
theorem
  for X be finite-dimensional RealNormSpace,
      Y be RealNormSpace
    st dim X <> 0
  holds LinearOperators(X,Y) = BoundedLinearOperators(X,Y)
  proof
    let X be finite-dimensional RealNormSpace,
        Y be RealNormSpace;
    assume
    A1: dim X <> 0;

    for f be object
    holds
      f in LinearOperators(X,Y)
        iff
      f in BoundedLinearOperators(X,Y)
    proof
      let f be object;
      hereby
        assume f in LinearOperators(X,Y);
        then f is Lipschitzian LinearOperator of X,Y
          by A1,Th2,LOPBAN_1:def 6;

        hence f in BoundedLinearOperators(X,Y)
          by LOPBAN_1:def 9;
      end;

      assume f in BoundedLinearOperators(X,Y);
      then f is Lipschitzian LinearOperator of X,Y
        by LOPBAN_1:def 9;
      hence f in LinearOperators(X,Y) by LOPBAN_1:def 6;
    end;
    hence thesis by TARSKI:2;
  end;
