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Hilbert-Schmidt norm

From Encyclopedia of Mathematics
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The norm of a linear operator acting from a Hilbert space into a Hilbert space , given by , where is an orthonormal basis in . The Hilbert–Schmidt norm satisfies all the axioms of a norm and is independent of the choice of the basis. Its properties are: , , , where is the norm of in the Hilbert space. If , then

References

[1] N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963)
[2] I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1968) (Translated from Russian)


Comments

References

[a1] N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian)
How to Cite This Entry:
Hilbert-Schmidt norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert-Schmidt_norm&oldid=42216
This article was adapted from an original article by V.B. Korotkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article