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Completeness and modular cross-symmetry in normed linear spaces. (English) Zbl 0786.46027
\(L_ c(X)\) denotes the lattice of all subspaces of a normed linear space \(X\). Various results are obtained, for example,
Corollary 2.4 (Holland): Let \(X\) be an inner product space and let \(L_ c(X)\) be cross-symmetric. Then \(X\) is a Hilbert space.
MSC:
46B99 Normed linear spaces and Banach spaces; Banach lattices
06C99 Modular lattices, complemented lattices
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References:
[1] S. S. Holland, Jr.: Partial solution to Mackey’s problem about modular pairs and completeness. Canad. J. Math. 21 (1969), 1518-1525. · Zbl 0188.43601
[2] G. W. Mackey: On infinite dimensional linear spaces. Trans. Amer. Math. Soc. 57 (1945), 155-207. · Zbl 0061.24301
[3] T. J. Marti: Introduction to the Theory of Bases. Springer Tracts in Natural Philosophy, 1969. · Zbl 0191.41301
[4] F. Maeda, S. Maeda: Theory of Symmetric Lattices. Springer-Verlag, Berlin-Heidelberg-New York, 1970. · Zbl 0219.06002
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