reserve V for RealLinearSpace;

theorem
  for V being RealNormSpace, X being Subspace of V, fi being
  linear-Functional of X st for x being VECTOR of X, v being VECTOR of V st x=v
holds fi.x <= ||.v.|| ex psi being linear-Functional of V st psi|the carrier of
  X=fi & for x being VECTOR of V holds psi.x <= ||.x.||
proof
  let V be RealNormSpace, X be Subspace of V, fi be linear-Functional of X
  such that
A1: for x being VECTOR of X, v being VECTOR of V st x=v holds fi.x <= ||.v.||;
  reconsider q = the normF of V as Banach-Functional of V by Th23;
  now
    let x be VECTOR of X, v be VECTOR of V such that
A2: x=v;
    q.v = ||.v.|| by NORMSP_0:def 1;
    hence fi.x <= q.v by A1,A2;
  end;
  then consider psi being linear-Functional of V such that
A3: psi|the carrier of X=fi and
A4: for x being VECTOR of V holds psi.x <= q.x by Th22;
  take psi;
  thus psi|the carrier of X=fi by A3;
  let x be VECTOR of V;
  q.x = ||.x.|| by NORMSP_0:def 1;
  hence thesis by A4;
end;
