
theorem
  for X be ComplexNormSpace, seq be sequence of X, rseq1 be
  Real_Sequence st ||.seq.|| is non-increasing & (for n be Nat holds
  rseq1.n = 2 to_power n *||.seq.||.(2 to_power n)) holds (seq is norm_summable
  iff rseq1 is summable)
proof
  let X be ComplexNormSpace;
  let seq be sequence of X;
  let rseq1 be Real_Sequence;
  assume ||.seq.|| is non-increasing & for n be Nat holds rseq1.n
  = 2 to_power n *||.seq.||.(2 to_power n);
  then
  for n be Nat holds ||.seq.|| is non-increasing & ||.seq.||.n
  >= 0 & rseq1.n = 2 to_power n *||.seq.||.(2 to_power n) by Th2;
  then ||.seq.|| is summable iff rseq1 is summable by SERIES_1:31;
  hence thesis;
end;
