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Summary:
Let $E$ be a real linear space. A vectorial inner product is a mapping from $E\times E$ into a real ordered vector space $Y$ with the properties of a usual inner product. Here we consider $Y$ to be a $\mathcal B$-regular Yosida space, that is a Dedekind complete Yosida space such that $\bigcap _{J\in {\mathcal B}}J=\lbrace 0 \rbrace $, where $\mathcal B$ is the set of all hypermaximal bands in $Y$. In Theorem 2.1.1 we assert that any $\mathcal B$-regular Yosida space is Riesz isomorphic to the space $B(A)$ of all bounded real-valued mappings on a certain set $A$. Next we prove Bessel Inequality and Parseval Identity for a vectorial inner product with range in the $\mathcal B$-regular and norm complete Yosida algebra $(B(A),\sup _{\alpha \in A}|x(\alpha )|)$.
References:
[1] E. Coimbra: Aproximação em Espaços V-Métricos. Ph.D. Thesis, Dept. Mathematics, FCT, UNL, 1979.
[2] W. A. J. Luxemburg, A. C. Zaanen: Riesz Spaces I. North-Holland, 1971.
[3] J. D. Marques: Normas Vectoriais e Espaços V-Métricos. P.A.P.C.C., FCT, UNL, 1988.
[4] J. D. Marques: Normas Vectoriais Hermíticas com Valores em Álgebras de Yosida $\mathcal B$-Regulares. Ph.D. Thesis, Dept. Mathematics, FCT, UNL, 1993.
[5] J. D. Marques: A Representation Theorem in Vectorially Normed Spaces. Trabalhos de Investigação - No. 1 Dept. Mathematics, FCT, UNL, 1995. MR 1377735 | Zbl 0851.46004
[6] F. Robert: Étude et Utilization de Normes Vectorielles en Analyse Numérique Linéaire. These Grenoble, 1968.
[7] A. C. Zaanen: Riesz Spaces II. North Holland, 1983. MR 0704021 | Zbl 0519.46001
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