Hilbert Spaces Induced by Hilbert Space Valued Functions

Let E be an arbitrary set and F(E) a linear space composed of all complex valued functions on E. Let H be a (possibly finite-dimensional) Hilbert space with inner product (,)H. Let h: E → H be a function and consider the linear mapping L from H into F(E) defined by (F,h(p))H. We let$\tilde{\mathscr{...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 1983, Vol.89 (1), p.74-78
Main Author: Saitoh, Saburou
Format: Article
Language:English
Subjects:
Hilbert spaces
Linear transformations
Mathematical functions
Mathematical integrals
Mathematical theorems
Mathematical vectors
Range of function
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Summary:Let E be an arbitrary set and F(E) a linear space composed of all complex valued functions on E. Let H be a (possibly finite-dimensional) Hilbert space with inner product (,)H. Let h: E → H be a function and consider the linear mapping L from H into F(E) defined by (F,h(p))H. We let$\tilde{\mathscr{H}}$denote the range of L. Then we assert that$\tilde{\mathscr{H}}$becomes a Hilbert space with a reproducing kernel composed of functions on E, and, moreover, it is uniquely determined by the mapping L, in a sense. Furthermore, we investigate several fundamental properties for the mapping L and its inverse.
ISSN:0002-9939
1088-6826
DOI:10.1090/s0002-9939-1983-0706514-9