reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem LM020:
  for I be LinearOperator of S, T
  st I is one-to-one onto isometric
  holds ex J be LinearOperator of T, S
  st J = I" & J is one-to-one onto isometric
  proof
    let I be LinearOperator of S, T;
    assume that
    AS0: I is one-to-one onto and
    AS1: I is isometric;
    P0: rng I = the carrier of T & dom I = the carrier of S
    by AS0,FUNCT_2:def 1;
    P1: rng I = dom(I") & dom I = rng(I") by AS0,FUNCT_1:33; then
    reconsider J = I" as Function of T,S by P0,FUNCT_2:1;
    X1: for v, w be Point of T holds J.(v+w) = J.v + J.w
    proof
      let v, w be Point of T;
      consider t be Point of S such that
      X1: v = I.t by AS0,FUNCT_2:113;
      consider s be Point of S such that
      X2: w = I.s by AS0,FUNCT_2:113;
      X3: J.(v+w) = J.(I.(t+s)) by X1,X2,VECTSP_1:def 20
      .= t+s by FUNCT_1:34,AS0,P0;
      J.w = s by X2,FUNCT_1:34,AS0,P0;
      hence thesis by AS0,P0,X1,X3,FUNCT_1:34;
    end;
    X2: for v be Point of T, r be Real holds J.(r*v) = r*(J.v)
    proof
      let v be Point of T,r be Real;
      consider t be Point of S such that
      X1: v = I.t by AS0,FUNCT_2:113;
      J.(r*v) = J.(I.(r*t)) by X1,LOPBAN_1:def 5
      .= r*t by FUNCT_1:34,AS0,P0;
      hence thesis by AS0,P0,X1,FUNCT_1:34;
    end;
    reconsider J as LinearOperator of T,S
    by X1,X2,LOPBAN_1:def 5,VECTSP_1:def 20;
    take J;
    thus J = I";
    thus J is one-to-one onto by AS0,P1,FUNCT_2:def 1;
    for v be Point of T holds ||. J.v .|| = ||.v.||
    proof
      let v be Point of T;
      consider t be Point of S such that
      X1: v = I.t by AS0,FUNCT_2:113;
      thus ||. J.v .|| = ||. t .|| by X1,FUNCT_1:34,AS0,P0
      .= ||. v .|| by X1,AS1,LMMAZU;
    end;
    hence thesis by LMMAZU;
  end;
