CC0SP2: Banach Algebra of Complex-Valued Continuous Functionals and Space of Complex-valued Continuous Functionals with Bounded Support

:: Banach Algebra of Complex-Valued Continuous Functionals and Space of Complex-valued Continuous Functionals with Bounded Support
:: by Katuhiko Kanazashi , Hiroyuki Okazaki and Yasunari Shidama
::
:: Received May 30, 2011
:: Copyright (c) 2011-2021 Association of Mizar Users

definition

let X be TopStruct ;
let f be Function of the carrier of X,COMPLEX;

attr f is continuous means :: CC0SP2:def 1
for Y being Subset of COMPLEX st Y is closed holds
f " Y is closed ;

end;

:: deftheorem defines continuous CC0SP2:def 1 :

definition

let X be 1-sorted ;
let y be Complex;

func X --> y -> Function of the carrier of X,COMPLEX equals :: CC0SP2:def 2
the carrier of X --> y;

proof end;

end;

:: deftheorem defines --> CC0SP2:def 2 :

theorem Th1: :: CC0SP2:1

for X being non empty TopSpace
for y being Complex
for f being Function of the carrier of X,COMPLEX st f = X --> y holds
f is continuous

proof end;

registration

let X be non empty TopSpace;
let y be Complex;

cluster X --> y -> continuous ;

coherence by Th1;

end;

registration

let X be non empty TopSpace;

cluster Relation-like the carrier of X -defined COMPLEX -valued non empty Function-like total quasi_total V172() continuous for Element of bool [: the carrier of X,COMPLEX:];

proof end;

end;

theorem Th2: :: CC0SP2:2

for X being non empty TopSpace
for f being Function of the carrier of X,COMPLEX holds
( f is continuous iff for Y being Subset of COMPLEX st Y is open holds
f " Y is open )

proof end;

theorem Th3: :: CC0SP2:3

for X being non empty TopSpace
for f being Function of the carrier of X,COMPLEX holds
( f is continuous iff for x being Point of X
for V being Subset of COMPLEX st f . x in V & V is open holds
ex W being Subset of X st
( x in W & W is open & f .: W c= V ) )

proof end;

theorem Th4: :: CC0SP2:4

for X being non empty TopSpace
for f, g being continuous Function of the carrier of X,COMPLEX holds f + g is continuous Function of the carrier of X,COMPLEX

proof end;

theorem Th5: :: CC0SP2:5

for X being non empty TopSpace
for a being Complex
for f being continuous Function of the carrier of X,COMPLEX holds a (#) f is continuous Function of the carrier of X,COMPLEX

proof end;

theorem :: CC0SP2:6

for X being non empty TopSpace
for f, g being continuous Function of the carrier of X,COMPLEX holds f - g is continuous Function of the carrier of X,COMPLEX

proof end;

theorem Th7: :: CC0SP2:7

for X being non empty TopSpace
for f, g being continuous Function of the carrier of X,COMPLEX holds f (#) g is continuous Function of the carrier of X,COMPLEX

proof end;

theorem Th8: :: CC0SP2:8

for X being non empty TopSpace
for f being continuous Function of the carrier of X,COMPLEX holds
( |.f.| is Function of the carrier of X,REAL & |.f.| is continuous )

proof end;

definition

let X be non empty TopSpace;

func CContinuousFunctions X -> Subset of (CAlgebra the carrier of X) equals :: CC0SP2:def 3
{ f where f is continuous Function of the carrier of X,COMPLEX : verum } ;

proof end;

end;

:: deftheorem defines CContinuousFunctions CC0SP2:def 3 :

registration

let X be non empty TopSpace;

cluster CContinuousFunctions X -> non empty ;

proof end;

end;

registration

let X be non empty TopSpace;

cluster CContinuousFunctions X -> multiplicatively-closed Cadditively-linearly-closed ;

proof end;

end;

definition

let X be non empty TopSpace;

func C_Algebra_of_ContinuousFunctions X -> ComplexAlgebra equals :: CC0SP2:def 4
ComplexAlgebraStr(# (CContinuousFunctions X),(mult_ ((CContinuousFunctions X),(CAlgebra the carrier of X))),(Add_ ((CContinuousFunctions X),(CAlgebra the carrier of X))),(Mult_ ((CContinuousFunctions X),(CAlgebra the carrier of X))),(One_ ((CContinuousFunctions X),(CAlgebra the carrier of X))),(Zero_ ((CContinuousFunctions X),(CAlgebra the carrier of X))) #);

coherence by CC0SP1:2;

end;

:: deftheorem defines C_Algebra_of_ContinuousFunctions CC0SP2:def 4 :

theorem :: CC0SP2:9

for X being non empty TopSpace holds C_Algebra_of_ContinuousFunctions X is ComplexSubAlgebra of CAlgebra the carrier of X by CC0SP1:2;

registration

let X be non empty TopSpace;

cluster C_Algebra_of_ContinuousFunctions X -> strict ;

end;

registration

let X be non empty TopSpace;

cluster C_Algebra_of_ContinuousFunctions X -> scalar-unital ;

proof end;

end;

theorem Th10: :: CC0SP2:10

for X being non empty TopSpace
for F, G, H being VECTOR of (C_Algebra_of_ContinuousFunctions X)
for f, g, h being Function of the carrier of X,COMPLEX st f = F & g = G & h = H holds
( H = F + G iff for x being Element of the carrier of X holds h . x = (f . x) + (g . x) )

proof end;

theorem Th11: :: CC0SP2:11

for X being non empty TopSpace
for F, G being VECTOR of (C_Algebra_of_ContinuousFunctions X)
for f, g being Function of the carrier of X,COMPLEX
for a being Complex st f = F & g = G holds
( G = a * F iff for x being Element of X holds g . x = a * (f . x) )

proof end;

theorem Th12: :: CC0SP2:12

for X being non empty TopSpace
for F, G, H being VECTOR of (C_Algebra_of_ContinuousFunctions X)
for f, g, h being Function of the carrier of X,COMPLEX st f = F & g = G & h = H holds
( H = F * G iff for x being Element of the carrier of X holds h . x = (f . x) * (g . x) )

proof end;

theorem Th13: :: CC0SP2:13

for X being non empty TopSpace holds 0. (C_Algebra_of_ContinuousFunctions X) = X --> 0c

proof end;

theorem Th14: :: CC0SP2:14

for X being non empty TopSpace holds 1_ (C_Algebra_of_ContinuousFunctions X) = X --> 1r

proof end;

theorem Th15: :: CC0SP2:15

for A being ComplexAlgebra
for A1, A2 being ComplexSubAlgebra of A st the carrier of A1 c= the carrier of A2 holds
A1 is ComplexSubAlgebra of A2

proof end;

Lm1: for X being non empty compact TopSpace
for x being set st x in CContinuousFunctions X holds
x in ComplexBoundedFunctions the carrier of X

proof end;

theorem :: CC0SP2:16

for X being non empty compact TopSpace holds C_Algebra_of_ContinuousFunctions X is ComplexSubAlgebra of C_Algebra_of_BoundedFunctions the carrier of X

proof end;

definition

let X be non empty compact TopSpace;

func CContinuousFunctionsNorm X -> Function of (CContinuousFunctions X),REAL equals :: CC0SP2:def 5
(ComplexBoundedFunctionsNorm the carrier of X) | (CContinuousFunctions X);

proof end;

end;

:: deftheorem defines CContinuousFunctionsNorm CC0SP2:def 5 :

definition

let X be non empty compact TopSpace;

func C_Normed_Algebra_of_ContinuousFunctions X -> Normed_Complex_AlgebraStr equals :: CC0SP2:def 6
Normed_Complex_AlgebraStr(# (CContinuousFunctions X),(mult_ ((CContinuousFunctions X),(CAlgebra the carrier of X))),(Add_ ((CContinuousFunctions X),(CAlgebra the carrier of X))),(Mult_ ((CContinuousFunctions X),(CAlgebra the carrier of X))),(One_ ((CContinuousFunctions X),(CAlgebra the carrier of X))),(Zero_ ((CContinuousFunctions X),(CAlgebra the carrier of X))),(CContinuousFunctionsNorm X) #);

end;

:: deftheorem defines C_Normed_Algebra_of_ContinuousFunctions CC0SP2:def 6 :

registration

let X be non empty compact TopSpace;

cluster C_Normed_Algebra_of_ContinuousFunctions X -> non empty strict ;

end;

Lm2: now :: thesis:

let X be non empty compact TopSpace; :: thesis:
set F = C_Normed_Algebra_of_ContinuousFunctions X;
let x, e be Element of (C_Normed_Algebra_of_ContinuousFunctions X); :: thesis:
assume A1: e = One_ ((CContinuousFunctions X),(CAlgebra the carrier of X)) ; :: thesis:
set X1 = CContinuousFunctions X;
reconsider f = x as Element of CContinuousFunctions X ;
1_ (CAlgebra the carrier of X) = X --> 1
.= 1_ (C_Algebra_of_ContinuousFunctions X) by Th14 ;
then A2: ( [f,(1_ (CAlgebra the carrier of X))] in [:(CContinuousFunctions X),(CContinuousFunctions X):] & [(1_ (CAlgebra the carrier of X)),f] in [:(CContinuousFunctions X),(CContinuousFunctions X):] ) by ZFMISC_1:87;
x * e = (mult_ ((CContinuousFunctions X),(CAlgebra the carrier of X))) . (f,(1_ (CAlgebra the carrier of X))) by A1, C0SP1:def 8;
then x * e = ( the multF of (CAlgebra the carrier of X) || (CContinuousFunctions X)) . (f,(1_ (CAlgebra the carrier of X))) by C0SP1:def 6;
then x * e = f * (1_ (CAlgebra the carrier of X)) by A2, FUNCT_1:49;
hence x * e = x ; :: thesis:
e * x = (mult_ ((CContinuousFunctions X),(CAlgebra the carrier of X))) . ((1_ (CAlgebra the carrier of X)),f) by A1, C0SP1:def 8;
then e * x = ( the multF of (CAlgebra the carrier of X) || (CContinuousFunctions X)) . ((1_ (CAlgebra the carrier of X)),f) by C0SP1:def 6;
then e * x = (1_ (CAlgebra the carrier of X)) * f by A2, FUNCT_1:49;
hence e * x = x ; :: thesis:

end;

registration

let X be non empty compact TopSpace;

cluster C_Normed_Algebra_of_ContinuousFunctions X -> unital ;

proof end;

end;

Lm3: for X being non empty compact TopSpace
for x being Point of (C_Normed_Algebra_of_ContinuousFunctions X)
for y being Point of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) st x = y holds
||.x.|| = ||.y.||
by FUNCT_1:49;

Lm4: for X being non empty compact TopSpace
for x1, x2 being Point of (C_Normed_Algebra_of_ContinuousFunctions X)
for y1, y2 being Point of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) st x1 = y1 & x2 = y2 holds
x1 + x2 = y1 + y2

proof end;

Lm5: for X being non empty compact TopSpace
for a being Complex
for x being Point of (C_Normed_Algebra_of_ContinuousFunctions X)
for y being Point of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) st x = y holds
a * x = a * y

proof end;

Lm6: for X being non empty compact TopSpace
for x1, x2 being Point of (C_Normed_Algebra_of_ContinuousFunctions X)
for y1, y2 being Point of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) st x1 = y1 & x2 = y2 holds
x1 * x2 = y1 * y2

proof end;

theorem Th17: :: CC0SP2:17

for X being non empty compact TopSpace holds C_Normed_Algebra_of_ContinuousFunctions X is ComplexAlgebra

proof end;

registration

let X be non empty compact TopSpace;

cluster C_Normed_Algebra_of_ContinuousFunctions X -> right_complementable Abelian add-associative right_zeroed right-distributive right_unital associative commutative vector-distributive scalar-distributive scalar-associative vector-associative ;

coherence by Th17;

end;

theorem :: CC0SP2:18

for X being non empty compact TopSpace
for F being Point of (C_Normed_Algebra_of_ContinuousFunctions X) holds (Mult_ ((CContinuousFunctions X),(CAlgebra the carrier of X))) . (1r,F) = F

proof end;

registration end;

theorem :: CC0SP2:19

for X being non empty compact TopSpace holds C_Normed_Algebra_of_ContinuousFunctions X is ComplexLinearSpace ;

theorem Th20: :: CC0SP2:20

for X being non empty compact TopSpace holds X --> 0 = 0. (C_Normed_Algebra_of_ContinuousFunctions X)

proof end;

Lm7: for X being non empty compact TopSpace holds 0. (C_Normed_Algebra_of_ContinuousFunctions X) = 0. (C_Normed_Algebra_of_BoundedFunctions the carrier of X)

proof end;

Lm8: for X being non empty compact TopSpace holds 1. (C_Normed_Algebra_of_ContinuousFunctions X) = 1. (C_Normed_Algebra_of_BoundedFunctions the carrier of X)

proof end;

theorem :: CC0SP2:21

for X being non empty compact TopSpace
for F being Point of (C_Normed_Algebra_of_ContinuousFunctions X) holds 0 <= ||.F.||

proof end;

theorem Th22: :: CC0SP2:22

for X being non empty compact TopSpace
for f, g, h being Function of the carrier of X,COMPLEX
for F, G, H being Point of (C_Normed_Algebra_of_ContinuousFunctions X) st f = F & g = G & h = H holds
( H = F + G iff for x being Element of X holds h . x = (f . x) + (g . x) )

proof end;

theorem :: CC0SP2:23

for a being Complex
for X being non empty compact TopSpace
for f, g being Function of the carrier of X,COMPLEX
for F, G being Point of (C_Normed_Algebra_of_ContinuousFunctions X) st f = F & g = G holds
( G = a * F iff for x being Element of X holds g . x = a * (f . x) )

proof end;

theorem :: CC0SP2:24

for X being non empty compact TopSpace
for f, g, h being Function of the carrier of X,COMPLEX
for F, G, H being Point of (C_Normed_Algebra_of_ContinuousFunctions X) st f = F & g = G & h = H holds
( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) )

proof end;

theorem Th25: :: CC0SP2:25

for X being non empty compact TopSpace holds ||.(0. (C_Normed_Algebra_of_ContinuousFunctions X)).|| = 0

proof end;

theorem Th26: :: CC0SP2:26

for X being non empty compact TopSpace
for F being Point of (C_Normed_Algebra_of_ContinuousFunctions X) st ||.F.|| = 0 holds
F = 0. (C_Normed_Algebra_of_ContinuousFunctions X)

proof end;

theorem Th27: :: CC0SP2:27

for a being Complex
for X being non empty compact TopSpace
for F being Point of (C_Normed_Algebra_of_ContinuousFunctions X) holds ||.(a * F).|| = |.a.| * ||.F.||

proof end;

theorem Th28: :: CC0SP2:28

for X being non empty compact TopSpace
for F, G being Point of (C_Normed_Algebra_of_ContinuousFunctions X) holds ||.(F + G).|| <= ||.F.|| + ||.G.||

proof end;

registration end;

Lm9: for X being non empty compact TopSpace
for x1, x2 being Point of (C_Normed_Algebra_of_ContinuousFunctions X)
for y1, y2 being Point of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) st x1 = y1 & x2 = y2 holds
x1 - x2 = y1 - y2

proof end;

theorem :: CC0SP2:29

for X being non empty compact TopSpace
for f, g, h being Function of the carrier of X,COMPLEX
for F, G, H being Point of (C_Normed_Algebra_of_ContinuousFunctions X) st f = F & g = G & h = H holds
( H = F - G iff for x being Element of X holds h . x = (f . x) - (g . x) )

proof end;

theorem Th30: :: CC0SP2:30

for X being ComplexBanachSpace
for Y being Subset of X
for seq being sequence of X st Y is closed & rng seq c= Y & seq is CCauchy holds
( seq is convergent & lim seq in Y ) by CLOPBAN1:def 13, NCFCONT1:def 3;

theorem Th31: :: CC0SP2:31

for X being non empty compact TopSpace
for Y being Subset of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) st Y = CContinuousFunctions X holds
Y is closed

proof end;

theorem Th32: :: CC0SP2:32

for X being non empty compact TopSpace
for seq being sequence of (C_Normed_Algebra_of_ContinuousFunctions X) st seq is CCauchy holds
seq is convergent

proof end;

registration

let X be non empty compact TopSpace;

cluster C_Normed_Algebra_of_ContinuousFunctions X -> complete ;

proof end;

end;

registration

let X be non empty compact TopSpace;

cluster C_Normed_Algebra_of_ContinuousFunctions X -> Banach_Algebra-like ;

proof end;

end;

:: Some properties of support

theorem Th33: :: CC0SP2:33

for X being non empty TopSpace
for f, g being Function of the carrier of X,COMPLEX holds support (f + g) c= (support f) \/ (support g)

proof end;

theorem Th34: :: CC0SP2:34

for X being non empty TopSpace
for a being Complex
for f being Function of the carrier of X,COMPLEX holds support (a (#) f) c= support f

proof end;

theorem :: CC0SP2:35

for X being non empty TopSpace
for f, g being Function of the carrier of X,COMPLEX holds support (f (#) g) c= (support f) \/ (support g)

proof end;

:: Space of Complex-Valued Continuous Functionals with bounded support

definition

let X be non empty TopSpace;

func CC_0_Functions X -> non empty Subset of (ComplexVectSpace the carrier of X) equals :: CC0SP2:def 7
{ f where f is Function of the carrier of X,COMPLEX : ( f is continuous & ex Y being non empty Subset of X st
( Y is compact & ( for A being Subset of X st A = support f holds
Cl A is Subset of Y ) ) ) } ;

proof end;

end;

:: deftheorem defines CC_0_Functions CC0SP2:def 7 :

theorem :: CC0SP2:36

for X being non empty TopSpace holds CC_0_Functions X is non empty Subset of (CAlgebra the carrier of X) ;

Lm10: for X being non empty TopSpace
for v, u being Element of (CAlgebra the carrier of X) st v in CC_0_Functions X & u in CC_0_Functions X holds
v + u in CC_0_Functions X

proof end;

Lm11: for X being non empty TopSpace
for a being Complex
for u being Element of (CAlgebra the carrier of X) st u in CC_0_Functions X holds
a * u in CC_0_Functions X

proof end;

Lm12: for X being non empty TopSpace
for u being Element of (CAlgebra the carrier of X) st u in CC_0_Functions X holds
- u in CC_0_Functions X

proof end;

theorem :: CC0SP2:37

for X being non empty TopSpace
for W being non empty Subset of (CAlgebra the carrier of X) st W = CC_0_Functions X holds
W is Cadditively-linearly-closed

proof end;

theorem Th38: :: CC0SP2:38

for X being non empty TopSpace holds CC_0_Functions X is add-closed

proof end;

theorem Th39: :: CC0SP2:39

for X being non empty TopSpace holds CC_0_Functions X is linearly-closed

proof end;

registration

let X be non empty TopSpace;

cluster CC_0_Functions X -> non empty linearly-closed ;

correctness by Th39;

end;

theorem Th40: :: CC0SP2:40

for V being ComplexLinearSpace
for V1 being Subset of V st V1 is linearly-closed & not V1 is empty holds
CLSStruct(# V1,(Zero_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)) #) is Subspace of V

proof end;

theorem Th41: :: CC0SP2:41

for X being non empty TopSpace holds CLSStruct(# (CC_0_Functions X),(Zero_ ((CC_0_Functions X),(ComplexVectSpace the carrier of X))),(Add_ ((CC_0_Functions X),(ComplexVectSpace the carrier of X))),(Mult_ ((CC_0_Functions X),(ComplexVectSpace the carrier of X))) #) is Subspace of ComplexVectSpace the carrier of X by Th40;

definition

let X be non empty TopSpace;

func C_VectorSpace_of_C_0_Functions X -> ComplexLinearSpace equals :: CC0SP2:def 8
CLSStruct(# (CC_0_Functions X),(Zero_ ((CC_0_Functions X),(ComplexVectSpace the carrier of X))),(Add_ ((CC_0_Functions X),(ComplexVectSpace the carrier of X))),(Mult_ ((CC_0_Functions X),(ComplexVectSpace the carrier of X))) #);

correctness by Th41;

end;

:: deftheorem defines C_VectorSpace_of_C_0_Functions CC0SP2:def 8 :

theorem Th42: :: CC0SP2:42

for X being non empty TopSpace
for x being set st x in CC_0_Functions X holds
x in ComplexBoundedFunctions the carrier of X

proof end;

definition

let X be non empty TopSpace;

func CC_0_FunctionsNorm X -> Function of (CC_0_Functions X),REAL equals :: CC0SP2:def 9
(ComplexBoundedFunctionsNorm the carrier of X) | (CC_0_Functions X);

proof end;

end;

:: deftheorem defines CC_0_FunctionsNorm CC0SP2:def 9 :

definition

let X be non empty TopSpace;

func C_Normed_Space_of_C_0_Functions X -> CNORMSTR equals :: CC0SP2:def 10
CNORMSTR(# (CC_0_Functions X),(Zero_ ((CC_0_Functions X),(ComplexVectSpace the carrier of X))),(Add_ ((CC_0_Functions X),(ComplexVectSpace the carrier of X))),(Mult_ ((CC_0_Functions X),(ComplexVectSpace the carrier of X))),(CC_0_FunctionsNorm X) #);

end;

:: deftheorem defines C_Normed_Space_of_C_0_Functions CC0SP2:def 10 :

registration

let X be non empty TopSpace;

cluster C_Normed_Space_of_C_0_Functions X -> non empty strict ;

end;

theorem :: CC0SP2:43

for X being non empty TopSpace
for x being set st x in CC_0_Functions X holds
x in CContinuousFunctions X

proof end;

theorem Th44: :: CC0SP2:44

for X being non empty TopSpace holds 0. (C_VectorSpace_of_C_0_Functions X) = X --> 0

proof end;

theorem Th45: :: CC0SP2:45

for X being non empty TopSpace holds 0. (C_Normed_Space_of_C_0_Functions X) = X --> 0

proof end;

Lm13: for X being non empty TopSpace
for x1, x2 being Point of (C_Normed_Space_of_C_0_Functions X)
for y1, y2 being Point of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) st x1 = y1 & x2 = y2 holds
x1 + x2 = y1 + y2

proof end;

Lm14: for X being non empty TopSpace
for a being Complex
for x being Point of (C_Normed_Space_of_C_0_Functions X)
for y being Point of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) st x = y holds
a * x = a * y

proof end;

theorem Th46: :: CC0SP2:46

for a being Complex
for X being non empty TopSpace
for F, G being Point of (C_Normed_Space_of_C_0_Functions X) holds
( ( ||.F.|| = 0 implies F = 0. (C_Normed_Space_of_C_0_Functions X) ) & ( F = 0. (C_Normed_Space_of_C_0_Functions X) implies ||.F.|| = 0 ) & ||.(a * F).|| = |.a.| * ||.F.|| & ||.(F + G).|| <= ||.F.|| + ||.G.|| )

proof end;

Lm15: for X being non empty TopSpace holds C_Normed_Space_of_C_0_Functions X is ComplexNormSpace-like
by Th46;

registration

let X be non empty TopSpace;

cluster C_Normed_Space_of_C_0_Functions X -> right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ;

proof end;

end;

theorem :: CC0SP2:47

for X being non empty TopSpace holds C_Normed_Space_of_C_0_Functions X is ComplexNormSpace ;