topological proof of the uniform boundedness theorem, Amer. Math. Monthly, 87, (1980), 217.

      [4] S. Holland, A Hilbert space proof of the Banach-Steinhaus theorem, Amer. Math. Monthly, 76, (1969), 40-41.

      [5] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1984.

      [6] A. Sokal, A relally simple elementary proof of the uniform boundedness theorem, Amer. Math. Monthly, 118, (2011), 450-452.

      [7] K. Yosida, Functional analysis, Springer, New York, 1980.


 

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Article ID: 5534
 
DOI: 10.14419/gjma.v4i1.5534




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