
theorem
  the carrier of Complex_l2_Space = the_set_of_l2ComplexSequences & (for
x be set holds x is Element of Complex_l2_Space iff x is Complex_Sequence & |.
seq_id(x).|(#)|.seq_id(x).| is summable) & (for x be set holds x is Element of
  Complex_l2_Space iff x is Complex_Sequence & seq_id(x)(#)(seq_id(x))*' is
  absolutely_summable ) & 0.Complex_l2_Space = CZeroseq & (for u be VECTOR of
  Complex_l2_Space holds u =seq_id(u)) & (for u,v be VECTOR of Complex_l2_Space
  holds u+v =seq_id(u)+seq_id(v)) & (for r be Complex for u be VECTOR of
  Complex_l2_Space holds r*u =r(#)seq_id(u)) & (for u be VECTOR of
  Complex_l2_Space holds -u =-seq_id(u) & seq_id(-u)=-seq_id(u)) & (for u,v be
VECTOR of Complex_l2_Space holds u-v =seq_id(u)-seq_id(v)) & for v,w be VECTOR
  of Complex_l2_Space holds |.seq_id(v).|(#)|.seq_id(w).| is summable & for v,w
  be VECTOR of Complex_l2_Space holds v.|.w = Sum(seq_id(v)(#)(seq_id(w))*')
by Lm9,Lm10,Lm12,Lm13,Lm14,Lm15,Lm16,Lm19,CSSPACE:def 17;
