CC0SP1: Banach Algebra of Bounded Complex-Valued Functionals

:: Banach Algebra of Bounded Complex-Valued Functionals
:: by Katuhiko Kanazashi , Hiroyuki Okazaki and Yasunari Shidama
::
:: Received November 20, 2010
:: Copyright (c) 2010-2021 Association of Mizar Users

definition

let V be ComplexAlgebra;

mode ComplexSubAlgebra of V -> ComplexAlgebra means :Def1: :: CC0SP1:def 1
( the carrier of it c= the carrier of V & the addF of it = the addF of V || the carrier of it & the multF of it = the multF of V || the carrier of it & the Mult of it = the Mult of V | [:COMPLEX, the carrier of it:] & 1. it = 1. V & 0. it = 0. V );

existence

proof end;

end;

:: deftheorem Def1 defines ComplexSubAlgebra CC0SP1:def 1 :

Lm1: now :: thesis:

let z be Complex; :: thesis:
let X be non empty set ; :: thesis:
let V be ComplexAlgebra; :: thesis:
let V1 be non empty Subset of V; :: thesis:
let d1, d2 be Element of X; :: thesis:
let A be BinOp of X; :: thesis:
let M be Function of [:X,X:],X; :: thesis:
let MR be Function of [:COMPLEX,X:],X; :: thesis:
assume that
A1: V1 = X and
A2: MR = the Mult of V | [:COMPLEX,V1:] ; :: thesis:
let W be non empty ComplexAlgebraStr ; :: thesis:
assume A3: W = ComplexAlgebraStr(# X,M,A,MR,d2,d1 #) ; :: thesis:
let w be VECTOR of W; :: thesis:
let v be Element of V; :: thesis:
assume A4: w = v ; :: thesis:
z in COMPLEX by XCMPLX_0:def 2;
then [z,w] in [:COMPLEX,V1:] by A1, A3, ZFMISC_1:def 2;
hence z * w = z * v by A4, A2, A3, FUNCT_1:49; :: thesis:

end;

theorem Th1: :: CC0SP1:1

for X being non empty set
for V being ComplexAlgebra
for V1 being non empty Subset of V
for d1, d2 being Element of X
for A being BinOp of X
for M being Function of [:X,X:],X
for MR being Function of [:COMPLEX,X:],X st V1 = X & d1 = 0. V & d2 = 1. V & A = the addF of V || V1 & M = the multF of V || V1 & MR = the Mult of V | [:COMPLEX,V1:] & V1 is having-inverse holds
ComplexAlgebraStr(# X,M,A,MR,d2,d1 #) is ComplexSubAlgebra of V

proof end;

registration

let V be ComplexAlgebra;

cluster non empty right_complementable Abelian add-associative right_zeroed right-distributive right_unital well-unital left_unital unital associative commutative vector-distributive scalar-distributive scalar-associative strict vector-associative for ComplexSubAlgebra of V;

existence

proof end;

end;

definition

let V be ComplexAlgebra;
let V1 be Subset of V;

attr V1 is Cadditively-linearly-closed means :Def2: :: CC0SP1:def 2
( V1 is add-closed & V1 is having-inverse & ( for a being Complex
for v being Element of V st v in V1 holds
a * v in V1 ) );

end;

:: deftheorem Def2 defines Cadditively-linearly-closed CC0SP1:def 2 :

definition

let V be ComplexAlgebra;
let V1 be Subset of V;
assume A1: ( V1 is Cadditively-linearly-closed & not V1 is empty ) ;

func Mult_ (V1,V) -> Function of [:COMPLEX,V1:],V1 equals :Def3: :: CC0SP1:def 3
the Mult of V | [:COMPLEX,V1:];

correctness

proof end;

end;

:: deftheorem Def3 defines Mult_ CC0SP1:def 3 :

definition

let X be non empty set ;

func ComplexBoundedFunctions X -> non empty Subset of (CAlgebra X) equals :: CC0SP1:def 4
{ f where f is Function of X,COMPLEX : f | X is bounded } ;

correctness

proof end;

end;

:: deftheorem defines ComplexBoundedFunctions CC0SP1:def 4 :

registration

let X be non empty set ;

cluster CAlgebra X -> scalar-unital ;

coherence by CFUNCDOM:12;

end;

registration

let X be non empty set ;

cluster ComplexBoundedFunctions X -> non empty multiplicatively-closed Cadditively-linearly-closed ;

coherence

proof end;

end;

registration

let V be ComplexAlgebra;

cluster non empty multiplicatively-closed Cadditively-linearly-closed for Element of bool the carrier of V;

existence

proof end;

end;

definition

let V be non empty CLSStruct ;

attr V is scalar-mult-cancelable means :: CC0SP1:def 5
for a being Complex
for v being Element of V holds
( not a * v = 0. V or a = 0 or v = 0. V );

end;

:: deftheorem defines scalar-mult-cancelable CC0SP1:def 5 :

theorem Th2: :: CC0SP1:2

for V being ComplexAlgebra
for V1 being non empty multiplicatively-closed Cadditively-linearly-closed Subset of V holds ComplexAlgebraStr(# V1,(mult_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)),(One_ (V1,V)),(Zero_ (V1,V)) #) is ComplexSubAlgebra of V

proof end;

theorem Th3: :: CC0SP1:3

for V being ComplexAlgebra
for V1 being ComplexSubAlgebra of V holds
( ( for v1, w1 being Element of V1
for v, w being Element of V st v1 = v & w1 = w holds
v1 + w1 = v + w ) & ( for v1, w1 being Element of V1
for v, w being Element of V st v1 = v & w1 = w holds
v1 * w1 = v * w ) & ( for v1 being Element of V1
for v being Element of V
for a being Complex st v1 = v holds
a * v1 = a * v ) & 1_ V1 = 1_ V & 0. V1 = 0. V )

proof end;

definition

let X be non empty set ;

func C_Algebra_of_BoundedFunctions X -> ComplexAlgebra equals :: CC0SP1:def 6
ComplexAlgebraStr(# (ComplexBoundedFunctions X),(mult_ ((ComplexBoundedFunctions X),(CAlgebra X))),(Add_ ((ComplexBoundedFunctions X),(CAlgebra X))),(Mult_ ((ComplexBoundedFunctions X),(CAlgebra X))),(One_ ((ComplexBoundedFunctions X),(CAlgebra X))),(Zero_ ((ComplexBoundedFunctions X),(CAlgebra X))) #);

coherence by Th2;

end;

:: deftheorem defines C_Algebra_of_BoundedFunctions CC0SP1:def 6 :

theorem :: CC0SP1:4

for X being non empty set holds C_Algebra_of_BoundedFunctions X is ComplexSubAlgebra of CAlgebra X by Th2;

registration

let X be non empty set ;

cluster C_Algebra_of_BoundedFunctions X -> scalar-unital ;

coherence

proof end;

end;

theorem Th5: :: CC0SP1:5

for X being non empty set
for F, G, H being VECTOR of (C_Algebra_of_BoundedFunctions X)
for f, g, h being Function of X,COMPLEX 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 Th6: :: CC0SP1:6

for X being non empty set
for a being Complex
for F, G being VECTOR of (C_Algebra_of_BoundedFunctions X)
for f, g being Function of X,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 Th7: :: CC0SP1:7

for X being non empty set
for F, G, H being VECTOR of (C_Algebra_of_BoundedFunctions X)
for f, g, h being Function of X,COMPLEX 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 Th8: :: CC0SP1:8

for X being non empty set holds 0. (C_Algebra_of_BoundedFunctions X) = X --> 0

proof end;

theorem Th9: :: CC0SP1:9

for X being non empty set holds 1_ (C_Algebra_of_BoundedFunctions X) = X --> 1r

proof end;

definition

let X be non empty set ;
let F be object ;
assume A1: F in ComplexBoundedFunctions X ;

func modetrans (F,X) -> Function of X,COMPLEX means :Def7: :: CC0SP1:def 7
( it = F & it | X is bounded );

correctness by A1;

end;

:: deftheorem Def7 defines modetrans CC0SP1:def 7 :

definition

let X be non empty set ;
let f be Function of X,COMPLEX;

func PreNorms f -> non empty Subset of REAL equals :: CC0SP1:def 8
{ |.(f . x).| where x is Element of X : verum } ;

coherence

proof end;

end;

:: deftheorem defines PreNorms CC0SP1:def 8 :

Lm2: for C being non empty set
for f being PartFunc of C,COMPLEX holds
( |.f.| is bounded iff f is bounded )

proof end;

theorem Th10: :: CC0SP1:10

for X being non empty set
for f being Function of X,COMPLEX st f | X is bounded holds
PreNorms f is bounded_above

proof end;

theorem Th11: :: CC0SP1:11

for X being non empty set
for f being Function of X,COMPLEX holds
( f | X is bounded iff PreNorms f is bounded_above )

proof end;

definition

let X be non empty set ;

func ComplexBoundedFunctionsNorm X -> Function of (ComplexBoundedFunctions X),REAL means :Def9: :: CC0SP1:def 9
for x being object st x in ComplexBoundedFunctions X holds
it . x = upper_bound (PreNorms (modetrans (x,X)));

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def9 defines ComplexBoundedFunctionsNorm CC0SP1:def 9 :

theorem Th12: :: CC0SP1:12

for X being non empty set
for f being Function of X,COMPLEX st f | X is bounded holds
modetrans (f,X) = f

proof end;

theorem Th13: :: CC0SP1:13

for X being non empty set
for f being Function of X,COMPLEX st f | X is bounded holds
(ComplexBoundedFunctionsNorm X) . f = upper_bound (PreNorms f)

proof end;

definition

let X be non empty set ;

func C_Normed_Algebra_of_BoundedFunctions X -> Normed_Complex_AlgebraStr equals :: CC0SP1:def 10
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions X),(mult_ ((ComplexBoundedFunctions X),(CAlgebra X))),(Add_ ((ComplexBoundedFunctions X),(CAlgebra X))),(Mult_ ((ComplexBoundedFunctions X),(CAlgebra X))),(One_ ((ComplexBoundedFunctions X),(CAlgebra X))),(Zero_ ((ComplexBoundedFunctions X),(CAlgebra X))),(ComplexBoundedFunctionsNorm X) #);

correctness ;

end;

:: deftheorem defines C_Normed_Algebra_of_BoundedFunctions CC0SP1:def 10 :

registration

let X be non empty set ;

cluster C_Normed_Algebra_of_BoundedFunctions X -> non empty ;

correctness ;

end;

Lm3: now :: thesis:

end;

registration

let X be non empty set ;

cluster C_Normed_Algebra_of_BoundedFunctions X -> unital ;

correctness

proof end;

end;

theorem Th14: :: CC0SP1:14

for W being Normed_Complex_AlgebraStr
for V being ComplexAlgebra st ComplexAlgebraStr(# 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 ComplexAlgebra

proof end;

theorem Th15: :: CC0SP1:15

for X being non empty set holds C_Normed_Algebra_of_BoundedFunctions X is ComplexAlgebra

proof end;

theorem :: CC0SP1:16

for X being non empty set
for F being Point of (C_Normed_Algebra_of_BoundedFunctions X) holds (Mult_ ((ComplexBoundedFunctions X),(CAlgebra X))) . (1r,F) = F

proof end;

theorem Th17: :: CC0SP1:17

for X being non empty set holds C_Normed_Algebra_of_BoundedFunctions X is ComplexLinearSpace

proof end;

theorem Th18: :: CC0SP1:18

for X being non empty set holds X --> 0 = 0. (C_Normed_Algebra_of_BoundedFunctions X)

proof end;

theorem Th19: :: CC0SP1:19

for X being non empty set
for x being Element of X
for f being Function of X,COMPLEX
for F being Point of (C_Normed_Algebra_of_BoundedFunctions X) st f = F & f | X is bounded holds
|.(f . x).| <= ||.F.||

proof end;

theorem :: CC0SP1:20

for X being non empty set
for F being Point of (C_Normed_Algebra_of_BoundedFunctions X) holds 0 <= ||.F.||

proof end;

theorem Th21: :: CC0SP1:21

for X being non empty set
for F being Point of (C_Normed_Algebra_of_BoundedFunctions X) st F = 0. (C_Normed_Algebra_of_BoundedFunctions X) holds
0 = ||.F.||

proof end;

theorem Th22: :: CC0SP1:22

for X being non empty set
for f, g, h being Function of X,COMPLEX
for F, G, H being Point of (C_Normed_Algebra_of_BoundedFunctions 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 Th23: :: CC0SP1:23

for X being non empty set
for a being Complex
for f, g being Function of X,COMPLEX
for F, G being Point of (C_Normed_Algebra_of_BoundedFunctions 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 Th24: :: CC0SP1:24

for X being non empty set
for f, g, h being Function of X,COMPLEX
for F, G, H being Point of (C_Normed_Algebra_of_BoundedFunctions 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: :: CC0SP1:25

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

proof end;

Lm4: for X being non empty set holds
( C_Normed_Algebra_of_BoundedFunctions X is reflexive & C_Normed_Algebra_of_BoundedFunctions X is discerning & C_Normed_Algebra_of_BoundedFunctions X is ComplexNormSpace-like )
by Th25;

registration

let X be non empty set ;

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

coherence by Th17, Lm4;

end;

theorem Th26: :: CC0SP1:26

for X being non empty set
for f, g, h being Function of X,COMPLEX
for F, G, H being Point of (C_Normed_Algebra_of_BoundedFunctions 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 Th27: :: CC0SP1:27

for X being non empty set
for seq being sequence of (C_Normed_Algebra_of_BoundedFunctions X) st seq is CCauchy holds
seq is convergent

proof end;

registration

let X be non empty set ;

cluster C_Normed_Algebra_of_BoundedFunctions X -> complete ;

coherence

proof end;

end;

theorem :: CC0SP1:28

for X being non empty set holds C_Normed_Algebra_of_BoundedFunctions X is Complex_Banach_Algebra

proof end;