:: Banach Space of Absolute Summable Complex Sequences
:: by Noboru Endou
::
:: Copyright (c) 2004-2021 Association of Mizar Users

definition
func the_set_of_l1ComplexSequences -> Subset of Linear_Space_of_ComplexSequences means :Def1: :: CSSPACE3:def 1
for x being object holds
( x in it iff ( x in the_set_of_ComplexSequences & seq_id x is absolutely_summable ) );
existence
ex b1 being Subset of Linear_Space_of_ComplexSequences st
for x being object holds
( x in b1 iff ( x in the_set_of_ComplexSequences & seq_id x is absolutely_summable ) )
proof end;
uniqueness
for b1, b2 being Subset of Linear_Space_of_ComplexSequences st ( for x being object holds
( x in b1 iff ( x in the_set_of_ComplexSequences & seq_id x is absolutely_summable ) ) ) & ( for x being object holds
( x in b2 iff ( x in the_set_of_ComplexSequences & seq_id x is absolutely_summable ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines the_set_of_l1ComplexSequences CSSPACE3:def 1 :
for b1 being Subset of Linear_Space_of_ComplexSequences holds
( b1 = the_set_of_l1ComplexSequences iff for x being object holds
( x in b1 iff ( x in the_set_of_ComplexSequences & seq_id x is absolutely_summable ) ) );

theorem Th1: :: CSSPACE3:1
for c being Complex
for seq being Complex_Sequence
for rseq being Real_Sequence st seq is convergent & ( for i being Nat holds rseq . i = |.((seq . i) - c).| ) holds
( rseq is convergent & lim rseq = |.((lim seq) - c).| )
proof end;

registration
coherence
proof end;
end;

registration end;

Lm1:
by CSSPACE:11;

definition
func cl_norm -> Function of the_set_of_l1ComplexSequences,REAL means :Def2: :: CSSPACE3:def 2
for x being object st x in the_set_of_l1ComplexSequences holds
it . x = Sum |.().|;
existence
ex b1 being Function of the_set_of_l1ComplexSequences,REAL st
for x being object st x in the_set_of_l1ComplexSequences holds
b1 . x = Sum |.().|
proof end;
uniqueness
for b1, b2 being Function of the_set_of_l1ComplexSequences,REAL st ( for x being object st x in the_set_of_l1ComplexSequences holds
b1 . x = Sum |.().| ) & ( for x being object st x in the_set_of_l1ComplexSequences holds
b2 . x = Sum |.().| ) holds
b1 = b2
proof end;
end;

:: deftheorem Def2 defines cl_norm CSSPACE3:def 2 :
for b1 being Function of the_set_of_l1ComplexSequences,REAL holds
( b1 = cl_norm iff for x being object st x in the_set_of_l1ComplexSequences holds
b1 . x = Sum |.().| );

registration
let X be non empty set ;
let Z be Element of X;
let A be BinOp of X;
let M be Function of ,X;
let N be Function of X,REAL;
cluster CNORMSTR(# X,Z,A,M,N #) -> non empty ;
coherence
not CNORMSTR(# X,Z,A,M,N #) is empty
;
end;

theorem :: CSSPACE3:2
for l being CNORMSTR st CLSStruct(# the carrier of l, the ZeroF of l, the addF of l, the Mult of l #) is ComplexLinearSpace holds
l is ComplexLinearSpace by CSSPACE:79;

theorem Th3: :: CSSPACE3:3
for cseq being Complex_Sequence st ( for n being Nat holds cseq . n = 0c ) holds
( cseq is absolutely_summable & Sum |.cseq.| = 0 )
proof end;

Lm2:
;

theorem :: CSSPACE3:4
canceled;

::\$CT
theorem Th4: :: CSSPACE3:5

theorem Th5: :: CSSPACE3:6
( the carrier of Complex_l1_Space = the_set_of_l1ComplexSequences & ( for x being set holds
( x is VECTOR of Complex_l1_Space iff ( x is Complex_Sequence & seq_id x is absolutely_summable ) ) ) & 0. Complex_l1_Space = CZeroseq & ( for u being VECTOR of Complex_l1_Space holds u = seq_id u ) & ( for u, v being VECTOR of Complex_l1_Space holds u + v = () + () ) & ( for p being Complex
for u being VECTOR of Complex_l1_Space holds p * u = p (#) () ) & ( for u being VECTOR of Complex_l1_Space holds
( - u = - () & seq_id (- u) = - () ) ) & ( for u, v being VECTOR of Complex_l1_Space holds u - v = () - () ) & ( for v being VECTOR of Complex_l1_Space holds seq_id v is absolutely_summable ) & ( for v being VECTOR of Complex_l1_Space holds = Sum |.().| ) )
proof end;

theorem Th6: :: CSSPACE3:7
for x, y being Point of Complex_l1_Space
for p being Complex holds
( ( = 0 implies x = 0. Complex_l1_Space ) & ( x = 0. Complex_l1_Space implies = 0 ) & 0 <= & ||.(x + y).|| <= + & ||.(p * x).|| = |.p.| * )
proof end;

registration
coherence by ;
end;

Lm3: for c being Complex
for seq being Complex_Sequence
for seq1 being Real_Sequence st seq is convergent & seq1 is convergent holds
for rseq being Real_Sequence st ( for i being Nat holds rseq . i = |.((seq . i) - c).| + (seq1 . i) ) holds
( rseq is convergent & lim rseq = |.((lim seq) - c).| + (lim seq1) )

proof end;

definition
let X be non empty CNORMSTR ;
let x, y be Point of X;
func dist (x,y) -> Real equals :: CSSPACE3:def 4
||.(x - y).||;
coherence
||.(x - y).|| is Real
;
end;

:: deftheorem defines dist CSSPACE3:def 4 :
for X being non empty CNORMSTR
for x, y being Point of X holds dist (x,y) = ||.(x - y).||;

definition
let CNRM be non empty CNORMSTR ;
let seqt be sequence of CNRM;
attr seqt is CCauchy means :: CSSPACE3:def 5
for r1 being Real st r1 > 0 holds
ex k1 being Nat st
for n1, m1 being Nat st n1 >= k1 & m1 >= k1 holds
dist ((seqt . n1),(seqt . m1)) < r1;
end;

:: deftheorem defines CCauchy CSSPACE3:def 5 :
for CNRM being non empty CNORMSTR
for seqt being sequence of CNRM holds
( seqt is CCauchy iff for r1 being Real st r1 > 0 holds
ex k1 being Nat st
for n1, m1 being Nat st n1 >= k1 & m1 >= k1 holds
dist ((seqt . n1),(seqt . m1)) < r1 );

notation
let CNRM be non empty CNORMSTR ;
let seq be sequence of CNRM;
synonym Cauchy_sequence_by_Norm seq for CCauchy ;
end;

theorem Th7: :: CSSPACE3:8
for NRM being ComplexNormSpace
for seq being sequence of NRM holds
( seq is Cauchy_sequence_by_Norm iff for r being Real st r > 0 holds
ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
||.((seq . n) - (seq . m)).|| < r )
proof end;

theorem :: CSSPACE3:9
for vseq being sequence of Complex_l1_Space st vseq is Cauchy_sequence_by_Norm holds
vseq is convergent
proof end;