A simple proof of the closed graph theorem
DOI:
https://doi.org/10.14419/gjma.v4i1.5534الكلمات المفتاحية:
Closed Graph Theorem، Closed Linear Operator، Uniform Boundedness Principle، New Short Proof of The Closed Graph Theoremالملخص
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.
المراجع
[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.
التنزيلات
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