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Let $E$ be a Banach space which is weakly sequentially complete (i.e. each weak Cauchy sequence converges weakly). Must $E^{**}$ be weakly sequentially complete either? Of course, this question is interesting only for non-reflexive spaces.

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up vote 3 down vote accepted

This is not true.

I reproduce Remark 3. on page 101 (in the section on Banach lattices) of Lindenstrauss-Tzafriri (in the old Springer Lecture Notes 338 edition):

Remark 3 of Lindenstrauss-Tzafriri

Reference [80] is:

William B. Johnson, A complementary universal conjugate Banach space and its relation to the approximation problem, Israel Journal of Mathematics 13(3-4) (1972), 301–310.

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Your answer is absolutely brilliant! – JoeHudson Nov 1 '11 at 14:31
Thanks, glad you liked it, but credit is fully due to Lindenstrauss-Tzafriri and Johnson :) – t.b. Nov 1 '11 at 19:34

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