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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
##### Keywords:
inner product space; cross-symmetric
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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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