On vectorial inner product spaces

Marques, João de Deus. "On vectorial inner product spaces." Czechoslovak Mathematical Journal 50.3 (2000): 539-550. <http://eudml.org/doc/30582>.

@article{Marques2000,
abstract = {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 $(B(A),\sup _\{\alpha \in A\}|x(\alpha )|)$.},
author = {Marques, João de Deus},
journal = {Czechoslovak Mathematical Journal},
keywords = {regular Yosida space; Bessel inequality; Parseval identity},
language = {eng},
number = {3},
pages = {539-550},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On vectorial inner product spaces},
url = {http://eudml.org/doc/30582},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Marques, João de Deus
TI - On vectorial inner product spaces
JO - Czechoslovak Mathematical Journal
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 50
IS - 3
SP - 539
EP - 550
AB - 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 $(B(A),\sup _{\alpha \in A}|x(\alpha )|)$.
LA - eng
KW - regular Yosida space; Bessel inequality; Parseval identity
UR - http://eudml.org/doc/30582
ER -