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
  for I be LinearOperator of S, T,
  Z be Subset of S
  st I is one-to-one onto isometric holds
  (Z is compact iff I.:Z is compact)
  proof
    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;
    consider J be LinearOperator of T, S such that
    P2: J = I" & J is one-to-one onto isometric by A1,A2,LM020;
P1: dom I = the carrier of S by FUNCT_2:def 1;
    thus Z is compact implies I.:Z is compact by LM026,A2;
    thus I.:Z is compact implies Z is compact
    proof
      assume X2: I.:Z is compact;
      J.:(I.:Z) = I"(I.:Z) by A1,P2,FUNCT_1:85
      .= Z by FUNCT_1:94,P1,A1;
      hence Z is compact by P2,X2,LM026;
    end;
  end;
