
theorem
for X,Y be RealNormSpace,
    L be Lipschitzian LinearOperator of X,Y,
    T be Lipschitzian LinearOperator of DualSp X, DualSp Y st
    L is isomorphism & T is isomorphism
  & for x be Point of DualSp X holds T.x = x*(L") holds
  ex S be Lipschitzian LinearOperator of DualSp Y, DualSp X
    st S is isomorphism
    & S = T"
    & for y be Point of DualSp Y holds S.y = y*L
proof
   let X,Y be RealNormSpace,
       L be Lipschitzian LinearOperator of X,Y,
       T be Lipschitzian LinearOperator of DualSp X, DualSp Y;
   assume AS1: L is isomorphism & T is isomorphism
             & for x be Point of DualSp X holds T.x = x*(L"); then
AS2: L is one-to-one onto
   & for x be Point of X holds ||.x.|| = ||. L.x .||;
   consider K be Lipschitzian LinearOperator of Y,X such that
AS3: K = L" & K is isomorphism by AS1,NORMSP_3:37;
AS4: T is one-to-one & T is onto
   & for x be Point of DualSp X holds ||.x.|| = ||. T.x .|| by AS1;
   consider S be Lipschitzian LinearOperator of DualSp Y, DualSp X such that
AS5: S is isomorphism
   & for y be Point of DualSp Y holds S.y = y*(K") by NISOM09,AS3;
   take S;
P2:K" = L by FUNCT_1:43,AS1,AS3;
   for y,x being object holds
        y in the carrier of DualSp Y & S.y = x
    iff x in the carrier of DualSp X & T.x = y
   proof
    let y,x be object;
    hereby assume P31: y in the carrier of DualSp Y & S.y = x;
     hence x in the carrier of DualSp X by FUNCT_2:5;
     reconsider yp=y as Point of DualSp Y by P31;
     reconsider xp=x as Point of DualSp X by P31,FUNCT_2:5;
     yp is linear-Functional of Y by DUALSP01:def 10; then
G6:  dom yp = the carrier of Y by FUNCT_2:def 1;
     thus T.x = xp*(L") by AS1
             .= (yp*L)*(L") by P2,AS5,P31
             .= yp*(L*L") by RELAT_1:36
             .= yp *(id rng L) by AS1,FUNCT_1:39
             .= y by G6,AS2,RELAT_1:51;
    end;
    assume P32: x in the carrier of DualSp X & T.x = y;
    hence y in the carrier of DualSp Y by FUNCT_2:5;
    reconsider yp=y as Point of DualSp Y by P32,FUNCT_2:5;
    reconsider xp=x as Point of DualSp X by P32;
G5: dom L = the carrier of X by FUNCT_2:def 1;
    xp is linear-Functional of X by DUALSP01:def 10; then
G6: dom xp = the carrier of X by FUNCT_2:def 1;
    thus S.y = yp*L by P2,AS5
            .= (xp*(L)")*L by AS1,P32
            .= xp*(L"*L) by RELAT_1:36
            .= xp *(id dom L) by AS1,FUNCT_1:39
            .= x by G6,G5,RELAT_1:51;
   end;
   hence thesis by AS5,P2,FUNCT_2:28,AS4;
end;
