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MR1350940 (97d:46034) 46E30 (46B10 47H09 47H10)
Dowling, P. N.; Lennard, C. J. Every nonreflexive subspace of $L\sb 1[0,1]$ fails the fixed point property. Proc. Amer. Math. Soc. 125 (1997), no. 2, 443–446.
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