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The weak convergence of unit vectors to zero in the Hilbert space is the convergence of one-dimensional subspaces in the order topology

Author: Vladimír Palko
Journal: Proc. Amer. Math. Soc. 123 (1995), 715-721
MSC: Primary 46C05; Secondary 06F30, 47N50, 81P10
MathSciNet review: 1231302
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Abstract: In this paper we deal with the (o)-convergence and the order topology in the hilbertian logic $ \mathcal{L}(H)$ of closed subspaces of a separable Hilbert space H. We compare the order topology on $ \mathcal{L}(H)$ with some other topologies. The main result is a theorem which asserts that the weak convergence of a sequence of unit vectors to zero in H is equivalent to the convergence of the sequence of one-dimensional subspaces generated by these vectors to the zero subspace in the order topology on $ \mathcal{L}(H)$.

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Keywords: Order topology, Hilbert space, quantum logic, weak convergence
Article copyright: © Copyright 1995 American Mathematical Society

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