Abstract
In a pre-Hilbert space, we can introduce the notion of orthogonality of two vectors. Thanks to this fact, a Hilbert space may be identified with its dual space, i.e., the space of bounded linear functionals. This result is the representation theorem of F. Riesz [1], and the whole theory of Hilbert spaces is founded on this theorem.
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References for Chapter III
Gelfand, I. M. (with I. M. SxLov) Generalized Functions, Vol. I—Iii, Moscow 1958.
Schwartz, J. See Dunford-Schwartz [4].
Nagy, B. Von SZ. Spektraldarstellung linearer Transformationen des Hilbertschen Raumes, Springer 1942.
Riesz, F. (with B. vox Sz. Nagy) Leçons d’Analyse Fonctionelle, Akad. Kiado, Budapest 1962.
Stone, M. H. Linear Transformations in Hilbert Space and Their Applications to Analysis. Colloq. Publ. Amer. Math. Soc., 1932.
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© 1968 Springer-Verlag Berlin Heidelberg
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Yosida, K. (1968). The Orthogonal Projection and F. Riesz’ Representation Theorem. In: Functional Analysis. Die Grundlehren der mathematischen Wissenschaften, vol 123. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-11791-0_4
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DOI: https://doi.org/10.1007/978-3-662-11791-0_4
Publisher Name: Springer, Berlin, Heidelberg
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