Geodesic
On vectorial inner product spaces
Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 539-550 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $(B(A),\sup _{\alpha \in A}|x(\alpha )|)$.
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 $(B(A),\sup _{\alpha \in A}|x(\alpha )|)$.
Classification : 46A19, 46C50 
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     author = {Marques, Jo\~ao de Deus},
     title = {On vectorial inner product spaces},
     journal = {Czechoslovak Mathematical Journal},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2000_50_3_a7/}
}
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Marques, João de Deus. On vectorial inner product spaces. Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 539-550. http://geodesic.mathdoc.fr/item/CMJ_2000_50_3_a7/

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