CSSPACE4: Complex Banach Space of Bounded Complex Sequences

:: Complex Banach Space of Bounded Complex Sequences
:: by Noboru Endou
::
:: Received March 18, 2004
:: Copyright (c) 2004-2021 Association of Mizar Users

Lm1: for rseq being Real_Sequence
for K being Real st ( for n being Nat holds rseq . n <= K ) holds
upper_bound (rng rseq) <= K

proof end;

Lm2: for rseq being Real_Sequence st rseq is bounded holds
for n being Nat holds rseq . n <= upper_bound (rng rseq)

proof end;

definition

func the_set_of_BoundedComplexSequences -> Subset of Linear_Space_of_ComplexSequences means :Def1: :: CSSPACE4:def 1
for x being object holds
( x in it iff ( x in the_set_of_ComplexSequences & seq_id x is bounded ) );

proof end;

proof end;

end;

:: deftheorem Def1 defines the_set_of_BoundedComplexSequences CSSPACE4:def 1 :

Lm3: for seq1, seq2 being Complex_Sequence st seq1 is bounded & seq2 is bounded holds
seq1 + seq2 is bounded

proof end;

reconsider jj = 1 as Real ;

Lm4: for c being Complex
for seq being Complex_Sequence st seq is bounded holds
c (#) seq is bounded

proof end;

registration

cluster the_set_of_BoundedComplexSequences -> non empty ;

proof end;

cluster the_set_of_BoundedComplexSequences -> linearly-closed ;

proof end;

end;

Lm5: CLSStruct(# the_set_of_BoundedComplexSequences,(Zero_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)) #) is Subspace of Linear_Space_of_ComplexSequences
by CSSPACE:11;

registration

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

coherence by CSSPACE:11;

end;

Lm6: ( CLSStruct(# the_set_of_BoundedComplexSequences,(Zero_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)) #) is Abelian & CLSStruct(# the_set_of_BoundedComplexSequences,(Zero_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)) #) is add-associative & CLSStruct(# the_set_of_BoundedComplexSequences,(Zero_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)) #) is right_zeroed & CLSStruct(# the_set_of_BoundedComplexSequences,(Zero_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)) #) is right_complementable & CLSStruct(# the_set_of_BoundedComplexSequences,(Zero_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)) #) is vector-distributive & CLSStruct(# the_set_of_BoundedComplexSequences,(Zero_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)) #) is scalar-distributive & CLSStruct(# the_set_of_BoundedComplexSequences,(Zero_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)) #) is scalar-associative & CLSStruct(# the_set_of_BoundedComplexSequences,(Zero_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Add_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)),(Mult_ (the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)) #) is scalar-unital )
;

definition

func Complex_linfty_norm -> Function of the_set_of_BoundedComplexSequences,REAL means :Def2: :: CSSPACE4:def 2
for x being object st x in the_set_of_BoundedComplexSequences holds
it . x = upper_bound (rng |.(seq_id x).|);

proof end;

proof end;

end;

:: deftheorem Def2 defines Complex_linfty_norm CSSPACE4:def 2 :

Lm7: for seq being Complex_Sequence st ( for n being Nat holds seq . n = 0c ) holds
( seq is bounded & upper_bound (rng |.seq.|) = 0 )

proof end;

Lm8: for seq being Complex_Sequence st seq is bounded holds
|.seq.| is bounded

proof end;

Lm9: for seq being Complex_Sequence st |.seq.| is bounded holds
seq is bounded

proof end;

Lm10: for seq being Complex_Sequence st seq is bounded & upper_bound (rng |.seq.|) = 0 holds
for n being Nat holds seq . n = 0c

proof end;

theorem :: CSSPACE4:1

for seq being Complex_Sequence holds
( ( seq is bounded & upper_bound (rng |.seq.|) = 0 ) iff for n being Nat holds seq . n = 0c ) by Lm7, Lm10;

registration

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

coherence by Lm6, CSSPACE3:2;

end;

definition end;

:: deftheorem defines Complex_linfty_Space CSSPACE4:def 3 :

theorem Th2: :: CSSPACE4:2

( the carrier of Complex_linfty_Space = the_set_of_BoundedComplexSequences & ( for x being set holds
( x is VECTOR of Complex_linfty_Space iff ( x is Complex_Sequence & seq_id x is bounded ) ) ) & 0. Complex_linfty_Space = CZeroseq & ( for u being VECTOR of Complex_linfty_Space holds u = seq_id u ) & ( for u, v being VECTOR of Complex_linfty_Space holds u + v = (seq_id u) + (seq_id v) ) & ( for c being Complex
for u being VECTOR of Complex_linfty_Space holds c * u = c (#) (seq_id u) ) & ( for u being VECTOR of Complex_linfty_Space holds
( - u = - (seq_id u) & seq_id (- u) = - (seq_id u) ) ) & ( for u, v being VECTOR of Complex_linfty_Space holds u - v = (seq_id u) - (seq_id v) ) & ( for v being VECTOR of Complex_linfty_Space holds seq_id v is bounded ) & ( for v being VECTOR of Complex_linfty_Space holds ||.v.|| = upper_bound (rng |.(seq_id v).|) ) )

proof end;

theorem Th3: :: CSSPACE4:3

for x, y being Point of Complex_linfty_Space
for c being Complex holds
( ( ||.x.|| = 0 implies x = 0. Complex_linfty_Space ) & ( x = 0. Complex_linfty_Space implies ||.x.|| = 0 ) & 0 <= ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(c * x).|| = |.c.| * ||.x.|| )

proof end;

registration end;

Lm11: for seq1, seq2, seq3 being Complex_Sequence holds
( seq1 = seq2 - seq3 iff for n being Nat holds seq1 . n = (seq2 . n) - (seq3 . n) )

proof end;

theorem Th4: :: CSSPACE4:4

for vseq being sequence of Complex_linfty_Space st vseq is Cauchy_sequence_by_Norm holds
vseq is convergent

proof end;

theorem :: CSSPACE4:5

Complex_linfty_Space is ComplexBanachSpace by Th4, CLOPBAN1:def 13;

definition

let X be non empty set ;
let Y be ComplexNormSpace;
let IT be Function of X, the carrier of Y;

attr IT is bounded means :Def4: :: CSSPACE4:def 4
ex K being Real st
( 0 <= K & ( for x being Element of X holds ||.(IT . x).|| <= K ) );

end;

:: deftheorem Def4 defines bounded CSSPACE4:def 4 :

theorem Th6: :: CSSPACE4:6

for X being non empty set
for Y being ComplexNormSpace
for f being Function of X, the carrier of Y st ( for x being Element of X holds f . x = 0. Y ) holds
f is bounded

proof end;

registration

let X be non empty set ;
let Y be ComplexNormSpace;

cluster Relation-like X -defined the carrier of Y -valued non empty Function-like V26(X) quasi_total bounded for Element of bool [:X, the carrier of Y:];

proof end;

end;

definition

let X be non empty set ;
let Y be ComplexNormSpace;

func ComplexBoundedFunctions (X,Y) -> Subset of (ComplexVectSpace (X,Y)) means :Def5: :: CSSPACE4:def 5
for x being set holds
( x in it iff x is bounded Function of X, the carrier of Y );

proof end;

proof end;

end;

:: deftheorem Def5 defines ComplexBoundedFunctions CSSPACE4:def 5 :

registration

let X be non empty set ;
let Y be ComplexNormSpace;

cluster ComplexBoundedFunctions (X,Y) -> non empty ;

proof end;

end;

theorem Th7: :: CSSPACE4:7

for X being non empty set
for Y being ComplexNormSpace holds ComplexBoundedFunctions (X,Y) is linearly-closed

proof end;

theorem :: CSSPACE4:8

for X being non empty set
for Y being ComplexNormSpace holds CLSStruct(# (ComplexBoundedFunctions (X,Y)),(Zero_ ((ComplexBoundedFunctions (X,Y)),(ComplexVectSpace (X,Y)))),(Add_ ((ComplexBoundedFunctions (X,Y)),(ComplexVectSpace (X,Y)))),(Mult_ ((ComplexBoundedFunctions (X,Y)),(ComplexVectSpace (X,Y)))) #) is Subspace of ComplexVectSpace (X,Y) by Th7, CSSPACE:11;

registration

let X be non empty set ;
let Y be ComplexNormSpace;

cluster CLSStruct(# (ComplexBoundedFunctions (X,Y)),(Zero_ ((ComplexBoundedFunctions (X,Y)),(ComplexVectSpace (X,Y)))),(Add_ ((ComplexBoundedFunctions (X,Y)),(ComplexVectSpace (X,Y)))),(Mult_ ((ComplexBoundedFunctions (X,Y)),(ComplexVectSpace (X,Y)))) #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;

coherence by Th7, CSSPACE:11;

end;

definition

let X be non empty set ;
let Y be ComplexNormSpace;

func C_VectorSpace_of_BoundedFunctions (X,Y) -> ComplexLinearSpace equals :: CSSPACE4:def 6
CLSStruct(# (ComplexBoundedFunctions (X,Y)),(Zero_ ((ComplexBoundedFunctions (X,Y)),(ComplexVectSpace (X,Y)))),(Add_ ((ComplexBoundedFunctions (X,Y)),(ComplexVectSpace (X,Y)))),(Mult_ ((ComplexBoundedFunctions (X,Y)),(ComplexVectSpace (X,Y)))) #);

end;

:: deftheorem defines C_VectorSpace_of_BoundedFunctions CSSPACE4:def 6 :

registration

let X be non empty set ;
let Y be ComplexNormSpace;

cluster C_VectorSpace_of_BoundedFunctions (X,Y) -> strict ;

end;

theorem Th9: :: CSSPACE4:9

for X being non empty set
for Y being ComplexNormSpace
for f, g, h being VECTOR of (C_VectorSpace_of_BoundedFunctions (X,Y))
for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds
( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )

proof end;

theorem Th10: :: CSSPACE4:10

for X being non empty set
for Y being ComplexNormSpace
for f, h being VECTOR of (C_VectorSpace_of_BoundedFunctions (X,Y))
for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds
for c being Complex holds
( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )

proof end;

theorem Th11: :: CSSPACE4:11

for X being non empty set
for Y being ComplexNormSpace holds 0. (C_VectorSpace_of_BoundedFunctions (X,Y)) = X --> (0. Y)

proof end;

definition

let X be non empty set ;
let Y be ComplexNormSpace;
let f be object ;
assume A1: f in ComplexBoundedFunctions (X,Y) ;

func modetrans (f,X,Y) -> bounded Function of X, the carrier of Y equals :Def7: :: CSSPACE4:def 7
f;

coherence by A1, Def5;

end;

:: deftheorem Def7 defines modetrans CSSPACE4:def 7 :

definition

let X be non empty set ;
let Y be ComplexNormSpace;
let u be Function of X, the carrier of Y;

func PreNorms u -> non empty Subset of REAL equals :: CSSPACE4:def 8
{ ||.(u . t).|| where t is Element of X : verum } ;

proof end;

end;

:: deftheorem defines PreNorms CSSPACE4:def 8 :

theorem Th12: :: CSSPACE4:12

for X being non empty set
for Y being ComplexNormSpace
for g being bounded Function of X, the carrier of Y holds PreNorms g is bounded_above

proof end;

theorem :: CSSPACE4:13

for X being non empty set
for Y being ComplexNormSpace
for g being Function of X, the carrier of Y holds
( g is bounded iff PreNorms g is bounded_above )

proof end;

definition

let X be non empty set ;
let Y be ComplexNormSpace;

func ComplexBoundedFunctionsNorm (X,Y) -> Function of (ComplexBoundedFunctions (X,Y)),REAL means :Def9: :: CSSPACE4:def 9
for x being object st x in ComplexBoundedFunctions (X,Y) holds
it . x = upper_bound (PreNorms (modetrans (x,X,Y)));

proof end;

proof end;

end;

:: deftheorem Def9 defines ComplexBoundedFunctionsNorm CSSPACE4:def 9 :

theorem Th14: :: CSSPACE4:14

for X being non empty set
for Y being ComplexNormSpace
for f being bounded Function of X, the carrier of Y holds modetrans (f,X,Y) = f

proof end;

theorem Th15: :: CSSPACE4:15

for X being non empty set
for Y being ComplexNormSpace
for f being bounded Function of X, the carrier of Y holds (ComplexBoundedFunctionsNorm (X,Y)) . f = upper_bound (PreNorms f)

proof end;

definition

let X be non empty set ;
let Y be ComplexNormSpace;

func C_NormSpace_of_BoundedFunctions (X,Y) -> non empty CNORMSTR equals :: CSSPACE4:def 10
CNORMSTR(# (ComplexBoundedFunctions (X,Y)),(Zero_ ((ComplexBoundedFunctions (X,Y)),(ComplexVectSpace (X,Y)))),(Add_ ((ComplexBoundedFunctions (X,Y)),(ComplexVectSpace (X,Y)))),(Mult_ ((ComplexBoundedFunctions (X,Y)),(ComplexVectSpace (X,Y)))),(ComplexBoundedFunctionsNorm (X,Y)) #);

end;

:: deftheorem defines C_NormSpace_of_BoundedFunctions CSSPACE4:def 10 :

theorem Th16: :: CSSPACE4:16

for X being non empty set
for Y being ComplexNormSpace holds X --> (0. Y) = 0. (C_NormSpace_of_BoundedFunctions (X,Y))

proof end;

theorem Th17: :: CSSPACE4:17

for X being non empty set
for Y being ComplexNormSpace
for f being Point of (C_NormSpace_of_BoundedFunctions (X,Y))
for g being bounded Function of X, the carrier of Y st g = f holds
for t being Element of X holds ||.(g . t).|| <= ||.f.||

proof end;

theorem :: CSSPACE4:18

for X being non empty set
for Y being ComplexNormSpace
for f being Point of (C_NormSpace_of_BoundedFunctions (X,Y)) holds 0 <= ||.f.||

proof end;

theorem Th19: :: CSSPACE4:19

for X being non empty set
for Y being ComplexNormSpace
for f being Point of (C_NormSpace_of_BoundedFunctions (X,Y)) st f = 0. (C_NormSpace_of_BoundedFunctions (X,Y)) holds
0 = ||.f.||

proof end;

theorem Th20: :: CSSPACE4:20

for X being non empty set
for Y being ComplexNormSpace
for f, g, h being Point of (C_NormSpace_of_BoundedFunctions (X,Y))
for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds
( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )

proof end;

theorem Th21: :: CSSPACE4:21

for X being non empty set
for Y being ComplexNormSpace
for f, h being Point of (C_NormSpace_of_BoundedFunctions (X,Y))
for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds
for c being Complex holds
( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )

proof end;

theorem Th22: :: CSSPACE4:22

for X being non empty set
for Y being ComplexNormSpace
for f, g being Point of (C_NormSpace_of_BoundedFunctions (X,Y))
for c being Complex holds
( ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedFunctions (X,Y)) ) & ( f = 0. (C_NormSpace_of_BoundedFunctions (X,Y)) implies ||.f.|| = 0 ) & ||.(c * f).|| = |.c.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )

proof end;

theorem Th23: :: CSSPACE4:23

for X being non empty set
for Y being ComplexNormSpace holds
( C_NormSpace_of_BoundedFunctions (X,Y) is reflexive & C_NormSpace_of_BoundedFunctions (X,Y) is discerning & C_NormSpace_of_BoundedFunctions (X,Y) is ComplexNormSpace-like )

proof end;

theorem Th24: :: CSSPACE4:24

for X being non empty set
for Y being ComplexNormSpace holds C_NormSpace_of_BoundedFunctions (X,Y) is ComplexNormSpace

proof end;

registration

let X be non empty set ;
let Y be ComplexNormSpace;

cluster C_NormSpace_of_BoundedFunctions (X,Y) -> non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ;

coherence by Th24;

end;

theorem Th25: :: CSSPACE4:25

for X being non empty set
for Y being ComplexNormSpace
for f, g, h being Point of (C_NormSpace_of_BoundedFunctions (X,Y))
for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds
( h = f - g iff for x being Element of X holds h9 . x = (f9 . x) - (g9 . x) )

proof end;

Lm12: for e being Real
for seq being Real_Sequence st seq is convergent & ex k being Nat st
for i being Nat st k <= i holds
seq . i <= e holds
lim seq <= e

proof end;

theorem Th26: :: CSSPACE4:26

for X being non empty set
for Y being ComplexNormSpace st Y is complete holds
for seq being sequence of (C_NormSpace_of_BoundedFunctions (X,Y)) st seq is Cauchy_sequence_by_Norm holds
seq is convergent

proof end;

theorem Th27: :: CSSPACE4:27

for X being non empty set
for Y being ComplexBanachSpace holds C_NormSpace_of_BoundedFunctions (X,Y) is ComplexBanachSpace

proof end;

registration

let X be non empty set ;
let Y be ComplexBanachSpace;

cluster C_NormSpace_of_BoundedFunctions (X,Y) -> non empty complete ;

coherence by Th27;

end;

theorem :: CSSPACE4:28

for seq1, seq2 being Complex_Sequence st seq1 is bounded & seq2 is bounded holds
seq1 + seq2 is bounded by Lm3;

theorem :: CSSPACE4:29

for c being Complex
for seq being Complex_Sequence st seq is bounded holds
c (#) seq is bounded by Lm4;

theorem :: CSSPACE4:30

for seq being Complex_Sequence holds
( seq is bounded iff |.seq.| is bounded ) by Lm8, Lm9;

theorem :: CSSPACE4:31

for seq1, seq2, seq3 being Complex_Sequence holds
( seq1 = seq2 - seq3 iff for n being Nat holds seq1 . n = (seq2 . n) - (seq3 . n) ) by Lm11;