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 Th38:
  for S,T be RealNormSpace,
      I be LinearOperator of S,T,
      Z be Subset of S
    st I is one-to-one onto isometric-like
  holds Z is compact iff I.:Z is compact
  proof
    let S, T be RealNormSpace;
    let I be LinearOperator of S,T,
        Z be Subset of S;

    assume that
    A1: I is one-to-one onto
          and
    A2: I is isometric-like;

    consider J be LinearOperator of T, S such that
    A3: J = I" & J is one-to-one onto isometric-like by A1,A2,Th29;

    A4: dom I = the carrier of S by FUNCT_2:def 1;
    thus Z is compact implies I.:Z is compact by A1,Lm5,A2;
    thus I.:Z is compact implies Z is compact
    proof
      assume
      A5: I.:Z is compact;
      J.:(I.:Z) = I"(I.:Z) by A1,A3,FUNCT_1:85
      .= Z by A1,A4,FUNCT_1:94;
      hence Z is compact by A3,A5,Lm5;
    end;
  end;
