:: Complex {B}anach Space of Bounded Linear Operators
:: by Noboru Endou
::
:: Copyright (c) 2004-2021 Association of Mizar Users

definition
let X be set ;
let Y be non empty set ;
let F be Function of ,Y;
let c be Complex;
let f be Function of X,Y;
:: original: [;]
redefine func F [;] (c,f) -> Element of Funcs (X,Y);
coherence
F [;] (c,f) is Element of Funcs (X,Y)
proof end;
end;

definition
let X be non empty set ;
let Y be ComplexLinearSpace;
func FuncExtMult (X,Y) -> Function of [:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)) means :Def1: :: CLOPBAN1:def 1
for c being Complex
for f being Element of Funcs (X, the carrier of Y)
for x being Element of X holds (it . [c,f]) . x = c * (f . x);
existence
ex b1 being Function of [:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)) st
for c being Complex
for f being Element of Funcs (X, the carrier of Y)
for x being Element of X holds (b1 . [c,f]) . x = c * (f . x)
proof end;
uniqueness
for b1, b2 being Function of [:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)) st ( for c being Complex
for f being Element of Funcs (X, the carrier of Y)
for x being Element of X holds (b1 . [c,f]) . x = c * (f . x) ) & ( for c being Complex
for f being Element of Funcs (X, the carrier of Y)
for x being Element of X holds (b2 . [c,f]) . x = c * (f . x) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines FuncExtMult CLOPBAN1:def 1 :
for X being non empty set
for Y being ComplexLinearSpace
for b3 being Function of [:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)) holds
( b3 = FuncExtMult (X,Y) iff for c being Complex
for f being Element of Funcs (X, the carrier of Y)
for x being Element of X holds (b3 . [c,f]) . x = c * (f . x) );

theorem Th1: :: CLOPBAN1:1
for X being non empty set
for Y being ComplexLinearSpace
for x being Element of X holds (FuncZero (X,Y)) . x = 0. Y
proof end;

theorem Th2: :: CLOPBAN1:2
for X being non empty set
for Y being ComplexLinearSpace
for f, h being Element of Funcs (X, the carrier of Y)
for a being Complex holds
( h = (FuncExtMult (X,Y)) . [a,f] iff for x being Element of X holds h . x = a * (f . x) )
proof end;

theorem Th3: :: CLOPBAN1:3
for X being non empty set
for Y being ComplexLinearSpace
for f, g being Element of Funcs (X, the carrier of Y) holds (FuncAdd (X,Y)) . (f,g) = (FuncAdd (X,Y)) . (g,f)
proof end;

theorem Th4: :: CLOPBAN1:4
for X being non empty set
for Y being ComplexLinearSpace
for f, g, h being Element of Funcs (X, the carrier of Y) holds (FuncAdd (X,Y)) . (f,((FuncAdd (X,Y)) . (g,h))) = (FuncAdd (X,Y)) . (((FuncAdd (X,Y)) . (f,g)),h)
proof end;

theorem Th5: :: CLOPBAN1:5
for X being non empty set
for Y being ComplexLinearSpace
for f being Element of Funcs (X, the carrier of Y) holds (FuncAdd (X,Y)) . ((FuncZero (X,Y)),f) = f
proof end;

theorem Th6: :: CLOPBAN1:6
for X being non empty set
for Y being ComplexLinearSpace
for f being Element of Funcs (X, the carrier of Y) holds (FuncAdd (X,Y)) . (f,((FuncExtMult (X,Y)) . [(),f])) = FuncZero (X,Y)
proof end;

theorem Th7: :: CLOPBAN1:7
for X being non empty set
for Y being ComplexLinearSpace
for f being Element of Funcs (X, the carrier of Y) holds (FuncExtMult (X,Y)) . [1r,f] = f
proof end;

theorem Th8: :: CLOPBAN1:8
for X being non empty set
for Y being ComplexLinearSpace
for f being Element of Funcs (X, the carrier of Y)
for a, b being Complex holds (FuncExtMult (X,Y)) . [a,((FuncExtMult (X,Y)) . [b,f])] = (FuncExtMult (X,Y)) . [(a * b),f]
proof end;

theorem Th9: :: CLOPBAN1:9
for X being non empty set
for Y being ComplexLinearSpace
for f being Element of Funcs (X, the carrier of Y)
for a, b being Complex holds (FuncAdd (X,Y)) . (((FuncExtMult (X,Y)) . [a,f]),((FuncExtMult (X,Y)) . [b,f])) = (FuncExtMult (X,Y)) . [(a + b),f]
proof end;

Lm1: for X being non empty set
for Y being ComplexLinearSpace
for f, g being Element of Funcs (X, the carrier of Y)
for a being Complex holds (FuncAdd (X,Y)) . (((FuncExtMult (X,Y)) . [a,f]),((FuncExtMult (X,Y)) . [a,g])) = (FuncExtMult (X,Y)) . [a,((FuncAdd (X,Y)) . (f,g))]

proof end;

theorem Th10: :: CLOPBAN1:10
for X being non empty set
for Y being ComplexLinearSpace holds CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) is ComplexLinearSpace
proof end;

definition
let X be non empty set ;
let Y be ComplexLinearSpace;
func ComplexVectSpace (X,Y) -> ComplexLinearSpace equals :: CLOPBAN1:def 2
CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #);
coherence
CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) is ComplexLinearSpace
by Th10;
end;

:: deftheorem defines ComplexVectSpace CLOPBAN1:def 2 :
for X being non empty set
for Y being ComplexLinearSpace holds ComplexVectSpace (X,Y) = CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #);

registration
let X be non empty set ;
let Y be ComplexLinearSpace;
cluster ComplexVectSpace (X,Y) -> strict ;
coherence
ComplexVectSpace (X,Y) is strict
;
end;

registration
let X be non empty set ;
let Y be ComplexLinearSpace;
coherence by MONOID_0:80;
end;

definition
let X be non empty set ;
let Y be ComplexLinearSpace;
let f be VECTOR of (ComplexVectSpace (X,Y));
let x be Element of X;
:: original: .
redefine func f . x -> VECTOR of Y;
coherence
f . x is VECTOR of Y
proof end;
end;

theorem :: CLOPBAN1:11
for X being non empty set
for Y being ComplexLinearSpace
for f, g, h being VECTOR of (ComplexVectSpace (X,Y)) holds
( h = f + g iff for x being Element of X holds h . x = (f . x) + (g . x) ) by LOPBAN_1:1;

theorem Th12: :: CLOPBAN1:12
for X being non empty set
for Y being ComplexLinearSpace
for f, h being VECTOR of (ComplexVectSpace (X,Y))
for c being Complex holds
( h = c * f iff for x being Element of X holds h . x = c * (f . x) )
proof end;

definition
let X, Y be non empty CLSStruct ;
let IT be Function of X,Y;
attr IT is homogeneous means :Def3: :: CLOPBAN1:def 3
for x being VECTOR of X
for r being Complex holds IT . (r * x) = r * (IT . x);
end;

:: deftheorem Def3 defines homogeneous CLOPBAN1:def 3 :
for X, Y being non empty CLSStruct
for IT being Function of X,Y holds
( IT is homogeneous iff for x being VECTOR of X
for r being Complex holds IT . (r * x) = r * (IT . x) );

registration
let X be non empty CLSStruct ;
let Y be ComplexLinearSpace;
cluster non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive homogeneous for Element of bool [: the carrier of X, the carrier of Y:];
existence
ex b1 being Function of X,Y st
( b1 is additive & b1 is homogeneous )
proof end;
end;

definition end;

definition
let X, Y be ComplexLinearSpace;
func LinearOperators (X,Y) -> Subset of (ComplexVectSpace ( the carrier of X,Y)) means :Def4: :: CLOPBAN1:def 4
for x being set holds
( x in it iff x is LinearOperator of X,Y );
existence
ex b1 being Subset of (ComplexVectSpace ( the carrier of X,Y)) st
for x being set holds
( x in b1 iff x is LinearOperator of X,Y )
proof end;
uniqueness
for b1, b2 being Subset of (ComplexVectSpace ( the carrier of X,Y)) st ( for x being set holds
( x in b1 iff x is LinearOperator of X,Y ) ) & ( for x being set holds
( x in b2 iff x is LinearOperator of X,Y ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines LinearOperators CLOPBAN1:def 4 :
for X, Y being ComplexLinearSpace
for b3 being Subset of (ComplexVectSpace ( the carrier of X,Y)) holds
( b3 = LinearOperators (X,Y) iff for x being set holds
( x in b3 iff x is LinearOperator of X,Y ) );

registration
let X, Y be ComplexLinearSpace;
cluster LinearOperators (X,Y) -> non empty functional ;
coherence
( not LinearOperators (X,Y) is empty & LinearOperators (X,Y) is functional )
proof end;
end;

theorem Th13: :: CLOPBAN1:13
for X, Y being ComplexLinearSpace holds LinearOperators (X,Y) is linearly-closed
proof end;

theorem :: CLOPBAN1:14
for X, Y being ComplexLinearSpace holds CLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))) #) is Subspace of ComplexVectSpace ( the carrier of X,Y) by ;

registration
let X, Y be ComplexLinearSpace;
cluster CLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))) #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
coherence
( CLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))) #) is Abelian & CLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))) #) is add-associative & CLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))) #) is right_zeroed & CLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))) #) is right_complementable & CLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))) #) is vector-distributive & CLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))) #) is scalar-distributive & CLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))) #) is scalar-associative & CLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))) #) is scalar-unital )
by ;
end;

definition
let X, Y be ComplexLinearSpace;
func C_VectorSpace_of_LinearOperators (X,Y) -> ComplexLinearSpace equals :: CLOPBAN1:def 5
CLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))) #);
coherence
CLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))) #) is ComplexLinearSpace
;
end;

:: deftheorem defines C_VectorSpace_of_LinearOperators CLOPBAN1:def 5 :
for X, Y being ComplexLinearSpace holds C_VectorSpace_of_LinearOperators (X,Y) = CLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(ComplexVectSpace ( the carrier of X,Y)))) #);

registration
let X, Y be ComplexLinearSpace;
coherence ;
end;

registration
let X, Y be ComplexLinearSpace;
coherence by MONOID_0:80;
end;

definition
let X, Y be ComplexLinearSpace;
let f be Element of ();
let v be VECTOR of X;
:: original: .
redefine func f . v -> VECTOR of Y;
coherence
f . v is VECTOR of Y
proof end;
end;

theorem Th15: :: CLOPBAN1:15
for X, Y being ComplexLinearSpace
for f, g, h being VECTOR of () holds
( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )
proof end;

theorem Th16: :: CLOPBAN1:16
for X, Y being ComplexLinearSpace
for f, h being VECTOR of ()
for c being Complex holds
( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) )
proof end;

theorem Th17: :: CLOPBAN1:17
for X, Y being ComplexLinearSpace holds 0. () = the carrier of X --> (0. Y)
proof end;

theorem Th18: :: CLOPBAN1:18
for X, Y being ComplexLinearSpace holds the carrier of X --> (0. Y) is LinearOperator of X,Y
proof end;

theorem Th19: :: CLOPBAN1:19
for X being ComplexNormSpace
for seq being sequence of X
for g being Point of X st seq is convergent & lim seq = g holds
( ||.seq.|| is convergent & lim ||.seq.|| = )
proof end;

definition
let X, Y be ComplexNormSpace;
let IT be LinearOperator of X,Y;
attr IT is Lipschitzian means :Def6: :: CLOPBAN1:def 6
ex K being Real st
( 0 <= K & ( for x being VECTOR of X holds ||.(IT . x).|| <= K * ) );
end;

:: deftheorem Def6 defines Lipschitzian CLOPBAN1:def 6 :
for X, Y being ComplexNormSpace
for IT being LinearOperator of X,Y holds
( IT is Lipschitzian iff ex K being Real st
( 0 <= K & ( for x being VECTOR of X holds ||.(IT . x).|| <= K * ) ) );

theorem Th20: :: CLOPBAN1:20
for X, Y being ComplexNormSpace
for f being LinearOperator of X,Y st ( for x being VECTOR of X holds f . x = 0. Y ) holds
f is Lipschitzian
proof end;

registration
let X, Y be ComplexNormSpace;
cluster non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive homogeneous Lipschitzian for Element of bool [: the carrier of X, the carrier of Y:];
existence
ex b1 being LinearOperator of X,Y st b1 is Lipschitzian
proof end;
end;

definition
let X, Y be ComplexNormSpace;
func BoundedLinearOperators (X,Y) -> Subset of () means :Def7: :: CLOPBAN1:def 7
for x being set holds
( x in it iff x is Lipschitzian LinearOperator of X,Y );
existence
ex b1 being Subset of () st
for x being set holds
( x in b1 iff x is Lipschitzian LinearOperator of X,Y )
proof end;
uniqueness
for b1, b2 being Subset of () st ( for x being set holds
( x in b1 iff x is Lipschitzian LinearOperator of X,Y ) ) & ( for x being set holds
( x in b2 iff x is Lipschitzian LinearOperator of X,Y ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def7 defines BoundedLinearOperators CLOPBAN1:def 7 :
for X, Y being ComplexNormSpace
for b3 being Subset of () holds
( b3 = BoundedLinearOperators (X,Y) iff for x being set holds
( x in b3 iff x is Lipschitzian LinearOperator of X,Y ) );

registration
let X, Y be ComplexNormSpace;
cluster BoundedLinearOperators (X,Y) -> non empty ;
coherence
not BoundedLinearOperators (X,Y) is empty
proof end;
end;

theorem Th21: :: CLOPBAN1:21
for X, Y being ComplexNormSpace holds BoundedLinearOperators (X,Y) is linearly-closed
proof end;

theorem :: CLOPBAN1:22
for X, Y being ComplexNormSpace holds CLSStruct(# (),(Zero_ ((),())),(Add_ ((),())),(Mult_ ((),())) #) is Subspace of C_VectorSpace_of_LinearOperators (X,Y) by ;

registration
let X, Y be ComplexNormSpace;
coherence
( CLSStruct(# (),(Zero_ ((),())),(Add_ ((),())),(Mult_ ((),())) #) is Abelian & CLSStruct(# (),(Zero_ ((),())),(Add_ ((),())),(Mult_ ((),())) #) is add-associative & CLSStruct(# (),(Zero_ ((),())),(Add_ ((),())),(Mult_ ((),())) #) is right_zeroed & CLSStruct(# (),(Zero_ ((),())),(Add_ ((),())),(Mult_ ((),())) #) is right_complementable & CLSStruct(# (),(Zero_ ((),())),(Add_ ((),())),(Mult_ ((),())) #) is vector-distributive & CLSStruct(# (),(Zero_ ((),())),(Add_ ((),())),(Mult_ ((),())) #) is scalar-distributive & CLSStruct(# (),(Zero_ ((),())),(Add_ ((),())),(Mult_ ((),())) #) is scalar-associative & CLSStruct(# (),(Zero_ ((),())),(Add_ ((),())),(Mult_ ((),())) #) is scalar-unital )
by ;
end;

definition
let X, Y be ComplexNormSpace;
func C_VectorSpace_of_BoundedLinearOperators (X,Y) -> ComplexLinearSpace equals :: CLOPBAN1:def 8
CLSStruct(# (),(Zero_ ((),())),(Add_ ((),())),(Mult_ ((),())) #);
coherence
CLSStruct(# (),(Zero_ ((),())),(Add_ ((),())),(Mult_ ((),())) #) is ComplexLinearSpace
;
end;

:: deftheorem defines C_VectorSpace_of_BoundedLinearOperators CLOPBAN1:def 8 :
for X, Y being ComplexNormSpace holds C_VectorSpace_of_BoundedLinearOperators (X,Y) = CLSStruct(# (),(Zero_ ((),())),(Add_ ((),())),(Mult_ ((),())) #);

registration
let X, Y be ComplexNormSpace;
coherence ;
end;

registration
let X, Y be ComplexNormSpace;
cluster -> Relation-like Function-like for Element of the carrier of ;
coherence
for b1 being Element of holds
( b1 is Function-like & b1 is Relation-like )
;
end;

definition
let X, Y be ComplexNormSpace;
let f be Element of ;
let v be VECTOR of X;
:: original: .
redefine func f . v -> VECTOR of Y;
coherence
f . v is VECTOR of Y
proof end;
end;

theorem Th23: :: CLOPBAN1:23
for X, Y being ComplexNormSpace
for f, g, h being VECTOR of holds
( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )
proof end;

theorem Th24: :: CLOPBAN1:24
for X, Y being ComplexNormSpace
for f, h being VECTOR of
for c being Complex holds
( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) )
proof end;

theorem Th25: :: CLOPBAN1:25
for X, Y being ComplexNormSpace holds 0. = the carrier of X --> (0. Y)
proof end;

definition
let X, Y be ComplexNormSpace;
let f be object ;
assume A1: f in BoundedLinearOperators (X,Y) ;
func modetrans (f,X,Y) -> Lipschitzian LinearOperator of X,Y equals :Def9: :: CLOPBAN1:def 9
f;
coherence
f is Lipschitzian LinearOperator of X,Y
by ;
end;

:: deftheorem Def9 defines modetrans CLOPBAN1:def 9 :
for X, Y being ComplexNormSpace
for f being object st f in BoundedLinearOperators (X,Y) holds
modetrans (f,X,Y) = f;

definition
let X, Y be ComplexNormSpace;
let u be LinearOperator of X,Y;
func PreNorms u -> non empty Subset of REAL equals :: CLOPBAN1:def 10
{ ||.(u . t).|| where t is VECTOR of X : <= 1 } ;
coherence
{ ||.(u . t).|| where t is VECTOR of X : <= 1 } is non empty Subset of REAL
proof end;
end;

:: deftheorem defines PreNorms CLOPBAN1:def 10 :
for X, Y being ComplexNormSpace
for u being LinearOperator of X,Y holds PreNorms u = { ||.(u . t).|| where t is VECTOR of X : <= 1 } ;

theorem Th26: :: CLOPBAN1:26
for X, Y being ComplexNormSpace
for g being Lipschitzian LinearOperator of X,Y holds PreNorms g is bounded_above
proof end;

theorem :: CLOPBAN1:27
for X, Y being ComplexNormSpace
for g being LinearOperator of X,Y holds
( g is Lipschitzian iff PreNorms g is bounded_above )
proof end;

definition
let X, Y be ComplexNormSpace;
func BoundedLinearOperatorsNorm (X,Y) -> Function of (),REAL means :Def11: :: CLOPBAN1:def 11
for x being object st x in BoundedLinearOperators (X,Y) holds
it . x = upper_bound (PreNorms (modetrans (x,X,Y)));
existence
ex b1 being Function of (),REAL st
for x being object st x in BoundedLinearOperators (X,Y) holds
b1 . x = upper_bound (PreNorms (modetrans (x,X,Y)))
proof end;
uniqueness
for b1, b2 being Function of (),REAL st ( for x being object st x in BoundedLinearOperators (X,Y) holds
b1 . x = upper_bound (PreNorms (modetrans (x,X,Y))) ) & ( for x being object st x in BoundedLinearOperators (X,Y) holds
b2 . x = upper_bound (PreNorms (modetrans (x,X,Y))) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def11 defines BoundedLinearOperatorsNorm CLOPBAN1:def 11 :
for X, Y being ComplexNormSpace
for b3 being Function of (),REAL holds
( b3 = BoundedLinearOperatorsNorm (X,Y) iff for x being object st x in BoundedLinearOperators (X,Y) holds
b3 . x = upper_bound (PreNorms (modetrans (x,X,Y))) );

theorem Th28: :: CLOPBAN1:28
for X, Y being ComplexNormSpace
for f being Lipschitzian LinearOperator of X,Y holds modetrans (f,X,Y) = f
proof end;

theorem Th29: :: CLOPBAN1:29
for X, Y being ComplexNormSpace
for f being Lipschitzian LinearOperator of X,Y holds () . f = upper_bound ()
proof end;

definition
let X, Y be ComplexNormSpace;
func C_NormSpace_of_BoundedLinearOperators (X,Y) -> non empty CNORMSTR equals :: CLOPBAN1:def 12
CNORMSTR(# (),(Zero_ ((),())),(Add_ ((),())),(Mult_ ((),())),() #);
coherence
CNORMSTR(# (),(Zero_ ((),())),(Add_ ((),())),(Mult_ ((),())),() #) is non empty CNORMSTR
;
end;

:: deftheorem defines C_NormSpace_of_BoundedLinearOperators CLOPBAN1:def 12 :
for X, Y being ComplexNormSpace holds C_NormSpace_of_BoundedLinearOperators (X,Y) = CNORMSTR(# (),(Zero_ ((),())),(Add_ ((),())),(Mult_ ((),())),() #);

theorem Th30: :: CLOPBAN1:30
for X, Y being ComplexNormSpace holds the carrier of X --> (0. Y) = 0.
proof end;

theorem Th31: :: CLOPBAN1:31
for X, Y being ComplexNormSpace
for f being Point of
for g being Lipschitzian LinearOperator of X,Y st g = f holds
for t being VECTOR of X holds ||.(g . t).|| <= *
proof end;

theorem Th32: :: CLOPBAN1:32
for X, Y being ComplexNormSpace
for f being Point of holds 0 <=
proof end;

theorem Th33: :: CLOPBAN1:33
for X, Y being ComplexNormSpace
for f being Point of st f = 0. holds
0 =
proof end;

registration
let X, Y be ComplexNormSpace;
cluster -> Relation-like Function-like for Element of the carrier of ;
coherence
for b1 being Element of holds
( b1 is Function-like & b1 is Relation-like )
;
end;

definition
let X, Y be ComplexNormSpace;
let f be Element of ;
let v be VECTOR of X;
:: original: .
redefine func f . v -> VECTOR of Y;
coherence
f . v is VECTOR of Y
proof end;
end;

theorem Th34: :: CLOPBAN1:34
for X, Y being ComplexNormSpace
for f, g, h being Point of holds
( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )
proof end;

theorem Th35: :: CLOPBAN1:35
for X, Y being ComplexNormSpace
for f, h being Point of
for c being Complex holds
( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) )
proof end;

theorem Th36: :: CLOPBAN1:36
for X, Y being ComplexNormSpace
for f, g being Point of
for c being Complex holds
( ( = 0 implies f = 0. ) & ( f = 0. implies = 0 ) & ||.(c * f).|| = |.c.| * & ||.(f + g).|| <= + )
proof end;

theorem Th37: :: CLOPBAN1:37
for X, Y being ComplexNormSpace holds
( C_NormSpace_of_BoundedLinearOperators (X,Y) is reflexive & C_NormSpace_of_BoundedLinearOperators (X,Y) is discerning & C_NormSpace_of_BoundedLinearOperators (X,Y) is ComplexNormSpace-like )
proof end;

theorem Th38: :: CLOPBAN1:38
for X, Y being ComplexNormSpace holds C_NormSpace_of_BoundedLinearOperators (X,Y) is ComplexNormSpace
proof end;

registration
let X, Y be ComplexNormSpace;
coherence by Th38;
end;

theorem Th39: :: CLOPBAN1:39
for X, Y being ComplexNormSpace
for f, g, h being Point of holds
( h = f - g iff for x being VECTOR of X holds h . x = (f . x) - (g . x) )
proof end;

definition
let X be ComplexNormSpace;
attr X is complete means :Def13: :: CLOPBAN1:def 13
for seq being sequence of X st seq is Cauchy_sequence_by_Norm holds
seq is convergent ;
end;

:: deftheorem Def13 defines complete CLOPBAN1:def 13 :
for X being ComplexNormSpace holds
( X is complete iff for seq being sequence of X st seq is Cauchy_sequence_by_Norm holds
seq is convergent );

registration
existence
ex b1 being ComplexNormSpace st b1 is complete
by ;
end;

definition end;

Lm2: 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 Th40: :: CLOPBAN1:40
for X being ComplexNormSpace
for seq being sequence of X st seq is convergent holds
( ||.seq.|| is convergent & lim ||.seq.|| = ||.(lim seq).|| )
proof end;

theorem Th41: :: CLOPBAN1:41
for X, Y being ComplexNormSpace st Y is complete holds
for seq being sequence of st seq is Cauchy_sequence_by_Norm holds
seq is convergent
proof end;

theorem Th42: :: CLOPBAN1:42
for X being ComplexNormSpace
for Y being ComplexBanachSpace holds C_NormSpace_of_BoundedLinearOperators (X,Y) is ComplexBanachSpace
proof end;

registration
let X be ComplexNormSpace;
let Y be ComplexBanachSpace;
coherence by Th42;
end;