ON MULTILINEAR MAPPINGS ATTAINING THEIR NORMS

We show, for any Banach spaces X and Y, the denseness of the set of bi linear forms on X x Y whose third Arens transpose attains its norm. We also prove the denseness of the set of norm attaining multilinear map pings in the class of multilinear mappings which are weakly continuous on bounded sets, under some additional assumptions on the Banach spac es, and give several examples of classical spaces satisfying these hyp otheses.