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The structure and geometry of Banach spaces with the property that £ ( 4 ) = £** + £ X J " are investigated: such spaces are called quotient reflexive spaces here. For these spaces, if £ is very smooth, E is also very smooth, and if £* is weakly locally uniformly rotund (WLUR), £ ( 4 ) is smooth on a certain (relatively) norm dense subset of E**. Consequently, for quotient reflexive spaces, WLUR and very-WLUR are equivalent in £*. 1980 Mathematics subject classification (Amer. Math. Soc): 46 B 99.doi:10.1017/s1446788700025751 fatcat:xyfrltfw2bavpieawzcucewp7y