C0SP2: Banach Algebra of Continuous Functionals and Space of Real-valued Continuous Functionals with Bounded Support

:: Banach Algebra of Continuous Functionals and Space of Real-valued Continuous Functionals with Bounded Support
:: by Katuhiko Kanazashi, Noboru Endou and Yasunari Shidama
::
:: Received October 20, 2009
:: Copyright (c) 2009-2021 Association of Mizar Users

definition

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

func X --> y -> RealMap of X equals :: C0SP2:def 1
the carrier of X --> y;

proof end;

end;

:: deftheorem defines --> C0SP2:def 1 :

registration

let X be TopSpace;
let y be Real;

cluster X --> y -> continuous ;

proof end;

end;

theorem Th1: :: C0SP2:1

for X being non empty TopSpace
for f being RealMap of X holds
( f is continuous iff for x being Point of X
for V being Subset of REAL 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;

definition

let X be non empty TopSpace;

func ContinuousFunctions X -> Subset of (RAlgebra the carrier of X) equals :: C0SP2:def 2
{ f where f is continuous RealMap of X : verum } ;

proof end;

end;

:: deftheorem defines ContinuousFunctions C0SP2:def 2 :

registration

let X be non empty TopSpace;

cluster ContinuousFunctions X -> non empty ;

proof end;

end;

registration

let X be non empty TopSpace;

cluster ContinuousFunctions X -> multiplicatively-closed additively-linearly-closed ;

proof end;

end;

definition

let X be non empty TopSpace;

func R_Algebra_of_ContinuousFunctions X -> AlgebraStr equals :: C0SP2:def 3
AlgebraStr(# (ContinuousFunctions X),(mult_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(Add_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(Mult_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(One_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(Zero_ ((ContinuousFunctions X),(RAlgebra the carrier of X))) #);

end;

:: deftheorem defines R_Algebra_of_ContinuousFunctions C0SP2:def 3 :

theorem :: C0SP2:2

for X being non empty TopSpace holds R_Algebra_of_ContinuousFunctions X is Subalgebra of RAlgebra the carrier of X by C0SP1:6;

registration

let X be non empty TopSpace;

cluster R_Algebra_of_ContinuousFunctions X -> non empty strict ;

end;

registration

let X be non empty TopSpace;

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

proof end;

end;

theorem Th3: :: C0SP2:3

for X being non empty TopSpace
for F, G, H being VECTOR of (R_Algebra_of_ContinuousFunctions X)
for f, g, h being RealMap of X 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 Th4: :: C0SP2:4

for X being non empty TopSpace
for F, G, H being VECTOR of (R_Algebra_of_ContinuousFunctions X)
for f, g, h being RealMap of X
for a being Real 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 Th5: :: C0SP2:5

for X being non empty TopSpace
for F, G, H being VECTOR of (R_Algebra_of_ContinuousFunctions X)
for f, g, h being RealMap of X 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 Th6: :: C0SP2:6

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

proof end;

theorem Th7: :: C0SP2:7

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

proof end;

theorem Th8: :: C0SP2:8

for A being Algebra
for A1, A2 being Subalgebra of A st the carrier of A1 c= the carrier of A2 holds
A1 is Subalgebra of A2

proof end;

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

proof end;

theorem :: C0SP2:9

for X being non empty compact TopSpace holds R_Algebra_of_ContinuousFunctions X is Subalgebra of R_Algebra_of_BoundedFunctions the carrier of X

proof end;

definition

let X be non empty compact TopSpace;

func ContinuousFunctionsNorm X -> Function of (ContinuousFunctions X),REAL equals :: C0SP2:def 4
(BoundedFunctionsNorm the carrier of X) | (ContinuousFunctions X);

proof end;

end;

:: deftheorem defines ContinuousFunctionsNorm C0SP2:def 4 :

definition

let X be non empty compact TopSpace;

func R_Normed_Algebra_of_ContinuousFunctions X -> Normed_AlgebraStr equals :: C0SP2:def 5
Normed_AlgebraStr(# (ContinuousFunctions X),(mult_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(Add_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(Mult_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(One_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(Zero_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(ContinuousFunctionsNorm X) #);

end;

:: deftheorem defines R_Normed_Algebra_of_ContinuousFunctions C0SP2:def 5 :

registration

let X be non empty compact TopSpace;

cluster R_Normed_Algebra_of_ContinuousFunctions X -> non empty strict ;

end;

Lm2: now :: thesis:

let X be non empty compact TopSpace; :: thesis:
set F = R_Normed_Algebra_of_ContinuousFunctions X;
let x, e be Element of (R_Normed_Algebra_of_ContinuousFunctions X); :: thesis:
assume A1: e = One_ ((ContinuousFunctions X),(RAlgebra the carrier of X)) ; :: thesis:
set X1 = ContinuousFunctions X;
reconsider f = x as Element of ContinuousFunctions X ;
1_ (RAlgebra the carrier of X) = X --> 1
.= 1_ (R_Algebra_of_ContinuousFunctions X) by Th7 ;
then A2: ( [f,(1_ (RAlgebra the carrier of X))] in [:(ContinuousFunctions X),(ContinuousFunctions X):] & [(1_ (RAlgebra the carrier of X)),f] in [:(ContinuousFunctions X),(ContinuousFunctions X):] ) by ZFMISC_1:87;
x * e = (mult_ ((ContinuousFunctions X),(RAlgebra the carrier of X))) . (f,(1_ (RAlgebra the carrier of X))) by A1, C0SP1:def 8;
then x * e = ( the multF of (RAlgebra the carrier of X) || (ContinuousFunctions X)) . (f,(1_ (RAlgebra the carrier of X))) by C0SP1:def 6;
then x * e = f * (1_ (RAlgebra the carrier of X)) by A2, FUNCT_1:49;
hence x * e = x ; :: thesis:
e * x = (mult_ ((ContinuousFunctions X),(RAlgebra the carrier of X))) . ((1_ (RAlgebra the carrier of X)),f) by A1, C0SP1:def 8;
then e * x = ( the multF of (RAlgebra the carrier of X) || (ContinuousFunctions X)) . ((1_ (RAlgebra the carrier of X)),f) by C0SP1:def 6;
then e * x = (1_ (RAlgebra 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 R_Normed_Algebra_of_ContinuousFunctions X -> unital ;

proof end;

end;

theorem Th10: :: C0SP2:10

for W being Normed_AlgebraStr
for V being Algebra st AlgebraStr(# the carrier of W, the multF of W, the addF of W, the Mult of W, the OneF of W, the ZeroF of W #) = V holds
W is Algebra

proof end;

Lm3: for X being non empty compact TopSpace
for x being Point of (R_Normed_Algebra_of_ContinuousFunctions X)
for y being Point of (R_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 (R_Normed_Algebra_of_ContinuousFunctions X)
for y1, y2 being Point of (R_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 Real
for x being Point of (R_Normed_Algebra_of_ContinuousFunctions X)
for y being Point of (R_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 (R_Normed_Algebra_of_ContinuousFunctions X)
for y1, y2 being Point of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) st x1 = y1 & x2 = y2 holds
x1 * x2 = y1 * y2

proof end;

registration

let X be non empty compact TopSpace;

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

proof end;

end;

reconsider jj = 1 as Element of REAL by XREAL_0:def 1;

theorem Th11: :: C0SP2:11

for X being non empty compact TopSpace
for F being Point of (R_Normed_Algebra_of_ContinuousFunctions X) holds (Mult_ ((ContinuousFunctions X),(RAlgebra the carrier of X))) . (1,F) = F

proof end;

registration end;

theorem Th12: :: C0SP2:12

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

proof end;

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

proof end;

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

proof end;

theorem :: C0SP2:13

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

proof end;

theorem :: C0SP2:14

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

proof end;

theorem Th15: :: C0SP2:15

for X being non empty compact TopSpace
for F, G, H being Point of (R_Normed_Algebra_of_ContinuousFunctions X)
for f, g, h being RealMap of 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 :: C0SP2:16

for a being Real
for X being non empty compact TopSpace
for F, G being Point of (R_Normed_Algebra_of_ContinuousFunctions X)
for f, g being RealMap of 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 :: C0SP2:17

for X being non empty compact TopSpace
for F, G, H being Point of (R_Normed_Algebra_of_ContinuousFunctions X)
for f, g, h being RealMap of 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 Th18: :: C0SP2:18

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

proof end;

registration

let X be non empty compact TopSpace;

cluster R_Normed_Algebra_of_ContinuousFunctions X -> discerning reflexive RealNormSpace-like ;

proof end;

end;

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

proof end;

theorem :: C0SP2:19

for X being non empty compact TopSpace
for F, G, H being Point of (R_Normed_Algebra_of_ContinuousFunctions X)
for f, g, h being RealMap of 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 Th20: :: C0SP2:20

for X being RealBanachSpace
for Y being Subset of X
for seq being sequence of X st Y is closed & rng seq c= Y & seq is Cauchy_sequence_by_Norm holds
( seq is convergent & lim seq in Y ) by LOPBAN_1:def 15, NFCONT_1:def 3;

theorem Th21: :: C0SP2:21

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

proof end;

theorem Th22: :: C0SP2:22

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

proof end;

registration

let X be non empty compact TopSpace;

cluster R_Normed_Algebra_of_ContinuousFunctions X -> complete ;

proof end;

end;

registration

let X be non empty compact TopSpace;

cluster R_Normed_Algebra_of_ContinuousFunctions X -> Banach_Algebra-like ;

proof end;

end;

theorem Th23: :: C0SP2:23

for X being non empty TopSpace
for f, g being RealMap of X holds support (f + g) c= (support f) \/ (support g)

proof end;

theorem Th24: :: C0SP2:24

for X being non empty TopSpace
for a being Real
for f being RealMap of X holds support (a (#) f) c= support f

proof end;

theorem :: C0SP2:25

for X being non empty TopSpace
for f, g being RealMap of X holds support (f (#) g) c= (support f) \/ (support g)

proof end;

definition

let X be non empty TopSpace;

func C_0_Functions X -> non empty Subset of (RealVectSpace the carrier of X) equals :: C0SP2:def 6
{ f where f is RealMap of X : ( 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 C_0_Functions C0SP2:def 6 :

theorem :: C0SP2:26

for X being non empty TopSpace holds C_0_Functions X is non empty Subset of (RAlgebra the carrier of X) ;

Lm10: for X being non empty TopSpace
for v, u being Element of (RAlgebra the carrier of X) st v in C_0_Functions X & u in C_0_Functions X holds
v + u in C_0_Functions X

proof end;

Lm11: for X being non empty TopSpace
for a being Real
for u being Element of (RAlgebra the carrier of X) st u in C_0_Functions X holds
a * u in C_0_Functions X

proof end;

Lm12: for X being non empty TopSpace
for u being Element of (RAlgebra the carrier of X) st u in C_0_Functions X holds
- u in C_0_Functions X

proof end;

theorem :: C0SP2:27

for X being non empty TopSpace
for W being non empty Subset of (RAlgebra the carrier of X) st W = C_0_Functions X holds
W is additively-linearly-closed

proof end;

theorem Th28: :: C0SP2:28

for X being non empty TopSpace holds C_0_Functions X is linearly-closed

proof end;

registration

let X be non empty TopSpace;

cluster C_0_Functions X -> non empty linearly-closed ;

correctness by Th28;

end;

definition

let X be non empty TopSpace;

func R_VectorSpace_of_C_0_Functions X -> RealLinearSpace equals :: C0SP2:def 7
RLSStruct(# (C_0_Functions X),(Zero_ ((C_0_Functions X),(RealVectSpace the carrier of X))),(Add_ ((C_0_Functions X),(RealVectSpace the carrier of X))),(Mult_ ((C_0_Functions X),(RealVectSpace the carrier of X))) #);

correctness by RSSPACE:11;

end;

:: deftheorem defines R_VectorSpace_of_C_0_Functions C0SP2:def 7 :

theorem :: C0SP2:29

for X being non empty TopSpace holds R_VectorSpace_of_C_0_Functions X is Subspace of RealVectSpace the carrier of X by RSSPACE:11;

theorem Th30: :: C0SP2:30

for X being non empty TopSpace
for x being set st x in C_0_Functions X holds
x in BoundedFunctions the carrier of X

proof end;

definition

let X be non empty TopSpace;

func C_0_FunctionsNorm X -> Function of (C_0_Functions X),REAL equals :: C0SP2:def 8
(BoundedFunctionsNorm the carrier of X) | (C_0_Functions X);

proof end;

end;

:: deftheorem defines C_0_FunctionsNorm C0SP2:def 8 :

definition

let X be non empty TopSpace;

func R_Normed_Space_of_C_0_Functions X -> non empty NORMSTR equals :: C0SP2:def 9
NORMSTR(# (C_0_Functions X),(Zero_ ((C_0_Functions X),(RealVectSpace the carrier of X))),(Add_ ((C_0_Functions X),(RealVectSpace the carrier of X))),(Mult_ ((C_0_Functions X),(RealVectSpace the carrier of X))),(C_0_FunctionsNorm X) #);

end;

:: deftheorem defines R_Normed_Space_of_C_0_Functions C0SP2:def 9 :

registration

let X be non empty TopSpace;

cluster R_Normed_Space_of_C_0_Functions X -> non empty strict ;

end;

theorem :: C0SP2:31

for X being non empty TopSpace
for x being set st x in C_0_Functions X holds
x in ContinuousFunctions X

proof end;

theorem Th32: :: C0SP2:32

for X being non empty TopSpace holds 0. (R_VectorSpace_of_C_0_Functions X) = X --> 0

proof end;

theorem Th33: :: C0SP2:33

for X being non empty TopSpace holds 0. (R_Normed_Space_of_C_0_Functions X) = X --> 0

proof end;

Lm13: for X being non empty TopSpace
for x1, x2 being Point of (R_Normed_Space_of_C_0_Functions X)
for y1, y2 being Point of (R_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 Real
for x being Point of (R_Normed_Space_of_C_0_Functions X)
for y being Point of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) st x = y holds
a * x = a * y

proof end;

theorem Th34: :: C0SP2:34

for a being Real
for X being non empty TopSpace
for F, G being Point of (R_Normed_Space_of_C_0_Functions X) holds
( ( ||.F.|| = 0 implies F = 0. (R_Normed_Space_of_C_0_Functions X) ) & ( F = 0. (R_Normed_Space_of_C_0_Functions X) implies ||.F.|| = 0 ) & ||.(a * F).|| = |.a.| * ||.F.|| & ||.(F + G).|| <= ||.F.|| + ||.G.|| )

proof end;

theorem Th35: :: C0SP2:35

for X being non empty TopSpace holds R_Normed_Space_of_C_0_Functions X is RealNormSpace-like

proof end;

registration

let X be non empty TopSpace;

cluster R_Normed_Space_of_C_0_Functions X -> non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ;

proof end;

end;

theorem :: C0SP2:36

for X being non empty TopSpace holds R_Normed_Space_of_C_0_Functions X is RealNormSpace ;