
theorem
  for X be RealBanachSpace, f be sequence of DualSp X holds
   f is weakly*-convergent
   iff ( ||.f.|| is bounded
    & ex X0 be Subset of X st
         X0 is dense
       & (for x be Point of X st x in X0 holds f#x is convergent)
     )
proof
  let X be RealBanachSpace,
      f be sequence of DualSp X;
  now assume AS: f is weakly*-convergent;
    hence ||.f.|| is bounded by Th711;
    set X0 = [#]X;
    take X0;
    thus X0 is dense;
    thus for x be Point of X st x in X0 holds f#x is convergent by AS;
  end;
  hence thesis by Th712A;
end;
