On complete-cocomplete subspaces of an inner product space

Buhagiar, David, and Chetcuti, Emmanuel. "On complete-cocomplete subspaces of an inner product space." Applications of Mathematics 50.2 (2005): 103-114. <http://eudml.org/doc/33210>.

@article{Buhagiar2005,
abstract = {In this note we give a measure-theoretic criterion for the completeness of an inner product space. We show that an inner product space $S$ is complete if and only if there exists a $\sigma $-additive state on $C(S)$, the orthomodular poset of complete-cocomplete subspaces of $S$. We then consider the problem of whether every state on $E(S)$, the class of splitting subspaces of $S$, can be extended to a Hilbertian state on $E(\bar\{S\})$; we show that for the dense hyperplane $S$ (of a separable Hilbert space) constructed by P. Pták and H. Weber in Proc. Am. Math. Soc. 129 (2001), 2111–2117, every state on $E(S)$ is a restriction of a state on $E(\bar\{S\})$.},
author = {Buhagiar, David, Chetcuti, Emmanuel},
journal = {Applications of Mathematics},
keywords = {Hilbert space; inner product space; orthogonally closed subspace; complete and cocomplete subspaces; finitely and $\sigma $-additive state; Hilbert space; inner product space; orthogonally closed subspace; complete and cocomplete subspaces; finitely and -additive state},
language = {eng},
number = {2},
pages = {103-114},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On complete-cocomplete subspaces of an inner product space},
url = {http://eudml.org/doc/33210},
volume = {50},
year = {2005},
}

TY - JOUR
AU - Buhagiar, David
AU - Chetcuti, Emmanuel
TI - On complete-cocomplete subspaces of an inner product space
JO - Applications of Mathematics
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 50
IS - 2
SP - 103
EP - 114
AB - In this note we give a measure-theoretic criterion for the completeness of an inner product space. We show that an inner product space $S$ is complete if and only if there exists a $\sigma $-additive state on $C(S)$, the orthomodular poset of complete-cocomplete subspaces of $S$. We then consider the problem of whether every state on $E(S)$, the class of splitting subspaces of $S$, can be extended to a Hilbertian state on $E(\bar{S})$; we show that for the dense hyperplane $S$ (of a separable Hilbert space) constructed by P. Pták and H. Weber in Proc. Am. Math. Soc. 129 (2001), 2111–2117, every state on $E(S)$ is a restriction of a state on $E(\bar{S})$.
LA - eng
KW - Hilbert space; inner product space; orthogonally closed subspace; complete and cocomplete subspaces; finitely and $\sigma $-additive state; Hilbert space; inner product space; orthogonally closed subspace; complete and cocomplete subspaces; finitely and -additive state
UR - http://eudml.org/doc/33210
ER -