let X be RealNormSpace-Sequence; :: thesis:
let Y be RealNormSpace; :: thesis:
assume A1: Y is complete ; :: thesis:
let vseq be sequence of (R_NormSpace_of_BoundedMultilinearOperators (X,Y)); :: thesis:
assume A2: vseq is Cauchy_sequence_by_Norm ; :: thesis:
defpred S₁[ set , set ] means ex xseq being sequence of Y st
( ( for n being Nat holds xseq . n = (modetrans ((vseq . n),X,Y)) . $1 ) & xseq is convergent & $2 = lim xseq );
A3: for x being Element of (product X) ex y being Element of Y st S₁[x,y]

let x be Element of (product X); :: thesis:
deffunc H₁( Nat) -> Element of the carrier of Y = (modetrans ((vseq . $1),X,Y)) . x;
consider xseq being sequence of Y such that
A4: for n being Element of NAT holds xseq . n = H₁(n) from FUNCT_2:sch 4();
A5: for n being Nat holds xseq . n = H₁(n)

let n be Nat; :: thesis:
n in NAT by ORDINAL1:def 12;
hence xseq . n = H₁(n) by A4; :: thesis:

end;

take lim xseq ; :: thesis:
A6: for m, k being Nat holds ||.((xseq . m) - (xseq . k)).|| <= ||.((vseq . m) - (vseq . k)).|| * (NrProduct x)

proof

let m, k be Nat; :: thesis:
reconsider h1 = (vseq . m) - (vseq . k) as Lipschitzian MultilinearOperator of X,Y by LOPBAN10:def 11;
k in NAT by ORDINAL1:def 12;
then A7: xseq . k = (modetrans ((vseq . k),X,Y)) . x by A4;
a8: vseq . m is Lipschitzian MultilinearOperator of X,Y by LOPBAN10:def 11;
a9: vseq . k is Lipschitzian MultilinearOperator of X,Y by LOPBAN10:def 11;
m in NAT by ORDINAL1:def 12;
then xseq . m = (modetrans ((vseq . m),X,Y)) . x by A4;
then (xseq . m) - (xseq . k) = h1 . x by A7, a8, a9, LOPBAN10:52;
hence ||.((xseq . m) - (xseq . k)).|| <= ||.((vseq . m) - (vseq . k)).|| * (NrProduct x) by LOPBAN10:45; :: thesis:

end;

now :: thesis:

let e be Real; :: thesis:
assume A10: e > 0 ; :: thesis:

now :: thesis:

per cases ;

case A11: ex i being Element of dom X st x . i = 0. (X . i) ; :: thesis:

reconsider k = 0 as Nat ;
take k = k; :: thesis:
thus for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e :: thesis:

proof

let n, m be Nat; :: thesis:
assume that
n >= k and
m >= k ; :: thesis:
m in NAT by ORDINAL1:def 12;
then A12: xseq . m = (modetrans ((vseq . m),X,Y)) . x by A4
.= 0. Y by A11, LOPBAN10:36 ;
n in NAT by ORDINAL1:def 12;
then xseq . n = (modetrans ((vseq . n),X,Y)) . x by A4
.= 0. Y by A11, LOPBAN10:36 ;
hence ||.((xseq . n) - (xseq . m)).|| < e by A10, A12; :: thesis:

end;

case for i being Element of dom X holds not x . i = 0. (X . i) ; :: thesis:

then A13: NrProduct x > 0 by LOPBAN10:27;
then consider k being Nat such that
A15: for n, m being Nat st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e / (NrProduct x) by A2, A10, RSSPACE3:8;
take k = k; :: thesis:
thus for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e :: thesis:

proof

let n, m be Nat; :: thesis:
assume that
A16: n >= k and
A17: m >= k ; :: thesis:
||.((vseq . n) - (vseq . m)).|| < e / (NrProduct x) by A15, A16, A17;
then A18: ||.((vseq . n) - (vseq . m)).|| * (NrProduct x) < (e / (NrProduct x)) * (NrProduct x) by A13, XREAL_1:68;
A19: (e / (NrProduct x)) * (NrProduct x) = e * (((NrProduct x) ") * (NrProduct x))
.= e * 1 by A13, XCMPLX_0:def 7
.= e ;
||.((xseq . n) - (xseq . m)).|| <= ||.((vseq . n) - (vseq . m)).|| * (NrProduct x) by A6;
hence ||.((xseq . n) - (xseq . m)).|| < e by A18, A19, XXREAL_0:2; :: thesis:

end;

hence ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e ; :: thesis:

end;

then xseq is Cauchy_sequence_by_Norm by RSSPACE3:8;
hence S₁[x, lim xseq] by A1, A5; :: thesis:

end;

consider f being Function of the carrier of (product X), the carrier of Y such that
A20: for x being Element of (product X) holds S₁[x,f . x] from FUNCT_2:sch 3(A3);
reconsider tseq = f as Function of (product X),Y ;
A21: for u being Point of (product X)
for i being Element of dom X
for x being Point of (X . i) ex xseqi being sequence of Y st
( ( for n being Nat holds xseqi . n = ((modetrans ((vseq . n),X,Y)) * (reproj (i,u))) . x ) & xseqi is convergent & (tseq * (reproj (i,u))) . x = lim xseqi )

proof

let u be Point of (product X); :: thesis:
let i be Element of dom X; :: thesis:
let x be Point of (X . i); :: thesis:
reconsider v = (reproj (i,u)) . x as Point of (product X) ;
consider xseq being sequence of Y such that
A22: ( ( for n being Nat holds xseq . n = (modetrans ((vseq . n),X,Y)) . v ) & xseq is convergent & tseq . v = lim xseq ) by A20;
A23: dom (reproj (i,u)) = the carrier of (X . i) by FUNCT_2:def 1;
take xseq ; :: thesis:
thus for n being Nat holds xseq . n = ((modetrans ((vseq . n),X,Y)) * (reproj (i,u))) . x :: thesis:

proof

let n be Nat; :: thesis:
thus xseq . n = (modetrans ((vseq . n),X,Y)) . v by A22
.= (vseq . n) . ((reproj (i,u)) . x) by LOPBAN10:def 13
.= ((vseq . n) * (reproj (i,u))) . x by A23, FUNCT_1:13
.= ((modetrans ((vseq . n),X,Y)) * (reproj (i,u))) . x by LOPBAN10:def 13 ; :: thesis:

end;

thus xseq is convergent by A22; :: thesis:
thus (tseq * (reproj (i,u))) . x = lim xseq by A22, A23, FUNCT_1:13; :: thesis:

end;

now :: thesis:

let i be Element of dom X; :: thesis:
let u be Point of (product X); :: thesis:
set tseqiu = tseq * (reproj (i,u));
deffunc H₁( Nat) -> Element of K16(K17( the carrier of (X . i), the carrier of Y)) = (modetrans ((vseq . $1),X,Y)) * (reproj (i,u));

A24: now :: thesis:

let x, y be Point of (X . i); :: thesis:
consider xseq being sequence of Y such that
A25: for n being Nat holds xseq . n = H₁(n) . x and
A26: xseq is convergent and
A27: (tseq * (reproj (i,u))) . x = lim xseq by A21;
consider zseq being sequence of Y such that
A28: for n being Nat holds zseq . n = H₁(n) . (x + y) and
zseq is convergent and
A29: (tseq * (reproj (i,u))) . (x + y) = lim zseq by A21;
consider yseq being sequence of Y such that
A30: for n being Nat holds yseq . n = H₁(n) . y and
A31: yseq is convergent and
A32: (tseq * (reproj (i,u))) . y = lim yseq by A21;

now :: thesis:

let n be Nat; :: thesis:
A33: H₁(n) is LinearOperator of (X . i),Y by LOPBAN10:def 6;
thus zseq . n = H₁(n) . (x + y) by A28
.= (H₁(n) . x) + (H₁(n) . y) by A33, VECTSP_1:def 20
.= (xseq . n) + (H₁(n) . y) by A25
.= (xseq . n) + (yseq . n) by A30 ; :: thesis:

end;

then zseq = xseq + yseq by NORMSP_1:def 2;
hence (tseq * (reproj (i,u))) . (x + y) = ((tseq * (reproj (i,u))) . x) + ((tseq * (reproj (i,u))) . y) by A26, A27, A29, A31, A32, NORMSP_1:25; :: thesis:

end;

now :: thesis:

let x be Point of (X . i); :: thesis:
let a be Real; :: thesis:
consider xseq being sequence of Y such that
A34: for n being Nat holds xseq . n = H₁(n) . x and
A35: xseq is convergent and
A36: (tseq * (reproj (i,u))) . x = lim xseq by A21;
consider zseq being sequence of Y such that
A37: for n being Nat holds zseq . n = H₁(n) . (a * x) and
zseq is convergent and
A38: (tseq * (reproj (i,u))) . (a * x) = lim zseq by A21;

now :: thesis:

let n be Nat; :: thesis:
A39: H₁(n) is LinearOperator of (X . i),Y by LOPBAN10:def 6;
thus zseq . n = H₁(n) . (a * x) by A37
.= a * (H₁(n) . x) by A39, LOPBAN_1:def 5
.= a * (xseq . n) by A34 ; :: thesis:

end;

then zseq = a * xseq by NORMSP_1:def 5;
hence (tseq * (reproj (i,u))) . (a * x) = a * ((tseq * (reproj (i,u))) . x) by A35, A36, A38, NORMSP_1:28; :: thesis:

end;

hence tseq * (reproj (i,u)) is LinearOperator of (X . i),Y by A24, LOPBAN_1:def 5, VECTSP_1:def 20; :: thesis:

end;

then reconsider tseq = tseq as MultilinearOperator of X,Y by LOPBAN10:def 6;

B39: now :: thesis:

let e1 be Real; :: thesis:
assume A40: e1 > 0 ; :: thesis:
reconsider e = e1 as Real ;
consider k being Nat such that
A41: for n, m being Nat st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e by A2, A40, RSSPACE3:8;
reconsider k = k as Nat ;
take k = k; :: thesis:

now :: thesis:

let m be Nat; :: thesis:
assume m >= k ; :: thesis:
then A42: ||.((vseq . m) - (vseq . k)).|| < e by A41;
A43: ||.(vseq . m).|| = ||.vseq.|| . m by NORMSP_0:def 4;
A44: ||.(vseq . k).|| = ||.vseq.|| . k by NORMSP_0:def 4;
|.(||.(vseq . m).|| - ||.(vseq . k).||).| <= ||.((vseq . m) - (vseq . k)).|| by NORMSP_1:9;
hence |.((||.vseq.|| . m) - (||.vseq.|| . k)).| < e1 by A42, A43, A44, XXREAL_0:2; :: thesis:

end;

hence for m being Nat st m >= k holds
|.((||.vseq.|| . m) - (||.vseq.|| . k)).| < e1 ; :: thesis:

end;

then A45: ||.vseq.|| is convergent by SEQ_4:41;
A46: tseq is Lipschitzian

proof

take lim ||.vseq.|| ; :: according to LOPBAN10:def 10 :: thesis:

A47: now :: thesis:

let x be Point of (product X); :: thesis:
consider xseq being sequence of Y such that
A48: for n being Nat holds xseq . n = (modetrans ((vseq . n),X,Y)) . x and
A49: xseq is convergent and
A50: tseq . x = lim xseq by A20;
A51: ||.(tseq . x).|| = lim ||.xseq.|| by A49, A50, LOPBAN_1:20;
A52: for m being Nat holds ||.(xseq . m).|| <= ||.(vseq . m).|| * (NrProduct x)

proof

let m be Nat; :: thesis:
A53: xseq . m = (modetrans ((vseq . m),X,Y)) . x by A48;
vseq . m is Lipschitzian MultilinearOperator of X,Y by LOPBAN10:def 11;
hence ||.(xseq . m).|| <= ||.(vseq . m).|| * (NrProduct x) by A53, LOPBAN10:45; :: thesis:

end;

A54: for n being Nat holds ||.xseq.|| . n <= ((NrProduct x) (#) ||.vseq.||) . n

proof

let n be Nat; :: thesis:
A55: ||.xseq.|| . n = ||.(xseq . n).|| by NORMSP_0:def 4;
A56: ||.(vseq . n).|| = ||.vseq.|| . n by NORMSP_0:def 4;
||.(xseq . n).|| <= ||.(vseq . n).|| * (NrProduct x) by A52;
hence ||.xseq.|| . n <= ((NrProduct x) (#) ||.vseq.||) . n by A55, A56, SEQ_1:9; :: thesis:

end;

A58: lim ((NrProduct x) (#) ||.vseq.||) = (lim ||.vseq.||) * (NrProduct x) by B39, SEQ_2:8, SEQ_4:41;
||.xseq.|| is convergent by A49, A50, LOPBAN_1:20;
hence ||.(tseq . x).|| <= (lim ||.vseq.||) * (NrProduct x) by A45, A51, A54, A58, SEQ_2:18; :: thesis:

end;

now :: thesis:

let n be Nat; :: thesis:
||.(vseq . n).|| >= 0 ;
hence ||.vseq.|| . n >= 0 by NORMSP_0:def 4; :: thesis:

end;

hence ( 0 <= lim ||.vseq.|| & ( for b₁ being Element of the carrier of (product X) holds ||.(tseq . b₁).|| <= (lim ||.vseq.||) * (NrProduct b₁) ) ) by B39, A47, SEQ_2:17, SEQ_4:41; :: thesis:

end;

A59: for e being Real st e > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
for x being Point of (product X) holds ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e * (NrProduct x)

proof

let e be Real; :: thesis:
assume e > 0 ; :: thesis:
then consider k being Nat such that
A60: for n, m being Nat st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e by A2, RSSPACE3:8;
take k ; :: thesis:

now :: thesis:

let n be Nat; :: thesis:
assume A61: n >= k ; :: thesis:

now :: thesis:

let x be Point of (product X); :: thesis:
consider xseq being sequence of Y such that
A62: for n being Nat holds xseq . n = (modetrans ((vseq . n),X,Y)) . x and
A63: xseq is convergent and
A64: tseq . x = lim xseq by A20;
A65: for m, k being Nat holds ||.((xseq . m) - (xseq . k)).|| <= ||.((vseq . m) - (vseq . k)).|| * (NrProduct x)

proof

let m, k be Nat; :: thesis:
reconsider h1 = (vseq . m) - (vseq . k) as Lipschitzian MultilinearOperator of X,Y by LOPBAN10:def 11;
A66: xseq . k = (modetrans ((vseq . k),X,Y)) . x by A62;
a67: vseq . m is Lipschitzian MultilinearOperator of X,Y by LOPBAN10:def 11;
a68: vseq . k is Lipschitzian MultilinearOperator of X,Y by LOPBAN10:def 11;
xseq . m = (modetrans ((vseq . m),X,Y)) . x by A62;
then (xseq . m) - (xseq . k) = h1 . x by A66, a67, a68, LOPBAN10:52;
hence ||.((xseq . m) - (xseq . k)).|| <= ||.((vseq . m) - (vseq . k)).|| * (NrProduct x) by LOPBAN10:45; :: thesis:

end;

A69: for m being Nat st m >= k holds
||.((xseq . n) - (xseq . m)).|| <= e * (NrProduct x)

proof

let m be Nat; :: thesis:
assume m >= k ; :: thesis:
then A70: ||.((vseq . n) - (vseq . m)).|| < e by A60, A61;
A71: ||.((xseq . n) - (xseq . m)).|| <= ||.((vseq . n) - (vseq . m)).|| * (NrProduct x) by A65;
||.((vseq . n) - (vseq . m)).|| * (NrProduct x) <= e * (NrProduct x) by A70, XREAL_1:64;
hence ||.((xseq . n) - (xseq . m)).|| <= e * (NrProduct x) by A71, XXREAL_0:2; :: thesis:

end;

||.((xseq . n) - (tseq . x)).|| <= e * (NrProduct x)

proof

deffunc H₁( Nat) -> object = ||.((xseq . $1) - (xseq . n)).||;
consider rseq being Real_Sequence such that
A72: for m being Nat holds rseq . m = H₁(m) from SEQ_1:sch 1();

now :: thesis:

let x be object ; :: thesis:
assume x in NAT ; :: thesis:
then reconsider k = x as Nat ;
thus rseq . x = ||.((xseq . k) - (xseq . n)).|| by A72
.= ||.((xseq - (xseq . n)) . k).|| by NORMSP_1:def 4
.= ||.(xseq - (xseq . n)).|| . x by NORMSP_0:def 4 ; :: thesis:

end;

then A73: rseq = ||.(xseq - (xseq . n)).|| by FUNCT_2:12;
A74: xseq - (xseq . n) is convergent by A63, NORMSP_1:21;
lim (xseq - (xseq . n)) = (tseq . x) - (xseq . n) by A63, A64, NORMSP_1:27;
then A75: lim rseq = ||.((tseq . x) - (xseq . n)).|| by A73, A74, LOPBAN_1:41;
for m being Nat st m >= k holds
rseq . m <= e * (NrProduct x)

proof

let m be Nat; :: thesis:
assume A76: m >= k ; :: thesis:
rseq . m = ||.((xseq . m) - (xseq . n)).|| by A72
.= ||.((xseq . n) - (xseq . m)).|| by NORMSP_1:7 ;
hence rseq . m <= e * (NrProduct x) by A69, A76; :: thesis:

end;

then lim rseq <= e * (NrProduct x) by A73, A74, LOPBAN_1:41, RSSPACE2:5;
hence ||.((xseq . n) - (tseq . x)).|| <= e * (NrProduct x) by A75, NORMSP_1:7; :: thesis:

end;

hence ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e * (NrProduct x) by A62; :: thesis:

end;

hence for x being Point of (product X) holds ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e * (NrProduct x) ; :: thesis:

end;

hence for n being Nat st n >= k holds
for x being Point of (product X) holds ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e * (NrProduct x) ; :: thesis:

end;

reconsider tseq = tseq as Lipschitzian MultilinearOperator of X,Y by A46;
reconsider tv = tseq as Point of (R_NormSpace_of_BoundedMultilinearOperators (X,Y)) by LOPBAN10:def 11;
A77: for e being Real st e > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
||.((vseq . n) - tv).|| <= e

proof

let e be Real; :: thesis:
assume A78: e > 0 ; :: thesis:
consider k being Nat such that
A79: for n being Nat st n >= k holds
for x being Point of (product X) holds ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e * (NrProduct x) by A59, A78;

now :: thesis:

set g1 = tseq;
let n be Nat; :: thesis:
assume A80: n >= k ; :: thesis:
reconsider h1 = (vseq . n) - tv as Lipschitzian MultilinearOperator of X,Y by LOPBAN10:def 11;
set f1 = modetrans ((vseq . n),X,Y);

A81: now :: thesis:

let t be Point of (product X); :: thesis:
assume for i being Element of dom X holds ||.(t . i).|| <= 1 ; :: thesis:
then ( 0 <= NrProduct t & NrProduct t <= 1 ) by LOPBAN10:35;
then A82: e * (NrProduct t) <= e * 1 by A78, XREAL_1:64;
A83: ||.(((modetrans ((vseq . n),X,Y)) . t) - (tseq . t)).|| <= e * (NrProduct t) by A79, A80;
vseq . n is Lipschitzian MultilinearOperator of X,Y by LOPBAN10:def 11;
then ||.(h1 . t).|| = ||.(((modetrans ((vseq . n),X,Y)) . t) - (tseq . t)).|| by LOPBAN10:52;
hence ||.(h1 . t).|| <= e by A82, A83, XXREAL_0:2; :: thesis:

end;

A84: now :: thesis:

let r be Real; :: thesis:
assume r in PreNorms h1 ; :: thesis:
then ex t being Point of (product X) st
( r = ||.(h1 . t).|| & ( for i being Element of dom X holds ||.(t . i).|| <= 1 ) ) ;
hence r <= e by A81; :: thesis:

end;

( ( for s being Real st s in PreNorms h1 holds
s <= e ) implies upper_bound (PreNorms h1) <= e ) by SEQ_4:45;
hence ||.((vseq . n) - tv).|| <= e by A84, LOPBAN10:43; :: thesis:

end;

hence ex k being Nat st
for n being Nat st n >= k holds
||.((vseq . n) - tv).|| <= e ; :: thesis:

end;

for e being Real st e > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((vseq . n) - tv).|| < e

proof

let e be Real; :: thesis:
assume A86: e > 0 ; :: thesis:
consider m being Nat such that
A87: for n being Nat st n >= m holds
||.((vseq . n) - tv).|| <= e / 2 by A77, A86;
A88: e / 2 < e by A86, XREAL_1:216;

now :: thesis:

let n be Nat; :: thesis:
assume n >= m ; :: thesis:
then ||.((vseq . n) - tv).|| <= e / 2 by A87;
hence ||.((vseq . n) - tv).|| < e by A88, XXREAL_0:2; :: thesis:

end;

hence ex m being Nat st
for n being Nat st n >= m holds
||.((vseq . n) - tv).|| < e ; :: thesis:

end;

hence vseq is convergent by NORMSP_1:def 6; :: thesis: