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MR2068983 46N40 (34B05 46B20 47A50 65J10)
Daugavet, I. K. Some problems of convergence in weak norms. (Russian) Algebra i Analiz 15 (2003), no. 4, 159--176; translation in St. Petersburg Math. J. 15 (2004), no. 4, 575–585
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