:: Banach Algebra of Bounded Complex-Valued Functionals
:: by Katuhiko Kanazashi , Hiroyuki Okazaki and Yasunari Shidama
::
:: Received November 20, 2010
:: Copyright (c) 2010-2019 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies STRUCT_0, FUNCOP_1, NUMBERS, PRE_TOPC, ORDINAL2, SUBSET_1,
RELAT_1, XBOOLE_0, FUNCT_1, TARSKI, CARD_1, ARYTM_3, ARYTM_1, COMPLEX1,
RLVECT_1, ALGSTR_0, FUNCSDOM, REAL_1, FUNCT_2, C0SP1, IDEAL_1, VALUED_1,
MESFUNC1, RSSPACE, BINOP_1, SUPINF_2, GROUP_1, REALSET1, ZFMISC_1,
XXREAL_2, XXREAL_0, NORMSP_1, REWRITE1, NAT_1, RSSPACE3, SEQ_2, PARTFUN1,
SEQ_1, LOPBAN_1, SEQ_4, CFUNCDOM, CLVECT_1, VECTSP_1, RELAT_2, CC0SP1,
XCMPLX_0, CLOPBAN2, LOPBAN_2, METRIC_1, COMSEQ_1, CSSPACE4;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, ORDINAL1,
RELSET_1, PARTFUN1, FUNCT_2, BINOP_1, DOMAIN_1, FUNCOP_1, NUMBERS,
XXREAL_0, XCMPLX_0, XREAL_0, REAL_1, COMPLEX1, XXREAL_2, REALSET1,
VALUED_1, SEQ_1, SEQ_2, SEQ_4, STRUCT_0, ALGSTR_0, PRE_TOPC, RLVECT_1,
GROUP_1, VECTSP_1, NORMSP_0, IDEAL_1, CLVECT_1, CFUNCDOM, CLOPBAN2,
C0SP1, CLOPBAN1, COMSEQ_1, CSSPACE3, COMSEQ_2;
constructors REAL_1, REALSET1, COMPLEX1, IDEAL_1, C0SP1, PARTFUN3, BINOP_2,
TOPMETR, SEQ_4, MEASURE6, CLOPBAN2, CSSPACE3, COMSEQ_2;
registrations XBOOLE_0, SUBSET_1, STRUCT_0, XREAL_0, REALSET1, ALGSTR_0,
NUMBERS, ORDINAL1, MEMBERED, VECTSP_1, VECTSP_2, VALUED_0, C0SP1,
XXREAL_0, FUNCT_2, VALUED_1, RFUNCT_1, COMPLEX1, FUNCOP_1, CFUNCDOM,
CLVECT_1, XCMPLX_0, COMSEQ_2, CLOPBAN2, RELSET_1, SEQ_2;
requirements NUMERALS, BOOLE, SUBSET, ARITHM, REAL;
begin :: Banach Algebra of Bounded Complex-Valued Functionals
definition
let V be ComplexAlgebra;
mode ComplexSubAlgebra of V -> ComplexAlgebra means
:: 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;
end;
theorem :: CC0SP1:1
for X being non empty set,
V being ComplexAlgebra,
V1 being non empty Subset of V,
d1,d2 being Element of X,
A being BinOp of X,
M being Function of [:X,X:],X,
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;
registration let V be ComplexAlgebra;
cluster strict for ComplexSubAlgebra of V;
end;
definition let V be ComplexAlgebra, V1 be Subset of V;
attr V1 is Cadditively-linearly-closed means
:: CC0SP1:def 2
V1 is add-closed having-inverse &
for a be Complex, v be Element of V st v in V1 holds a * v in V1;
end;
definition let V be ComplexAlgebra, V1 be Subset of V such that
V1 is Cadditively-linearly-closed non empty;
func Mult_(V1,V) -> Function of [:COMPLEX,V1:], V1 equals
:: CC0SP1:def 3
(the Mult of V) | [:COMPLEX,V1:];
end;
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 };
end;
registration let X be non empty set;
cluster CAlgebra X -> scalar-unital;
end;
registration let X be non empty set;
cluster ComplexBoundedFunctions(X) -> Cadditively-linearly-closed
multiplicatively-closed;
end;
registration let V be ComplexAlgebra;
cluster Cadditively-linearly-closed multiplicatively-closed
for non empty Subset of V;
end;
definition let V be non empty CLSStruct;
attr V is scalar-mult-cancelable means
:: CC0SP1:def 5
for a be Complex, v be Element of V st a*v=0.V holds a=0 or v=0.V;
end;
theorem :: CC0SP1:2
for V being ComplexAlgebra,
V1 being Cadditively-linearly-closed multiplicatively-closed
non empty 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;
theorem :: CC0SP1:3
for V be ComplexAlgebra, V1 be ComplexSubAlgebra of V holds
( for v1,w1 be Element of V1, v,w be Element of V st
v1=v & w1=w holds v1+w1=v+w ) &
( for v1,w1 be Element of V1, v,w be Element of V st
v1=v & w1=w holds v1*w1=v*w ) &
( for v1 be Element of V1, v be Element of V, a be Complex st
v1=v holds a*v1=a*v ) &
1_V1 = 1_V & 0.V1=0.V;
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)) #);
end;
theorem :: CC0SP1:4
for X being non empty set holds C_Algebra_of_BoundedFunctions(X) is
ComplexSubAlgebra of CAlgebra(X);
registration let X be non empty set;
cluster C_Algebra_of_BoundedFunctions(X) ->
vector-distributive scalar-unital;
end;
theorem :: CC0SP1:5
for X being non empty set,
F, G, H being VECTOR of C_Algebra_of_BoundedFunctions(X),
f, g, h being Function of X,COMPLEX st f=F & g=G & h=H holds
( H = F+G iff (for x be Element of X holds h.x = (f.x) + (g.x)) );
theorem :: CC0SP1:6
for X being non empty set, a being Complex,
F, G being VECTOR of C_Algebra_of_BoundedFunctions(X),
f, g being Function of X,COMPLEX st f=F & g=G holds
( G = a*F iff for x be Element of X holds g.x = a*(f.x) );
theorem :: CC0SP1:7
for X being non empty set,
F, G, H being VECTOR of C_Algebra_of_BoundedFunctions(X),
f, g, h being Function of X,COMPLEX st f=F & g=G & h=H holds
( H = F*G iff for x be Element of X holds h.x = (f.x)*(g.x));
theorem :: CC0SP1:8
for X being non empty set holds
0.C_Algebra_of_BoundedFunctions X = X -->0;
theorem :: CC0SP1:9
for X being non empty set holds
1_C_Algebra_of_BoundedFunctions X = X --> 1r;
definition let X be non empty set, F be object;
assume
F in ComplexBoundedFunctions(X);
func modetrans(F,X) -> Function of X,COMPLEX means
:: CC0SP1:def 7
it = F & it|X is bounded;
end;
definition let X be non empty set, f be Function of X,COMPLEX;
func PreNorms f -> non empty Subset of REAL equals
:: CC0SP1:def 8
the set of all |.f.x.| where x is Element of X ;
end;
theorem :: CC0SP1:10
for X being non empty set,
f being Function of X,COMPLEX st f | X is bounded holds
PreNorms f is bounded_above;
theorem :: CC0SP1:11
for X being non empty set,
f being Function of X,COMPLEX holds
f|X is bounded iff PreNorms f is bounded_above;
definition let X be non empty set;
func ComplexBoundedFunctionsNorm(X) ->
Function of ComplexBoundedFunctions(X),REAL means
:: CC0SP1:def 9
for x be object st x in ComplexBoundedFunctions(X) holds
it.x = upper_bound PreNorms(modetrans(x,X));
end;
theorem :: CC0SP1:12
for X being non empty set,
f being Function of X,COMPLEX st f|X is bounded holds
modetrans(f,X) = f;
theorem :: CC0SP1:13
for X being non empty set,
f being Function of X,COMPLEX st f|X is bounded holds
(ComplexBoundedFunctionsNorm(X)).f = upper_bound PreNorms f;
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) #);
end;
registration let X be non empty set;
cluster C_Normed_Algebra_of_BoundedFunctions(X) -> non empty;
end;
registration let X be non empty set;
cluster C_Normed_Algebra_of_BoundedFunctions(X) -> unital;
end;
theorem :: CC0SP1:14
for W being Normed_Complex_AlgebraStr,
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;
theorem :: CC0SP1:15
for X being non empty set holds
C_Normed_Algebra_of_BoundedFunctions(X) is ComplexAlgebra;
theorem :: CC0SP1:16
for X being non empty set,
F being Point of C_Normed_Algebra_of_BoundedFunctions(X) holds
Mult_(ComplexBoundedFunctions(X),CAlgebra(X)).(1r,F) = F;
theorem :: CC0SP1:17
for X being non empty set holds
C_Normed_Algebra_of_BoundedFunctions(X) is ComplexLinearSpace;
theorem :: CC0SP1:18
for X being non empty set holds
X --> 0 = 0.(C_Normed_Algebra_of_BoundedFunctions(X));
theorem :: CC0SP1:19
for X being non empty set,
x being Element of X,
f being Function of X,COMPLEX,
F being Point of C_Normed_Algebra_of_BoundedFunctions(X) st
f = F & f | X is bounded holds |.f.x.| <= ||.F.||;
theorem :: CC0SP1:20
for X being non empty set,
F being Point of C_Normed_Algebra_of_BoundedFunctions(X) holds
0 <= ||.F.||;
theorem :: CC0SP1:21
for X being non empty set,
F being Point of C_Normed_Algebra_of_BoundedFunctions(X) st
F = 0.(C_Normed_Algebra_of_BoundedFunctions X) holds
0 = ||.F.||;
theorem :: CC0SP1:22
for X being non empty set,
f, g, h being Function of X,COMPLEX,
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));
theorem :: CC0SP1:23
for X being non empty set,
a being Complex,
f, g being Function of X,COMPLEX,
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));
theorem :: CC0SP1:24
for X being non empty set,
f, g, h being Function of X,COMPLEX,
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));
theorem :: CC0SP1:25
for X being non empty set,
a being Complex,
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.||);
registration let X be non empty set;
cluster C_Normed_Algebra_of_BoundedFunctions(X) -> right_complementable
Abelian add-associative right_zeroed vector-distributive
scalar-distributive scalar-associative scalar-unital discerning
reflexive ComplexNormSpace-like;
end;
theorem :: CC0SP1:26
for X being non empty set,
f, g, h being Function of X,COMPLEX,
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));
theorem :: CC0SP1:27
for X being non empty set,
seq being sequence of C_Normed_Algebra_of_BoundedFunctions(X)
st seq is CCauchy holds seq is convergent;
registration let X be non empty set;
cluster C_Normed_Algebra_of_BoundedFunctions(X) -> complete;
end;
theorem :: CC0SP1:28
for X being non empty set holds
C_Normed_Algebra_of_BoundedFunctions(X) is Complex_Banach_Algebra;