reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th29:
  for V,W be RealNormSpace,
      L be LinearOperator of V,W
    st L is one-to-one onto isometric-like
  holds
    ex K be LinearOperator of W,V
    st K = L"
     & K is one-to-one onto isometric-like
  proof
    let V,W be RealNormSpace,
        L be LinearOperator of V,W;
    assume
    A1: L is one-to-one onto isometric-like;
    consider K be LinearOperator of W,V such that
    A2: K = L" & K is one-to-one onto by A1,REAL_NS2:85;

    take K;
    thus K = L" by A2;
    thus K is one-to-one onto by A2;
    consider k1,k2 be Real such that
    A3: 0 <= k1 & 0 <= k2
      & for x be Element of V
        holds ||.L.x.|| <= k1 * ||.x.||
      & ||.x.|| <= k2 * ||.L.x.|| by A1;

    for y be Element of W
    holds ||.K.y.|| <= k2 * ||.y.||
        & ||.y.|| <= k1 * ||.K.y.||
    proof
      let y be Element of W;
      the carrier of W = rng L by A1,FUNCT_2:def 3;
      then
      consider x be Element of the carrier of V such that
      A4: y = L.x by FUNCT_2:113;
      A5: K.y = x by A1,A2,A4,FUNCT_2:26;
      hence ||.K.y.|| <= k2 * ||.y.|| by A3,A4;
      thus ||.y.|| <= k1 * ||.K.y.|| by A3,A4,A5;
    end;
    hence K is isometric-like by A3;
  end;
