J. Aust. Math. Soc.  78 (2005), 199-210
Boundedness of sign-preserving charges, regularity, and the completeness of inner product spaces

Emmanuel Chetcuti
  Mathematical Institute
  Slovak Academy of Sciences
  Stefánikova 49
  SK-814 73 Bratislava
  Slovakia
  chetcuti@mat.savba.sk
and
Anatolij Dvurecenskij
  Mathematical Institute
  Slovak Academy of Sciences
  Stefánikova 49
  SK-814 73 Bratislava
  Slovakia
  dvurecen@mat.savba.sk


Abstract
We introduce sign-preserving charges on the system of all orthogonally closed subspaces, $F(S)$, of an inner product space $S$, and we show that it is always bounded on all the finite-dimensional subspaces whenever $\dim S = \infty$. When $S$ is finite-dimensional this is not true. This fact is used for a new completeness criterion showing that $S$ is complete whenever $F(S)$ admits at least one non-zero sign-preserving regular charge. In particular, every such charge is always completely additive.
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