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
Achieser, N. I. (with I. M. Glazman) Theorie der linearen Operatoren im Hilbert-Raum, Akademie-Verlag 1954.
Dunford, N. Uniformity in linear spaces. Trans. Amer. Math. Soc. 44, 305–356 (1938).
Nagy, B. von Sz. Spektraldarstellung linearer Transformationen des Hilbertschen Raumes, Springer 1942.
Riesz, F. (with B. von Sz. Nagy) Leçons d’Analyse Fonctionelle, Akad. Kiado, Budapest 1952.
Stone, M. H. Linear Transformations in Hilbert Space and Their Applications to Analysis. Colloq. Publ. Amer. Math. Soc., 1932.
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© 1995 Springer-Verlag Berlin Heidelberg
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Yosida, K. (1995). The Orthogonal Projection and F. Riesz’ Representation Theorem. In: Functional Analysis. Classics in Mathematics, vol 123. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61859-8_4
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DOI: https://doi.org/10.1007/978-3-642-61859-8_4
Publisher Name: Springer, Berlin, Heidelberg
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