
theorem LM6:
  for X be RealUnitarySpace holds
    BoundedLinearFunctionals X = BoundedLinearFunctionals (RUSp2RNSp X)
proof
  let X be RealUnitarySpace;
  set Y = RUSp2RNSp X;
  now let x be object;
    assume x in BoundedLinearFunctionals X; then
A1: x is Lipschitzian linear-Functional of X by Def10; then
    x is linear-Functional of Y by LM6A; then
    x is Lipschitzian linear-Functional of Y by A1,LM6B;
    hence x in BoundedLinearFunctionals Y by DUALSP01:def 10;
  end; then
A2: BoundedLinearFunctionals X c= BoundedLinearFunctionals Y;
  now let x be object;
    assume x in BoundedLinearFunctionals Y; then
    x is Lipschitzian linear-Functional of Y by DUALSP01:def 10; then
    x is Lipschitzian linear-Functional of X by LM6B,LM6A;
    hence x in BoundedLinearFunctionals X by Def10;
  end; then
  BoundedLinearFunctionals Y c= BoundedLinearFunctionals X;
  hence thesis by A2;
end;
