:: by Yasunari Shidama

::

:: Received January 26, 2004

:: Copyright (c) 2004-2018 Association of Mizar Users

theorem Th1: :: LOPBAN_2:1

for X, Y, Z being RealLinearSpace

for f being LinearOperator of X,Y

for g being LinearOperator of Y,Z holds g * f is LinearOperator of X,Z

for f being LinearOperator of X,Y

for g being LinearOperator of Y,Z holds g * f is LinearOperator of X,Z

proof end;

theorem Th2: :: LOPBAN_2:2

for X, Y, Z being RealNormSpace

for f being Lipschitzian LinearOperator of X,Y

for g being Lipschitzian LinearOperator of Y,Z holds

( g * f is Lipschitzian LinearOperator of X,Z & ( for x being VECTOR of X holds

( ||.((g * f) . x).|| <= (((BoundedLinearOperatorsNorm (Y,Z)) . g) * ((BoundedLinearOperatorsNorm (X,Y)) . f)) * ||.x.|| & (BoundedLinearOperatorsNorm (X,Z)) . (g * f) <= ((BoundedLinearOperatorsNorm (Y,Z)) . g) * ((BoundedLinearOperatorsNorm (X,Y)) . f) ) ) )

for f being Lipschitzian LinearOperator of X,Y

for g being Lipschitzian LinearOperator of Y,Z holds

( g * f is Lipschitzian LinearOperator of X,Z & ( for x being VECTOR of X holds

( ||.((g * f) . x).|| <= (((BoundedLinearOperatorsNorm (Y,Z)) . g) * ((BoundedLinearOperatorsNorm (X,Y)) . f)) * ||.x.|| & (BoundedLinearOperatorsNorm (X,Z)) . (g * f) <= ((BoundedLinearOperatorsNorm (Y,Z)) . g) * ((BoundedLinearOperatorsNorm (X,Y)) . f) ) ) )

proof end;

definition

let X be RealNormSpace;

let f, g be Lipschitzian LinearOperator of X,X;

:: original: *

redefine func g * f -> Lipschitzian LinearOperator of X,X;

correctness

coherence

g * f is Lipschitzian LinearOperator of X,X;

by Th2;

end;
let f, g be Lipschitzian LinearOperator of X,X;

:: original: *

redefine func g * f -> Lipschitzian LinearOperator of X,X;

correctness

coherence

g * f is Lipschitzian LinearOperator of X,X;

by Th2;

definition

let X be RealNormSpace;

let f, g be Element of BoundedLinearOperators (X,X);

coherence

(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) . (f,g) is Element of BoundedLinearOperators (X,X);

;

end;
let f, g be Element of BoundedLinearOperators (X,X);

func f + g -> Element of BoundedLinearOperators (X,X) equals :: LOPBAN_2:def 1

(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) . (f,g);

correctness (Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) . (f,g);

coherence

(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) . (f,g) is Element of BoundedLinearOperators (X,X);

;

:: deftheorem defines + LOPBAN_2:def 1 :

for X being RealNormSpace

for f, g being Element of BoundedLinearOperators (X,X) holds f + g = (Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) . (f,g);

for X being RealNormSpace

for f, g being Element of BoundedLinearOperators (X,X) holds f + g = (Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) . (f,g);

definition

let X be RealNormSpace;

let f, g be Element of BoundedLinearOperators (X,X);

coherence

(modetrans (g,X,X)) * (modetrans (f,X,X)) is Element of BoundedLinearOperators (X,X);

by LOPBAN_1:def 9;

end;
let f, g be Element of BoundedLinearOperators (X,X);

func g * f -> Element of BoundedLinearOperators (X,X) equals :: LOPBAN_2:def 2

(modetrans (g,X,X)) * (modetrans (f,X,X));

correctness (modetrans (g,X,X)) * (modetrans (f,X,X));

coherence

(modetrans (g,X,X)) * (modetrans (f,X,X)) is Element of BoundedLinearOperators (X,X);

by LOPBAN_1:def 9;

:: deftheorem defines * LOPBAN_2:def 2 :

for X being RealNormSpace

for f, g being Element of BoundedLinearOperators (X,X) holds g * f = (modetrans (g,X,X)) * (modetrans (f,X,X));

for X being RealNormSpace

for f, g being Element of BoundedLinearOperators (X,X) holds g * f = (modetrans (g,X,X)) * (modetrans (f,X,X));

definition

let X be RealNormSpace;

let f be Element of BoundedLinearOperators (X,X);

let a be Real;

coherence

(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) . (a,f) is Element of BoundedLinearOperators (X,X);

end;
let f be Element of BoundedLinearOperators (X,X);

let a be Real;

func a * f -> Element of BoundedLinearOperators (X,X) equals :: LOPBAN_2:def 3

(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) . (a,f);

correctness (Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) . (a,f);

coherence

(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) . (a,f) is Element of BoundedLinearOperators (X,X);

proof end;

:: deftheorem defines * LOPBAN_2:def 3 :

for X being RealNormSpace

for f being Element of BoundedLinearOperators (X,X)

for a being Real holds a * f = (Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) . (a,f);

for X being RealNormSpace

for f being Element of BoundedLinearOperators (X,X)

for a being Real holds a * f = (Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) . (a,f);

definition

let X be RealNormSpace;

ex b_{1} being BinOp of (BoundedLinearOperators (X,X)) st

for f, g being Element of BoundedLinearOperators (X,X) holds b_{1} . (f,g) = f * g

for b_{1}, b_{2} being BinOp of (BoundedLinearOperators (X,X)) st ( for f, g being Element of BoundedLinearOperators (X,X) holds b_{1} . (f,g) = f * g ) & ( for f, g being Element of BoundedLinearOperators (X,X) holds b_{2} . (f,g) = f * g ) holds

b_{1} = b_{2}

end;
func FuncMult X -> BinOp of (BoundedLinearOperators (X,X)) means :Def4: :: LOPBAN_2:def 4

for f, g being Element of BoundedLinearOperators (X,X) holds it . (f,g) = f * g;

existence for f, g being Element of BoundedLinearOperators (X,X) holds it . (f,g) = f * g;

ex b

for f, g being Element of BoundedLinearOperators (X,X) holds b

proof end;

uniqueness for b

b

proof end;

:: deftheorem Def4 defines FuncMult LOPBAN_2:def 4 :

for X being RealNormSpace

for b_{2} being BinOp of (BoundedLinearOperators (X,X)) holds

( b_{2} = FuncMult X iff for f, g being Element of BoundedLinearOperators (X,X) holds b_{2} . (f,g) = f * g );

for X being RealNormSpace

for b

( b

definition

let X be RealNormSpace;

id the carrier of X is Element of BoundedLinearOperators (X,X)

end;
func FuncUnit X -> Element of BoundedLinearOperators (X,X) equals :: LOPBAN_2:def 5

id the carrier of X;

coherence id the carrier of X;

id the carrier of X is Element of BoundedLinearOperators (X,X)

proof end;

:: deftheorem defines FuncUnit LOPBAN_2:def 5 :

for X being RealNormSpace holds FuncUnit X = id the carrier of X;

for X being RealNormSpace holds FuncUnit X = id the carrier of X;

theorem Th4: :: LOPBAN_2:4

for X being RealNormSpace

for f, g, h being Lipschitzian LinearOperator of X,X holds

( h = f * g iff for x being VECTOR of X holds h . x = f . (g . x) )

for f, g, h being Lipschitzian LinearOperator of X,X holds

( h = f * g iff for x being VECTOR of X holds h . x = f . (g . x) )

proof end;

theorem Th5: :: LOPBAN_2:5

for X being RealNormSpace

for f, g, h being Lipschitzian LinearOperator of X,X holds f * (g * h) = (f * g) * h

for f, g, h being Lipschitzian LinearOperator of X,X holds f * (g * h) = (f * g) * h

proof end;

theorem Th6: :: LOPBAN_2:6

for X being RealNormSpace

for f being Lipschitzian LinearOperator of X,X holds

( f * (id the carrier of X) = f & (id the carrier of X) * f = f )

for f being Lipschitzian LinearOperator of X,X holds

( f * (id the carrier of X) = f & (id the carrier of X) * f = f )

proof end;

theorem Th7: :: LOPBAN_2:7

for X being RealNormSpace

for f, g, h being Element of BoundedLinearOperators (X,X) holds f * (g * h) = (f * g) * h

for f, g, h being Element of BoundedLinearOperators (X,X) holds f * (g * h) = (f * g) * h

proof end;

theorem Th8: :: LOPBAN_2:8

for X being RealNormSpace

for f being Element of BoundedLinearOperators (X,X) holds

( f * (FuncUnit X) = f & (FuncUnit X) * f = f )

for f being Element of BoundedLinearOperators (X,X) holds

( f * (FuncUnit X) = f & (FuncUnit X) * f = f )

proof end;

theorem Th9: :: LOPBAN_2:9

for X being RealNormSpace

for f, g, h being Element of BoundedLinearOperators (X,X) holds f * (g + h) = (f * g) + (f * h)

for f, g, h being Element of BoundedLinearOperators (X,X) holds f * (g + h) = (f * g) + (f * h)

proof end;

theorem Th10: :: LOPBAN_2:10

for X being RealNormSpace

for f, g, h being Element of BoundedLinearOperators (X,X) holds (g + h) * f = (g * f) + (h * f)

for f, g, h being Element of BoundedLinearOperators (X,X) holds (g + h) * f = (g * f) + (h * f)

proof end;

theorem Th11: :: LOPBAN_2:11

for X being RealNormSpace

for f, g being Element of BoundedLinearOperators (X,X)

for a, b being Real holds (a * b) * (f * g) = (a * f) * (b * g)

for f, g being Element of BoundedLinearOperators (X,X)

for a, b being Real holds (a * b) * (f * g) = (a * f) * (b * g)

proof end;

theorem Th12: :: LOPBAN_2:12

for X being RealNormSpace

for f, g being Element of BoundedLinearOperators (X,X)

for a being Real holds a * (f * g) = (a * f) * g

for f, g being Element of BoundedLinearOperators (X,X)

for a being Real holds a * (f * g) = (a * f) * g

proof end;

definition

let X be RealNormSpace;

coherence

doubleLoopStr(# (BoundedLinearOperators (X,X)),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncMult X),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) is doubleLoopStr ;

;

end;
func Ring_of_BoundedLinearOperators X -> doubleLoopStr equals :: LOPBAN_2:def 6

doubleLoopStr(# (BoundedLinearOperators (X,X)),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncMult X),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #);

correctness doubleLoopStr(# (BoundedLinearOperators (X,X)),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncMult X),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #);

coherence

doubleLoopStr(# (BoundedLinearOperators (X,X)),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncMult X),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) is doubleLoopStr ;

;

:: deftheorem defines Ring_of_BoundedLinearOperators LOPBAN_2:def 6 :

for X being RealNormSpace holds Ring_of_BoundedLinearOperators X = doubleLoopStr(# (BoundedLinearOperators (X,X)),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncMult X),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #);

for X being RealNormSpace holds Ring_of_BoundedLinearOperators X = doubleLoopStr(# (BoundedLinearOperators (X,X)),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncMult X),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #);

registration

let X be RealNormSpace;

coherence

( not Ring_of_BoundedLinearOperators X is empty & Ring_of_BoundedLinearOperators X is strict ) ;

end;
coherence

( not Ring_of_BoundedLinearOperators X is empty & Ring_of_BoundedLinearOperators X is strict ) ;

Lm1: now :: thesis: for X being RealNormSpace

for x, e being Element of (Ring_of_BoundedLinearOperators X) st e = FuncUnit X holds

( x * e = x & e * x = x )

for x, e being Element of (Ring_of_BoundedLinearOperators X) st e = FuncUnit X holds

( x * e = x & e * x = x )

let X be RealNormSpace; :: thesis: for x, e being Element of (Ring_of_BoundedLinearOperators X) st e = FuncUnit X holds

( x * e = x & e * x = x )

set F = Ring_of_BoundedLinearOperators X;

let x, e be Element of (Ring_of_BoundedLinearOperators X); :: thesis: ( e = FuncUnit X implies ( x * e = x & e * x = x ) )

reconsider f = x as Element of BoundedLinearOperators (X,X) ;

assume A1: e = FuncUnit X ; :: thesis: ( x * e = x & e * x = x )

hence x * e = f * (FuncUnit X) by Def4

.= x by Th8 ;

:: thesis: e * x = x

thus e * x = (FuncUnit X) * f by A1, Def4

.= x by Th8 ; :: thesis: verum

end;
( x * e = x & e * x = x )

set F = Ring_of_BoundedLinearOperators X;

let x, e be Element of (Ring_of_BoundedLinearOperators X); :: thesis: ( e = FuncUnit X implies ( x * e = x & e * x = x ) )

reconsider f = x as Element of BoundedLinearOperators (X,X) ;

assume A1: e = FuncUnit X ; :: thesis: ( x * e = x & e * x = x )

hence x * e = f * (FuncUnit X) by Def4

.= x by Th8 ;

:: thesis: e * x = x

thus e * x = (FuncUnit X) * f by A1, Def4

.= x by Th8 ; :: thesis: verum

registration
end;

theorem Th13: :: LOPBAN_2:13

for X being RealNormSpace

for x, y, z being Element of (Ring_of_BoundedLinearOperators X) holds

( x + y = y + x & (x + y) + z = x + (y + z) & x + (0. (Ring_of_BoundedLinearOperators X)) = x & ex t being Element of (Ring_of_BoundedLinearOperators X) st x + t = 0. (Ring_of_BoundedLinearOperators X) & (x * y) * z = x * (y * z) & x * (1. (Ring_of_BoundedLinearOperators X)) = x & (1. (Ring_of_BoundedLinearOperators X)) * x = x & x * (y + z) = (x * y) + (x * z) & (y + z) * x = (y * x) + (z * x) )

for x, y, z being Element of (Ring_of_BoundedLinearOperators X) holds

( x + y = y + x & (x + y) + z = x + (y + z) & x + (0. (Ring_of_BoundedLinearOperators X)) = x & ex t being Element of (Ring_of_BoundedLinearOperators X) st x + t = 0. (Ring_of_BoundedLinearOperators X) & (x * y) * z = x * (y * z) & x * (1. (Ring_of_BoundedLinearOperators X)) = x & (1. (Ring_of_BoundedLinearOperators X)) * x = x & x * (y + z) = (x * y) + (x * z) & (y + z) * x = (y * x) + (z * x) )

proof end;

registration

let X be RealNormSpace;

( Ring_of_BoundedLinearOperators X is Abelian & Ring_of_BoundedLinearOperators X is add-associative & Ring_of_BoundedLinearOperators X is right_zeroed & Ring_of_BoundedLinearOperators X is right_complementable & Ring_of_BoundedLinearOperators X is associative & Ring_of_BoundedLinearOperators X is left_unital & Ring_of_BoundedLinearOperators X is right_unital & Ring_of_BoundedLinearOperators X is distributive & Ring_of_BoundedLinearOperators X is well-unital ) by Th14;

end;
cluster Ring_of_BoundedLinearOperators X -> right_complementable Abelian add-associative right_zeroed right_unital well-unital distributive left_unital associative ;

coherence ( Ring_of_BoundedLinearOperators X is Abelian & Ring_of_BoundedLinearOperators X is add-associative & Ring_of_BoundedLinearOperators X is right_zeroed & Ring_of_BoundedLinearOperators X is right_complementable & Ring_of_BoundedLinearOperators X is associative & Ring_of_BoundedLinearOperators X is left_unital & Ring_of_BoundedLinearOperators X is right_unital & Ring_of_BoundedLinearOperators X is distributive & Ring_of_BoundedLinearOperators X is well-unital ) by Th14;

definition

let X be RealNormSpace;

coherence

AlgebraStr(# (BoundedLinearOperators (X,X)),(FuncMult X),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) is AlgebraStr ;

;

end;
func R_Algebra_of_BoundedLinearOperators X -> AlgebraStr equals :: LOPBAN_2:def 7

AlgebraStr(# (BoundedLinearOperators (X,X)),(FuncMult X),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #);

correctness AlgebraStr(# (BoundedLinearOperators (X,X)),(FuncMult X),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #);

coherence

AlgebraStr(# (BoundedLinearOperators (X,X)),(FuncMult X),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) is AlgebraStr ;

;

:: deftheorem defines R_Algebra_of_BoundedLinearOperators LOPBAN_2:def 7 :

for X being RealNormSpace holds R_Algebra_of_BoundedLinearOperators X = AlgebraStr(# (BoundedLinearOperators (X,X)),(FuncMult X),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #);

for X being RealNormSpace holds R_Algebra_of_BoundedLinearOperators X = AlgebraStr(# (BoundedLinearOperators (X,X)),(FuncMult X),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #);

registration

let X be RealNormSpace;

coherence

( not R_Algebra_of_BoundedLinearOperators X is empty & R_Algebra_of_BoundedLinearOperators X is strict ) ;

end;
coherence

( not R_Algebra_of_BoundedLinearOperators X is empty & R_Algebra_of_BoundedLinearOperators X is strict ) ;

Lm2: now :: thesis: for X being RealNormSpace

for x, e being Element of (R_Algebra_of_BoundedLinearOperators X) st e = FuncUnit X holds

( x * e = x & e * x = x )

for x, e being Element of (R_Algebra_of_BoundedLinearOperators X) st e = FuncUnit X holds

( x * e = x & e * x = x )

let X be RealNormSpace; :: thesis: for x, e being Element of (R_Algebra_of_BoundedLinearOperators X) st e = FuncUnit X holds

( x * e = x & e * x = x )

set F = R_Algebra_of_BoundedLinearOperators X;

let x, e be Element of (R_Algebra_of_BoundedLinearOperators X); :: thesis: ( e = FuncUnit X implies ( x * e = x & e * x = x ) )

reconsider f = x as Element of BoundedLinearOperators (X,X) ;

assume A1: e = FuncUnit X ; :: thesis: ( x * e = x & e * x = x )

hence x * e = f * (FuncUnit X) by Def4

.= x by Th8 ;

:: thesis: e * x = x

thus e * x = (FuncUnit X) * f by A1, Def4

.= x by Th8 ; :: thesis: verum

end;
( x * e = x & e * x = x )

set F = R_Algebra_of_BoundedLinearOperators X;

let x, e be Element of (R_Algebra_of_BoundedLinearOperators X); :: thesis: ( e = FuncUnit X implies ( x * e = x & e * x = x ) )

reconsider f = x as Element of BoundedLinearOperators (X,X) ;

assume A1: e = FuncUnit X ; :: thesis: ( x * e = x & e * x = x )

hence x * e = f * (FuncUnit X) by Def4

.= x by Th8 ;

:: thesis: e * x = x

thus e * x = (FuncUnit X) * f by A1, Def4

.= x by Th8 ; :: thesis: verum

registration
end;

theorem :: LOPBAN_2:15

for X being RealNormSpace

for x, y, z being Element of (R_Algebra_of_BoundedLinearOperators X)

for a, b being Real holds

( x + y = y + x & (x + y) + z = x + (y + z) & x + (0. (R_Algebra_of_BoundedLinearOperators X)) = x & ex t being Element of (R_Algebra_of_BoundedLinearOperators X) st x + t = 0. (R_Algebra_of_BoundedLinearOperators X) & (x * y) * z = x * (y * z) & x * (1. (R_Algebra_of_BoundedLinearOperators X)) = x & (1. (R_Algebra_of_BoundedLinearOperators X)) * x = x & x * (y + z) = (x * y) + (x * z) & (y + z) * x = (y * x) + (z * x) & a * (x * y) = (a * x) * y & a * (x + y) = (a * x) + (a * y) & (a + b) * x = (a * x) + (b * x) & (a * b) * x = a * (b * x) & (a * b) * (x * y) = (a * x) * (b * y) )

for x, y, z being Element of (R_Algebra_of_BoundedLinearOperators X)

for a, b being Real holds

( x + y = y + x & (x + y) + z = x + (y + z) & x + (0. (R_Algebra_of_BoundedLinearOperators X)) = x & ex t being Element of (R_Algebra_of_BoundedLinearOperators X) st x + t = 0. (R_Algebra_of_BoundedLinearOperators X) & (x * y) * z = x * (y * z) & x * (1. (R_Algebra_of_BoundedLinearOperators X)) = x & (1. (R_Algebra_of_BoundedLinearOperators X)) * x = x & x * (y + z) = (x * y) + (x * z) & (y + z) * x = (y * x) + (z * x) & a * (x * y) = (a * x) * y & a * (x + y) = (a * x) + (a * y) & (a + b) * x = (a * x) + (b * x) & (a * b) * x = a * (b * x) & (a * b) * (x * y) = (a * x) * (b * y) )

proof end;

definition

mode BLAlgebra is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative vector-associative right-distributive right_unital associative AlgebraStr ;

end;
registration

let X be RealNormSpace;

( R_Algebra_of_BoundedLinearOperators X is Abelian & R_Algebra_of_BoundedLinearOperators X is add-associative & R_Algebra_of_BoundedLinearOperators X is right_zeroed & R_Algebra_of_BoundedLinearOperators X is right_complementable & R_Algebra_of_BoundedLinearOperators X is associative & R_Algebra_of_BoundedLinearOperators X is right_unital & R_Algebra_of_BoundedLinearOperators X is right-distributive & R_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Algebra_of_BoundedLinearOperators X is vector-associative & R_Algebra_of_BoundedLinearOperators X is strict )

end;
cluster R_Algebra_of_BoundedLinearOperators X -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative strict vector-associative right-distributive right_unital associative ;

coherence ( R_Algebra_of_BoundedLinearOperators X is Abelian & R_Algebra_of_BoundedLinearOperators X is add-associative & R_Algebra_of_BoundedLinearOperators X is right_zeroed & R_Algebra_of_BoundedLinearOperators X is right_complementable & R_Algebra_of_BoundedLinearOperators X is associative & R_Algebra_of_BoundedLinearOperators X is right_unital & R_Algebra_of_BoundedLinearOperators X is right-distributive & R_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Algebra_of_BoundedLinearOperators X is vector-associative & R_Algebra_of_BoundedLinearOperators X is strict )

proof end;

theorem :: LOPBAN_2:16

registration

not for b_{1} being RealBanachSpace holds b_{1} is trivial
end;

cluster non empty non trivial left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V138() discerning reflexive RealNormSpace-like complete for RealBanachSpace;

existence not for b

proof end;

theorem Th18: :: LOPBAN_2:18

for X being non trivial RealNormSpace holds (BoundedLinearOperatorsNorm (X,X)) . (id the carrier of X) = 1

proof end;

definition

attr c_{1} is strict ;

struct Normed_AlgebraStr -> AlgebraStr , NORMSTR ;

aggr Normed_AlgebraStr(# carrier, multF, addF, Mult, OneF, ZeroF, normF #) -> Normed_AlgebraStr ;

end;
struct Normed_AlgebraStr -> AlgebraStr , NORMSTR ;

aggr Normed_AlgebraStr(# carrier, multF, addF, Mult, OneF, ZeroF, normF #) -> Normed_AlgebraStr ;

definition

let X be RealNormSpace;

coherence

Normed_AlgebraStr(# (BoundedLinearOperators (X,X)),(FuncMult X),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(BoundedLinearOperatorsNorm (X,X)) #) is Normed_AlgebraStr ;

;

end;
func R_Normed_Algebra_of_BoundedLinearOperators X -> Normed_AlgebraStr equals :: LOPBAN_2:def 8

Normed_AlgebraStr(# (BoundedLinearOperators (X,X)),(FuncMult X),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(BoundedLinearOperatorsNorm (X,X)) #);

correctness Normed_AlgebraStr(# (BoundedLinearOperators (X,X)),(FuncMult X),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(BoundedLinearOperatorsNorm (X,X)) #);

coherence

Normed_AlgebraStr(# (BoundedLinearOperators (X,X)),(FuncMult X),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(BoundedLinearOperatorsNorm (X,X)) #) is Normed_AlgebraStr ;

;

:: deftheorem defines R_Normed_Algebra_of_BoundedLinearOperators LOPBAN_2:def 8 :

for X being RealNormSpace holds R_Normed_Algebra_of_BoundedLinearOperators X = Normed_AlgebraStr(# (BoundedLinearOperators (X,X)),(FuncMult X),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(BoundedLinearOperatorsNorm (X,X)) #);

for X being RealNormSpace holds R_Normed_Algebra_of_BoundedLinearOperators X = Normed_AlgebraStr(# (BoundedLinearOperators (X,X)),(FuncMult X),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(FuncUnit X),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(BoundedLinearOperatorsNorm (X,X)) #);

registration

let X be RealNormSpace;

coherence

( not R_Normed_Algebra_of_BoundedLinearOperators X is empty & R_Normed_Algebra_of_BoundedLinearOperators X is strict ) ;

end;
coherence

( not R_Normed_Algebra_of_BoundedLinearOperators X is empty & R_Normed_Algebra_of_BoundedLinearOperators X is strict ) ;

Lm3: now :: thesis: for X being RealNormSpace

for x, e being Element of (R_Normed_Algebra_of_BoundedLinearOperators X) st e = FuncUnit X holds

( x * e = x & e * x = x )

for x, e being Element of (R_Normed_Algebra_of_BoundedLinearOperators X) st e = FuncUnit X holds

( x * e = x & e * x = x )

let X be RealNormSpace; :: thesis: for x, e being Element of (R_Normed_Algebra_of_BoundedLinearOperators X) st e = FuncUnit X holds

( x * e = x & e * x = x )

set F = R_Normed_Algebra_of_BoundedLinearOperators X;

let x, e be Element of (R_Normed_Algebra_of_BoundedLinearOperators X); :: thesis: ( e = FuncUnit X implies ( x * e = x & e * x = x ) )

reconsider f = x as Element of BoundedLinearOperators (X,X) ;

assume A1: e = FuncUnit X ; :: thesis: ( x * e = x & e * x = x )

hence x * e = f * (FuncUnit X) by Def4

.= x by Th8 ;

:: thesis: e * x = x

thus e * x = (FuncUnit X) * f by A1, Def4

.= x by Th8 ; :: thesis: verum

end;
( x * e = x & e * x = x )

set F = R_Normed_Algebra_of_BoundedLinearOperators X;

let x, e be Element of (R_Normed_Algebra_of_BoundedLinearOperators X); :: thesis: ( e = FuncUnit X implies ( x * e = x & e * x = x ) )

reconsider f = x as Element of BoundedLinearOperators (X,X) ;

assume A1: e = FuncUnit X ; :: thesis: ( x * e = x & e * x = x )

hence x * e = f * (FuncUnit X) by Def4

.= x by Th8 ;

:: thesis: e * x = x

thus e * x = (FuncUnit X) * f by A1, Def4

.= x by Th8 ; :: thesis: verum

registration
end;

theorem Th19: :: LOPBAN_2:19

for X being RealNormSpace

for x, y, z being Element of (R_Normed_Algebra_of_BoundedLinearOperators X)

for a, b being Real holds

( x + y = y + x & (x + y) + z = x + (y + z) & x + (0. (R_Normed_Algebra_of_BoundedLinearOperators X)) = x & ex t being Element of (R_Normed_Algebra_of_BoundedLinearOperators X) st x + t = 0. (R_Normed_Algebra_of_BoundedLinearOperators X) & (x * y) * z = x * (y * z) & x * (1. (R_Normed_Algebra_of_BoundedLinearOperators X)) = x & (1. (R_Normed_Algebra_of_BoundedLinearOperators X)) * x = x & x * (y + z) = (x * y) + (x * z) & (y + z) * x = (y * x) + (z * x) & a * (x * y) = (a * x) * y & (a * b) * (x * y) = (a * x) * (b * y) & a * (x + y) = (a * x) + (a * y) & (a + b) * x = (a * x) + (b * x) & (a * b) * x = a * (b * x) & 1 * x = x )

for x, y, z being Element of (R_Normed_Algebra_of_BoundedLinearOperators X)

for a, b being Real holds

( x + y = y + x & (x + y) + z = x + (y + z) & x + (0. (R_Normed_Algebra_of_BoundedLinearOperators X)) = x & ex t being Element of (R_Normed_Algebra_of_BoundedLinearOperators X) st x + t = 0. (R_Normed_Algebra_of_BoundedLinearOperators X) & (x * y) * z = x * (y * z) & x * (1. (R_Normed_Algebra_of_BoundedLinearOperators X)) = x & (1. (R_Normed_Algebra_of_BoundedLinearOperators X)) * x = x & x * (y + z) = (x * y) + (x * z) & (y + z) * x = (y * x) + (z * x) & a * (x * y) = (a * x) * y & (a * b) * (x * y) = (a * x) * (b * y) & a * (x + y) = (a * x) + (a * y) & (a + b) * x = (a * x) + (b * x) & (a * b) * x = a * (b * x) & 1 * x = x )

proof end;

theorem Th20: :: LOPBAN_2:20

for X being RealNormSpace holds

( R_Normed_Algebra_of_BoundedLinearOperators X is reflexive & R_Normed_Algebra_of_BoundedLinearOperators X is discerning & R_Normed_Algebra_of_BoundedLinearOperators X is RealNormSpace-like & R_Normed_Algebra_of_BoundedLinearOperators X is Abelian & R_Normed_Algebra_of_BoundedLinearOperators X is add-associative & R_Normed_Algebra_of_BoundedLinearOperators X is right_zeroed & R_Normed_Algebra_of_BoundedLinearOperators X is right_complementable & R_Normed_Algebra_of_BoundedLinearOperators X is associative & R_Normed_Algebra_of_BoundedLinearOperators X is right_unital & R_Normed_Algebra_of_BoundedLinearOperators X is right-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Normed_Algebra_of_BoundedLinearOperators X is vector-associative & R_Normed_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-unital )

( R_Normed_Algebra_of_BoundedLinearOperators X is reflexive & R_Normed_Algebra_of_BoundedLinearOperators X is discerning & R_Normed_Algebra_of_BoundedLinearOperators X is RealNormSpace-like & R_Normed_Algebra_of_BoundedLinearOperators X is Abelian & R_Normed_Algebra_of_BoundedLinearOperators X is add-associative & R_Normed_Algebra_of_BoundedLinearOperators X is right_zeroed & R_Normed_Algebra_of_BoundedLinearOperators X is right_complementable & R_Normed_Algebra_of_BoundedLinearOperators X is associative & R_Normed_Algebra_of_BoundedLinearOperators X is right_unital & R_Normed_Algebra_of_BoundedLinearOperators X is right-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Normed_Algebra_of_BoundedLinearOperators X is vector-associative & R_Normed_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-unital )

proof end;

registration

ex b_{1} being non empty Normed_AlgebraStr st

( b_{1} is reflexive & b_{1} is discerning & b_{1} is RealNormSpace-like & b_{1} is Abelian & b_{1} is add-associative & b_{1} is right_zeroed & b_{1} is right_complementable & b_{1} is associative & b_{1} is right_unital & b_{1} is right-distributive & b_{1} is vector-distributive & b_{1} is scalar-distributive & b_{1} is scalar-associative & b_{1} is vector-associative & b_{1} is vector-distributive & b_{1} is scalar-distributive & b_{1} is scalar-associative & b_{1} is scalar-unital & b_{1} is strict )
end;

cluster non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative discerning reflexive RealNormSpace-like right-distributive right_unital associative strict for Normed_AlgebraStr ;

existence ex b

( b

proof end;

definition

mode Normed_Algebra is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative discerning reflexive RealNormSpace-like right-distributive right_unital associative Normed_AlgebraStr ;

end;
registration

let X be RealNormSpace;

coherence

( R_Normed_Algebra_of_BoundedLinearOperators X is reflexive & R_Normed_Algebra_of_BoundedLinearOperators X is discerning & R_Normed_Algebra_of_BoundedLinearOperators X is RealNormSpace-like & R_Normed_Algebra_of_BoundedLinearOperators X is Abelian & R_Normed_Algebra_of_BoundedLinearOperators X is add-associative & R_Normed_Algebra_of_BoundedLinearOperators X is right_zeroed & R_Normed_Algebra_of_BoundedLinearOperators X is right_complementable & R_Normed_Algebra_of_BoundedLinearOperators X is associative & R_Normed_Algebra_of_BoundedLinearOperators X is right_unital & R_Normed_Algebra_of_BoundedLinearOperators X is right-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Normed_Algebra_of_BoundedLinearOperators X is vector-associative & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-unital );

by Th20;

end;
cluster R_Normed_Algebra_of_BoundedLinearOperators X -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative discerning reflexive RealNormSpace-like right-distributive right_unital associative ;

correctness coherence

( R_Normed_Algebra_of_BoundedLinearOperators X is reflexive & R_Normed_Algebra_of_BoundedLinearOperators X is discerning & R_Normed_Algebra_of_BoundedLinearOperators X is RealNormSpace-like & R_Normed_Algebra_of_BoundedLinearOperators X is Abelian & R_Normed_Algebra_of_BoundedLinearOperators X is add-associative & R_Normed_Algebra_of_BoundedLinearOperators X is right_zeroed & R_Normed_Algebra_of_BoundedLinearOperators X is right_complementable & R_Normed_Algebra_of_BoundedLinearOperators X is associative & R_Normed_Algebra_of_BoundedLinearOperators X is right_unital & R_Normed_Algebra_of_BoundedLinearOperators X is right-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Normed_Algebra_of_BoundedLinearOperators X is vector-associative & R_Normed_Algebra_of_BoundedLinearOperators X is scalar-unital );

by Th20;

definition

let X be non empty Normed_AlgebraStr ;

end;
attr X is Banach_Algebra-like_1 means :: LOPBAN_2:def 9

for x, y being Element of X holds ||.(x * y).|| <= ||.x.|| * ||.y.||;

for x, y being Element of X holds ||.(x * y).|| <= ||.x.|| * ||.y.||;

attr X is Banach_Algebra-like_3 means :: LOPBAN_2:def 11

for a being Real

for x, y being Element of X holds a * (x * y) = x * (a * y);

for a being Real

for x, y being Element of X holds a * (x * y) = x * (a * y);

:: deftheorem defines Banach_Algebra-like_1 LOPBAN_2:def 9 :

for X being non empty Normed_AlgebraStr holds

( X is Banach_Algebra-like_1 iff for x, y being Element of X holds ||.(x * y).|| <= ||.x.|| * ||.y.|| );

for X being non empty Normed_AlgebraStr holds

( X is Banach_Algebra-like_1 iff for x, y being Element of X holds ||.(x * y).|| <= ||.x.|| * ||.y.|| );

:: deftheorem defines Banach_Algebra-like_2 LOPBAN_2:def 10 :

for X being non empty Normed_AlgebraStr holds

( X is Banach_Algebra-like_2 iff ||.(1. X).|| = 1 );

for X being non empty Normed_AlgebraStr holds

( X is Banach_Algebra-like_2 iff ||.(1. X).|| = 1 );

:: deftheorem defines Banach_Algebra-like_3 LOPBAN_2:def 11 :

for X being non empty Normed_AlgebraStr holds

( X is Banach_Algebra-like_3 iff for a being Real

for x, y being Element of X holds a * (x * y) = x * (a * y) );

for X being non empty Normed_AlgebraStr holds

( X is Banach_Algebra-like_3 iff for a being Real

for x, y being Element of X holds a * (x * y) = x * (a * y) );

definition

let X be Normed_Algebra;

end;
attr X is Banach_Algebra-like means :: LOPBAN_2:def 12

( X is Banach_Algebra-like_1 & X is Banach_Algebra-like_2 & X is Banach_Algebra-like_3 & X is left_unital & X is left-distributive & X is complete );

( X is Banach_Algebra-like_1 & X is Banach_Algebra-like_2 & X is Banach_Algebra-like_3 & X is left_unital & X is left-distributive & X is complete );

:: deftheorem defines Banach_Algebra-like LOPBAN_2:def 12 :

for X being Normed_Algebra holds

( X is Banach_Algebra-like iff ( X is Banach_Algebra-like_1 & X is Banach_Algebra-like_2 & X is Banach_Algebra-like_3 & X is left_unital & X is left-distributive & X is complete ) );

for X being Normed_Algebra holds

( X is Banach_Algebra-like iff ( X is Banach_Algebra-like_1 & X is Banach_Algebra-like_2 & X is Banach_Algebra-like_3 & X is left_unital & X is left-distributive & X is complete ) );

registration

for b_{1} being Normed_Algebra st b_{1} is Banach_Algebra-like holds

( b_{1} is Banach_Algebra-like_1 & b_{1} is Banach_Algebra-like_2 & b_{1} is Banach_Algebra-like_3 & b_{1} is left-distributive & b_{1} is left_unital & b_{1} is complete )
;

for b_{1} being Normed_Algebra st b_{1} is Banach_Algebra-like_1 & b_{1} is Banach_Algebra-like_2 & b_{1} is Banach_Algebra-like_3 & b_{1} is left-distributive & b_{1} is left_unital & b_{1} is complete holds

b_{1} is Banach_Algebra-like
;

end;

cluster non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative discerning reflexive RealNormSpace-like right-distributive right_unital associative Banach_Algebra-like -> left-distributive left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 for Normed_Algebra;

coherence for b

( b

cluster non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative discerning reflexive RealNormSpace-like right-distributive left-distributive right_unital left_unital complete associative Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 -> Banach_Algebra-like for Normed_Algebra;

coherence for b

b

registration

let X be non trivial RealBanachSpace;

coherence

R_Normed_Algebra_of_BoundedLinearOperators X is Banach_Algebra-like

end;
coherence

R_Normed_Algebra_of_BoundedLinearOperators X is Banach_Algebra-like

proof end;

registration

ex b_{1} being Normed_Algebra st b_{1} is Banach_Algebra-like
end;

cluster non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V138() vector-associative discerning reflexive RealNormSpace-like right-distributive right_unital associative Banach_Algebra-like for Normed_Algebra;

existence ex b

proof end;

theorem :: LOPBAN_2:21

theorem :: LOPBAN_2:22

theorem :: LOPBAN_2:23