
theorem
  for X be RealNormSpace holds
   X is Reflexive iff Im (BidualFunc X) = DualSp DualSp X
proof
   let X be RealNormSpace;
   thus X is Reflexive implies Im(BidualFunc X) =
     DualSp DualSp X by NORMSP_3:46;
   assume
A1: Im BidualFunc X = DualSp DualSp X;
   dom BidualFunc X <> {} by FUNCT_2:def 1; then
   rng BidualFunc X <> {} & rng BidualFunc X is linearly-closed
     by NORMSP_3:35,RELAT_1:42; then
   the carrier of Lin rng BidualFunc X = rng BidualFunc X by NORMSP_3:31;
     then
   BidualFunc X is onto by A1;
   hence X is Reflexive;
end;
