
theorem Th1:
  the carrier of l2_Space = the_set_of_l2RealSequences &
  (for x be set holds x is VECTOR of l2_Space iff x is Real_Sequence &
    seq_id(x)(#)seq_id(x)is summable) &
  0.l2_Space = Zeroseq &
  (for u be VECTOR of l2_Space holds u = seq_id(u)) &
  (for u,v be VECTOR of l2_Space holds u+v =seq_id(u)+seq_id(v)) &
  (for r be Real for u be VECTOR of l2_Space holds r*u =r(#)seq_id(u)) &
  (for u be VECTOR of l2_Space holds -u =-seq_id(u) & seq_id(-u)=-seq_id(u)) &
  (for u,v be VECTOR of l2_Space holds u-v =seq_id(u)-seq_id(v)) &
  (for v,w be VECTOR of l2_Space holds seq_id(v)(#)seq_id(w) is summable) &
  for v,w be VECTOR of l2_Space holds v.|.w = Sum(seq_id(v)(#)seq_id(w))
proof
  thus the carrier of l2_Space = the_set_of_l2RealSequences;
  thus for x be set holds x is Element of l2_Space iff x is Real_Sequence &
  seq_id(x)(#)seq_id(x) is summable
  proof
    let x be set;
    x in the_set_of_RealSequences iff x is Real_Sequence by FUNCT_2:8,66;
    hence thesis by RSSPACE:def 11;
  end;
  thus 0.l2_Space = 0.Linear_Space_of_RealSequences by RSSPACE:def 10
    .= Zeroseq;
  thus for u be VECTOR of l2_Space holds u = seq_id(u);
    set W = the_set_of_l2RealSequences;
    set L1 = Linear_Space_of_RealSequences;
  thus
A10: for u,v be VECTOR of l2_Space holds u+v =seq_id(u)+seq_id(v)
  proof
    let u,v be VECTOR of l2_Space;
    reconsider u1=u,v1=v as VECTOR of Linear_Space_of_RealSequences by
RLSUB_1:10,RSSPACE:12;
    dom (the addF of L1) = [:the carrier of L1,the carrier of L1:] by
FUNCT_2:def 1;
    then dom ((the addF of Linear_Space_of_RealSequences)||W) =[:W,W:] by
RELAT_1:62,ZFMISC_1:96;
    then
A11: [u,v] in dom ((the addF of Linear_Space_of_RealSequences)||W)
by ZFMISC_1:87;
    u+v =((the addF of Linear_Space_of_RealSequences)||W).[u,v] by
RSSPACE:def 8
      .=u1+v1 by A11,FUNCT_1:47;
    hence thesis by RSSPACE:2;
  end;
  thus
A12: for r be Real for u be VECTOR of l2_Space holds r*u =r(#)seq_id(u)
  proof
    let r be Real;
    reconsider r as Element of REAL by XREAL_0:def 1;
    let u be VECTOR of l2_Space;
    reconsider u1=u as VECTOR of Linear_Space_of_RealSequences by RLSUB_1:10
,RSSPACE:12;
    dom (the Mult of L1) = [:REAL,the carrier of L1:] by FUNCT_2:def 1;
    then dom ((the Mult of Linear_Space_of_RealSequences) | [:REAL,W:]) =[:
    REAL,W:] by RELAT_1:62,ZFMISC_1:96;
    then
A13: [r,u] in dom ((the Mult of Linear_Space_of_RealSequences)|[:REAL,W :]
    ) by ZFMISC_1:87;
    r*u =((the Mult of Linear_Space_of_RealSequences)|[:REAL,W:]).[r,u]
    by RSSPACE:def 9
      .=r*u1 by A13,FUNCT_1:47;
    hence thesis by RSSPACE:3;
  end;
  thus
A15: for u be VECTOR of l2_Space holds -u =-seq_id(u) & seq_id(-u)=-seq_id(u )
  proof
    let u be VECTOR of l2_Space;
    -u = (-1)*u by RLVECT_1:16
      .= (-1)(#)seq_id(u) by A12
      .= -seq_id(u) by SEQ_1:17;
    hence thesis;
  end;
  thus for u,v be VECTOR of l2_Space holds u-v =seq_id(u)-seq_id(v)
  proof
    let u,v be VECTOR of l2_Space;
    thus u-v = seq_id(u)+seq_id(-v) by A10
      .= seq_id(u)+(-seq_id(v)) by A15
      .= seq_id(u)-seq_id(v) by SEQ_1:11;
  end;
  thus for u,v be VECTOR of l2_Space holds seq_id(u)(#)seq_id(v) is summable
  proof
    set q0 = 1/2;
    let u,v be VECTOR of l2_Space;
    set p = (seq_id(v))(#)(seq_id(v));
    set q = (seq_id(u))(#)(seq_id(u));
    set r = abs( (seq_id(u))(#)(seq_id(v)));
A2: for n be Nat holds 0<=(2(#)r).n
    proof
      set rr=(seq_id u)(#)(seq_id v);
      let n be Nat;
      reconsider tt=|.rr.n.| as Real;
A3:   0 <= tt by COMPLEX1:46;
      (2(#)r).n = 2*r.n by SEQ_1:9
        .= 2*|.rr.n.| by SEQ_1:12;
      hence thesis by A3;
    end;
A4: for n be Nat holds (2(#)r).n <=(p+q).n
    proof
      set s = seq_id v, t = seq_id u;
      let n be Nat;
      reconsider sn=s.n,tn=t.n as Real;
      reconsider ss=|.sn.|,tt=|.tn.| as Real;
A5:   (p+q).n = p.n +q.n by SEQ_1:7
        .= (s.n*s.n) + q.n by SEQ_1:8
        .= sn^2 + (t.n*t.n) by SEQ_1:8
        .=(|.sn.|)^2 + tn^2 by COMPLEX1:75
        .=(|.sn.|)^2 + (|.tn.|)^2 by COMPLEX1:75;
A6:   0 <= (|.sn.|-|.tn.|)^2 by XREAL_1:63;
      (2(#)r).n =2*r.n by SEQ_1:9
        .=2*|.((seq_id u)(#)(seq_id v)).n .| by SEQ_1:12
        .=2*(|.(seq_id u).n * (seq_id v).n.|) by SEQ_1:8
        .= 2*(ss*tt) by COMPLEX1:65
        .= 2*|.sn.|*|.tn.|;
      then 0 + (2(#)r).n <= (p+q).n - (2(#)r).n + (2(#)r).n by A5,A6,XREAL_1:7;
      hence thesis;
    end;
A7: q0(#)(2(#)r)=(q0*2)(#)r by SEQ_1:23
      .=r by SEQ_1:27;
    (seq_id(v))(#)(seq_id(v)) is summable & (seq_id(u))(#)(seq_id(u)) is
    summable by RSSPACE:def 11;
    then p+q is summable by SERIES_1:7;
    then 2(#)r is summable by A2,A4,SERIES_1:20;
    then r is summable by A7,SERIES_1:10;
    then (seq_id(u))(#)(seq_id(v)) is absolutely_summable by SERIES_1:def 4;
    hence thesis;
  end;
  thus for v,w be VECTOR of l2_Space holds v.|.w = Sum(seq_id(v)(#)seq_id(w))
  proof
    let v,w be VECTOR of l2_Space;
    thus v.|.w = (the scalar of l2_Space).(v,w) by BHSP_1:def 1
      .= Sum(seq_id(v)(#)seq_id(w)) by RSSPACE:def 12;
  end;
end;
