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MR1231302 (95d:46016) 46C05 (06F30 47N50 81P10)
Palko, Vladimír The weak convergence of unit vectors to zero in the Hilbert space is the convergence of one-dimensional subspaces in the order topology. Proc. Amer. Math. Soc. 123 (1995), no. 3, 715–721.
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