RSSPACE4: Banach Space of Bounded Real Sequences

:: Banach Space of Bounded Real Sequences
:: by Yasumasa Suzuki
::
:: Received January 6, 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_BoundedRealSequences -> Subset of Linear_Space_of_RealSequences means :Def1: :: RSSPACE4:def 1
for x being object holds
( x in it iff ( x in the_set_of_RealSequences & seq_id x is bounded ) );

proof end;

proof end;

end;

:: deftheorem Def1 defines the_set_of_BoundedRealSequences RSSPACE4:def 1 :

registration

cluster the_set_of_BoundedRealSequences -> non empty ;

proof end;

cluster the_set_of_BoundedRealSequences -> linearly-closed ;

proof end;

end;

Lm3: RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is Subspace of Linear_Space_of_RealSequences
by RSSPACE:11;

registration

cluster RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;

coherence by RSSPACE:11;

end;

Lm4: ( RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is Abelian & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is add-associative & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is right_zeroed & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is right_complementable & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is vector-distributive & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is scalar-distributive & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is scalar-associative & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is scalar-unital )
;

definition

func linfty_norm -> Function of the_set_of_BoundedRealSequences,REAL means :Def2: :: RSSPACE4:def 2
for x being object st x in the_set_of_BoundedRealSequences holds
it . x = upper_bound (rng (abs (seq_id x)));

proof end;

proof end;

end;

:: deftheorem Def2 defines linfty_norm RSSPACE4:def 2 :

Lm5: for rseq being Real_Sequence st ( for n being Nat holds rseq . n = In (0,REAL) ) holds
( rseq is bounded & upper_bound (rng (abs rseq)) = 0 )

proof end;

Lm6: for rseq being Real_Sequence st rseq is bounded & upper_bound (rng (abs rseq)) = 0 holds
for n being Nat holds rseq . n = 0

proof end;

theorem :: RSSPACE4:1

for rseq being Real_Sequence holds
( ( rseq is bounded & upper_bound (rng (abs rseq)) = 0 ) iff for n being Nat holds rseq . n = 0 ) by Lm5, Lm6;

registration

cluster NORMSTR(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),linfty_norm #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;

coherence by Lm4, RSSPACE3:2;

end;

definition end;

:: deftheorem defines linfty_Space RSSPACE4:def 3 :

theorem Th2: :: RSSPACE4:2

( the carrier of linfty_Space = the_set_of_BoundedRealSequences & ( for x being set holds
( x is VECTOR of linfty_Space iff ( x is Real_Sequence & seq_id x is bounded ) ) ) & 0. linfty_Space = Zeroseq & ( for u being VECTOR of linfty_Space holds u = seq_id u ) & ( for u, v being VECTOR of linfty_Space holds u + v = (seq_id u) + (seq_id v) ) & ( for r being Real
for u being VECTOR of linfty_Space holds r * u = r (#) (seq_id u) ) & ( for u being VECTOR of linfty_Space holds
( - u = - (seq_id u) & seq_id (- u) = - (seq_id u) ) ) & ( for u, v being VECTOR of linfty_Space holds u - v = (seq_id u) - (seq_id v) ) & ( for v being VECTOR of linfty_Space holds seq_id v is bounded ) & ( for v being VECTOR of linfty_Space holds ||.v.|| = upper_bound (rng (abs (seq_id v))) ) )

proof end;

theorem Th3: :: RSSPACE4:3

for x, y being Point of linfty_Space
for a being Real holds
( ( ||.x.|| = 0 implies x = 0. linfty_Space ) & ( x = 0. linfty_Space implies ||.x.|| = 0 ) & 0 <= ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(a * x).|| = |.a.| * ||.x.|| )

proof end;

registration end;

theorem :: RSSPACE4:4

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

proof end;

definition

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

attr IT is bounded means :Def4: :: RSSPACE4:def 4
ex K being Real st
( 0 <= K & ( for x being Element of X holds ||.(IT . x).|| <= K ) );

end;

:: deftheorem Def4 defines bounded RSSPACE4:def 4 :

theorem Th5: :: RSSPACE4:5

for X being non empty set
for Y being RealNormSpace
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 RealNormSpace;

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 RealNormSpace;

func BoundedFunctions (X,Y) -> Subset of (RealVectSpace (X,Y)) means :Def5: :: RSSPACE4: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 BoundedFunctions RSSPACE4:def 5 :

registration

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

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

proof end;

end;

theorem Th6: :: RSSPACE4:6

for X being non empty set
for Y being RealNormSpace holds BoundedFunctions (X,Y) is linearly-closed

proof end;

theorem :: RSSPACE4:7

for X being non empty set
for Y being RealNormSpace holds RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #) is Subspace of RealVectSpace (X,Y) by Th6, RSSPACE:11;

registration

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

cluster RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;

coherence by Th6, RSSPACE:11;

end;

definition

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

func R_VectorSpace_of_BoundedFunctions (X,Y) -> RealLinearSpace equals :: RSSPACE4:def 6
RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #);

end;

:: deftheorem defines R_VectorSpace_of_BoundedFunctions RSSPACE4:def 6 :

registration

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

cluster R_VectorSpace_of_BoundedFunctions (X,Y) -> strict ;

end;

theorem Th8: :: RSSPACE4:8

for X being non empty set
for Y being RealNormSpace
for f, g, h being VECTOR of (R_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 Th9: :: RSSPACE4:9

for X being non empty set
for Y being RealNormSpace
for f, h being VECTOR of (R_VectorSpace_of_BoundedFunctions (X,Y))
for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds
for a being Real holds
( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )

proof end;

theorem Th10: :: RSSPACE4:10

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

proof end;

definition

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

func modetrans (f,X,Y) -> bounded Function of X, the carrier of Y equals :Def7: :: RSSPACE4:def 7
f;

coherence by A1, Def5;

end;

:: deftheorem Def7 defines modetrans RSSPACE4:def 7 :

definition

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

func PreNorms u -> non empty Subset of REAL equals :: RSSPACE4:def 8
{ ||.(u . t).|| where t is Element of X : verum } ;

proof end;

end;

:: deftheorem defines PreNorms RSSPACE4:def 8 :

theorem Th11: :: RSSPACE4:11

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

proof end;

theorem :: RSSPACE4:12

for X being non empty set
for Y being RealNormSpace
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 RealNormSpace;

func BoundedFunctionsNorm (X,Y) -> Function of (BoundedFunctions (X,Y)),REAL means :Def9: :: RSSPACE4:def 9
for x being object st x in BoundedFunctions (X,Y) holds
it . x = upper_bound (PreNorms (modetrans (x,X,Y)));

proof end;

proof end;

end;

:: deftheorem Def9 defines BoundedFunctionsNorm RSSPACE4:def 9 :

theorem Th13: :: RSSPACE4:13

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

proof end;

theorem Th14: :: RSSPACE4:14

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

proof end;

definition

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

func R_NormSpace_of_BoundedFunctions (X,Y) -> non empty NORMSTR equals :: RSSPACE4:def 10
NORMSTR(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(BoundedFunctionsNorm (X,Y)) #);

end;

:: deftheorem defines R_NormSpace_of_BoundedFunctions RSSPACE4:def 10 :

theorem Th15: :: RSSPACE4:15

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

proof end;

theorem Th16: :: RSSPACE4:16

for X being non empty set
for Y being RealNormSpace
for f being Point of (R_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 :: RSSPACE4:17

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

proof end;

theorem Th18: :: RSSPACE4:18

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

proof end;

theorem Th19: :: RSSPACE4:19

for X being non empty set
for Y being RealNormSpace
for f, g, h being Point of (R_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 Th20: :: RSSPACE4:20

for X being non empty set
for Y being RealNormSpace
for f, h being Point of (R_NormSpace_of_BoundedFunctions (X,Y))
for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds
for a being Real holds
( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )

proof end;

theorem Th21: :: RSSPACE4:21

for X being non empty set
for Y being RealNormSpace
for f, g being Point of (R_NormSpace_of_BoundedFunctions (X,Y))
for a being Real holds
( ( ||.f.|| = 0 implies f = 0. (R_NormSpace_of_BoundedFunctions (X,Y)) ) & ( f = 0. (R_NormSpace_of_BoundedFunctions (X,Y)) implies ||.f.|| = 0 ) & ||.(a * f).|| = |.a.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )

proof end;

theorem Th22: :: RSSPACE4:22

for X being non empty set
for Y being RealNormSpace holds
( R_NormSpace_of_BoundedFunctions (X,Y) is reflexive & R_NormSpace_of_BoundedFunctions (X,Y) is discerning & R_NormSpace_of_BoundedFunctions (X,Y) is RealNormSpace-like ) by Th21;

theorem Th23: :: RSSPACE4:23

for X being non empty set
for Y being RealNormSpace holds R_NormSpace_of_BoundedFunctions (X,Y) is RealNormSpace

proof end;

registration

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

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

coherence by Th23;

end;

theorem Th24: :: RSSPACE4:24

for X being non empty set
for Y being RealNormSpace
for f, g, h being Point of (R_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;

Lm7: 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 Th25: :: RSSPACE4:25

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

proof end;

theorem Th26: :: RSSPACE4:26

for X being non empty set
for Y being RealBanachSpace holds R_NormSpace_of_BoundedFunctions (X,Y) is RealBanachSpace

proof end;

registration

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

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

coherence by Th26;

end;