
theorem
  for X be RealBanachSpace, Y be Subset of X
  holds ClNLin(Y) is RealBanachSpace
  proof
    let X be RealBanachSpace, Y be Subset of X;
    ex Z be Subset of X
    st Z = the carrier of Lin(Y) & ClNLin(Y) = NLin(Cl(Z))
     & Cl(Z) is linearly-closed & Cl(Z) <> {} by Th35;
    hence thesis by NORMSP_3:49;
  end;
