:: by Noboru Endou

::

:: Received February 24, 2004

:: Copyright (c) 2004-2021 Association of Mizar Users

definition

ex b_{1} being Subset of Linear_Space_of_ComplexSequences st

for x being object holds

( x in b_{1} iff ( x in the_set_of_ComplexSequences & seq_id x is absolutely_summable ) )

for b_{1}, b_{2} being Subset of Linear_Space_of_ComplexSequences st ( for x being object holds

( x in b_{1} iff ( x in the_set_of_ComplexSequences & seq_id x is absolutely_summable ) ) ) & ( for x being object holds

( x in b_{2} iff ( x in the_set_of_ComplexSequences & seq_id x is absolutely_summable ) ) ) holds

b_{1} = b_{2}
end;

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 for x being object holds

( x in it iff ( x in the_set_of_ComplexSequences & seq_id x is absolutely_summable ) );

ex b

for x being object holds

( x in b

proof end;

uniqueness for b

( x in b

( x in b

b

proof end;

:: deftheorem Def1 defines the_set_of_l1ComplexSequences CSSPACE3:def 1 :

for b_{1} being Subset of Linear_Space_of_ComplexSequences holds

( b_{1} = the_set_of_l1ComplexSequences iff for x being object holds

( x in b_{1} iff ( x in the_set_of_ComplexSequences & seq_id x is absolutely_summable ) ) );

for b

( b

( x in b

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).| )

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;

Lm1: CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is Subspace of Linear_Space_of_ComplexSequences

by CSSPACE:11;

registration

( CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is Abelian & CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is add-associative & CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is right_zeroed & CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is right_complementable & CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is vector-distributive & CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is scalar-distributive & CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is scalar-associative & CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is scalar-unital ) by CSSPACE:11;

end;

cluster CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;

coherence ( CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is Abelian & CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is add-associative & CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is right_zeroed & CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is right_complementable & CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is vector-distributive & CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is scalar-distributive & CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is scalar-associative & CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is scalar-unital ) by CSSPACE:11;

definition

ex b_{1} being Function of the_set_of_l1ComplexSequences,REAL st

for x being object st x in the_set_of_l1ComplexSequences holds

b_{1} . x = Sum |.(seq_id x).|

for b_{1}, b_{2} being Function of the_set_of_l1ComplexSequences,REAL st ( for x being object st x in the_set_of_l1ComplexSequences holds

b_{1} . x = Sum |.(seq_id x).| ) & ( for x being object st x in the_set_of_l1ComplexSequences holds

b_{2} . x = Sum |.(seq_id x).| ) holds

b_{1} = b_{2}
end;

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 |.(seq_id x).|;

existence for x being object st x in the_set_of_l1ComplexSequences holds

it . x = Sum |.(seq_id x).|;

ex b

for x being object st x in the_set_of_l1ComplexSequences holds

b

proof end;

uniqueness for b

b

b

b

proof end;

:: deftheorem Def2 defines cl_norm CSSPACE3:def 2 :

for b_{1} being Function of the_set_of_l1ComplexSequences,REAL holds

( b_{1} = cl_norm iff for x being object st x in the_set_of_l1ComplexSequences holds

b_{1} . x = Sum |.(seq_id x).| );

for b

( b

b

registration

let X be non empty set ;

let Z be Element of X;

let A be BinOp of X;

let M be Function of [:COMPLEX,X:],X;

let N be Function of X,REAL;

coherence

not CNORMSTR(# X,Z,A,M,N #) is empty ;

end;
let Z be Element of X;

let A be BinOp of X;

let M be Function of [:COMPLEX,X:],X;

let N be Function of X,REAL;

coherence

not CNORMSTR(# X,Z,A,M,N #) is empty ;

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;

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 )

( cseq is absolutely_summable & Sum |.cseq.| = 0 )

proof end;

Lm2: CLSStruct(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)) #) is ComplexLinearSpace

;

definition

CNORMSTR(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),cl_norm #) is non empty CNORMSTR ;

end;

func Complex_l1_Space -> non empty CNORMSTR equals :: CSSPACE3:def 3

CNORMSTR(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),cl_norm #);

coherence CNORMSTR(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),cl_norm #);

CNORMSTR(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),cl_norm #) is non empty CNORMSTR ;

:: deftheorem defines Complex_l1_Space CSSPACE3:def 3 :

Complex_l1_Space = CNORMSTR(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),cl_norm #);

Complex_l1_Space = CNORMSTR(# the_set_of_l1ComplexSequences,(Zero_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_l1ComplexSequences,Linear_Space_of_ComplexSequences)),cl_norm #);

::$CT

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 = (seq_id u) + (seq_id v) ) & ( for p being Complex

for u being VECTOR of Complex_l1_Space holds p * u = p (#) (seq_id u) ) & ( for u being VECTOR of Complex_l1_Space holds

( - u = - (seq_id u) & seq_id (- u) = - (seq_id u) ) ) & ( for u, v being VECTOR of Complex_l1_Space holds u - v = (seq_id u) - (seq_id 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 ||.v.|| = Sum |.(seq_id v).| ) )

( 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 = (seq_id u) + (seq_id v) ) & ( for p being Complex

for u being VECTOR of Complex_l1_Space holds p * u = p (#) (seq_id u) ) & ( for u being VECTOR of Complex_l1_Space holds

( - u = - (seq_id u) & seq_id (- u) = - (seq_id u) ) ) & ( for u, v being VECTOR of Complex_l1_Space holds u - v = (seq_id u) - (seq_id 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 ||.v.|| = Sum |.(seq_id v).| ) )

proof end;

theorem Th6: :: CSSPACE3:7

for x, y being Point of Complex_l1_Space

for p being Complex holds

( ( ||.x.|| = 0 implies x = 0. Complex_l1_Space ) & ( x = 0. Complex_l1_Space implies ||.x.|| = 0 ) & 0 <= ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(p * x).|| = |.p.| * ||.x.|| )

for p being Complex holds

( ( ||.x.|| = 0 implies x = 0. Complex_l1_Space ) & ( x = 0. Complex_l1_Space implies ||.x.|| = 0 ) & 0 <= ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(p * x).|| = |.p.| * ||.x.|| )

proof end;

registration

( Complex_l1_Space is reflexive & Complex_l1_Space is discerning & Complex_l1_Space is ComplexNormSpace-like & Complex_l1_Space is vector-distributive & Complex_l1_Space is scalar-distributive & Complex_l1_Space is scalar-associative & Complex_l1_Space is scalar-unital & Complex_l1_Space is Abelian & Complex_l1_Space is add-associative & Complex_l1_Space is right_zeroed & Complex_l1_Space is right_complementable ) by Lm2, Th6, CLVECT_1:def 13, CSSPACE:79;

end;

cluster Complex_l1_Space -> non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ;

coherence ( Complex_l1_Space is reflexive & Complex_l1_Space is discerning & Complex_l1_Space is ComplexNormSpace-like & Complex_l1_Space is vector-distributive & Complex_l1_Space is scalar-distributive & Complex_l1_Space is scalar-associative & Complex_l1_Space is scalar-unital & Complex_l1_Space is Abelian & Complex_l1_Space is add-associative & Complex_l1_Space is right_zeroed & Complex_l1_Space is right_complementable ) by Lm2, Th6, CLVECT_1:def 13, CSSPACE:79;

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
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).||;

for X being non empty CNORMSTR

for x, y being Point of X holds dist (x,y) = ||.(x - y).||;

:: 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 );

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;
let seq be sequence of CNRM;

synonym Cauchy_sequence_by_Norm seq for CCauchy ;

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 )

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

vseq is convergent

proof end;