
theorem Th23:
  for X be RealNormSpace for seq be sequence of X holds
  Partial_Sums ||.seq.|| is bounded_above iff seq is norm_summable
proof
  let X be RealNormSpace;
  let seq be sequence of X;
  for n be Nat holds 0 <=||.seq.||.n by Th2;
  then Partial_Sums(||.seq.||) is bounded_above iff ||.seq.|| is summable by
SERIES_1:17;
  hence thesis;
end;
