Download e-book for iPad: Applications of Harmonic Analysis by I. M. Gel'fand and N. Ya. Vilenkin (Auth.)

By I. M. Gel'fand and N. Ya. Vilenkin (Auth.)

ISBN-10: 1483229742

ISBN-13: 9781483229744

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Glazman, ' T h e o r y of Linear Operators in Hilbert S p a c e / * Chapter 2 , § 2 7 . Moscow-Leningrad, 1 9 5 0 . (English translation by Ungar, New York, 1 9 6 2 ) . If one considers linear operators acting in some Hilbert space / / , then properties ( l ) - ( 3 ) can be formulated briefly by saying that the completely continuous operators form an ideal in the ring with an involution consisting of aU continuous Hnear operators. , a completely continuous operator A such that {Af, g) = {f, Ag) for every fyge H, has a particularly simple structure.

If and only if φ = Ο, a scalar product in Φ. With every scalar product ( , ψ\η and is a fundamental sequence in the norm || ψ ||„, then it also converges to zero in the norm ll

We will not linger over the proofs of these assertions, since they have been proved in the case which is of interest to us, namely Hilbert space, and, as we said, the concept of nuclearity for operators in a Banach space is apparently not sufficiently worthwhile. 3. N u c l e a r Spaces. 1 Nuclear Spaces 57 normed spaces, is that of distinguishing a class of spaces which is defined by sufficiently simple requirements and is of service in analysis. We maintain that one such class of spaces is the class of nuclear spaces, which will be studied in this section.

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Applications of Harmonic Analysis by I. M. Gel'fand and N. Ya. Vilenkin (Auth.)


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