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Hilbert spaces induced by Hilbert space valued functions


Author: Saburou Saitoh
Journal: Proc. Amer. Math. Soc. 89 (1983), 74-78
MSC: Primary 44A05; Secondary 30C40, 47A05
DOI: https://doi.org/10.1090/S0002-9939-1983-0706514-9
MathSciNet review: 706514
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Abstract: Let $ E$ be an arbitrary set and $ \mathcal{F}(E)$ a linear space composed of all complex valued functions on $ E$. Let $ \mathcal{H}$ be a (possibly finite-dimensional) Hilbert space with inner product $ {(,)_\mathcal{H}}$. Let $ {\mathbf{h}}:E \to \mathcal{H}$ be a function and consider the linear mapping $ L$ from $ \mathcal{H}$ into $ \mathcal{F}(E)$ defined by $ {({\mathbf{F}},{\mathbf{h}}(p))_\mathcal{H}}$. We let $ \tilde{\mathcal{H}}$ denote the range of $ L$. Then we assert that $ \tilde{\mathcal{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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0706514-9
Keywords: Hilbert space, linear mapping, linear operator, integral transform, reproducing kernel, isometry, completeness of functions
Article copyright: © Copyright 1983 American Mathematical Society

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