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If we have a sequence ( x_n )_(n >= 1) in l^p that converges weakly to zero then that implies the x_n are uniformly bounded and that ((x^{m})_n) -> 0 in C (as n -> oo) for each fixed m.

(where l^p is the space of p-summable sequences of complex numbers, and I wrote x_n = ((x^{m})_n )_(m >= 1) , couldn't think of a better notation...)

Is the converse true? Why would that be? It doesn't look too obvious... at least not for me.