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Quasi‐splitting subspaces in a pre‐Hilbert space
Author(s) -
Buhagiar David,
Chetcuti Emmanuel
Publication year - 2007
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200410496
Subject(s) - linear subspace , mathematics , hilbert space , lattice (music) , algebraic number , pure mathematics , space (punctuation) , combinatorics , discrete mathematics , mathematical analysis , physics , computer science , acoustics , operating system
Let S be a pre‐Hilbert space. Two classes of closed subspaces of S that can naturally replace the lattice of projections in a Hilbert space are E ( S ) and F ( S ), the classes of splitting subspaces and orthogonally closed subspaces of S respectively. It is well‐known that in general the algebraic structure of E ( S ) differs considerably from that of F ( S ) and the two coalesce if and only if S is a Hilbert space. In the present note we introduce the class E q ( S ) of quasi‐splitting subspaces of S . First it is shown that E q ( S ) falls between E ( S ) and F ( S ). It is also shown that, in contrast to the other two classes, E q ( S ) can sometimes be a complete lattice (without S being complete) and yet, in other examples E q ( S ) is not a lattice. At the end, the algebraic structure of E q ( S ) is used to characterize Hilbert spaces. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)