:: LOPBAN_2 semantic presentation  Show TPTP formulae Show IDV graph for whole article:: Showing IDV graph ... (Click the Palm Trees again to close it)

definition
let X be non empty set ;
let f, g be Element of Funcs X,X;
:: original: *
redefine func g * f -> Element of Funcs X,X;
coherence
g * f is Element of Funcs X,X
proof end;
end;

theorem Th1: :: LOPBAN_2:1  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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
proof end;

theorem Th2: :: LOPBAN_2:2  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X, Y, Z being RealNormSpace
for f being bounded LinearOperator of X,Y
for g being bounded LinearOperator of Y,Z holds
( g * f is bounded 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 bounded LinearOperator of X,X;
:: original: *
redefine func g * f -> bounded LinearOperator of X,X;
correctness
coherence
g * f is bounded LinearOperator of X,X;
by Th2;
end;

definition
let X be RealNormSpace;
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
coherence
(Add_ (BoundedLinearOperators X,X),(R_VectorSpace_of_LinearOperators X,X)) . f,g is Element of BoundedLinearOperators X,X;
;
end;

:: 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;

definition
let X be RealNormSpace;
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
coherence
(modetrans g,X,X) * (modetrans f,X,X) is Element of BoundedLinearOperators X,X;
by LOPBAN_1:def 10;
end;

:: 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);

definition
let X be RealNormSpace;
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
coherence
(Mult_ (BoundedLinearOperators X,X),(R_VectorSpace_of_LinearOperators X,X)) . a,f is Element of BoundedLinearOperators X,X;
;
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;

definition
let X be RealNormSpace;
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
ex b1 being BinOp of BoundedLinearOperators X,X st
for f, g being Element of BoundedLinearOperators X,X holds b1 . f,g = f * g
proof end;
uniqueness
for b1, b2 being BinOp of BoundedLinearOperators X,X st ( for f, g being Element of BoundedLinearOperators X,X holds b1 . f,g = f * g ) & ( for f, g being Element of BoundedLinearOperators X,X holds b2 . f,g = f * g ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines FuncMult LOPBAN_2:def 4 :
for X being RealNormSpace
for b2 being BinOp of BoundedLinearOperators X,X holds
( b2 = FuncMult X iff for f, g being Element of BoundedLinearOperators X,X holds b2 . f,g = f * g );

theorem Th3: :: LOPBAN_2:3  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being RealNormSpace holds id the carrier of X is bounded LinearOperator of X,X
proof end;

definition
let X be RealNormSpace;
func FuncUnit X -> Element of BoundedLinearOperators X,X equals :: LOPBAN_2:def 5
 id the carrier of X;
coherence
id the carrier of X is Element of BoundedLinearOperators X,X
proof end;
end;

:: deftheorem defines FuncUnit LOPBAN_2:def 5 :
for X being RealNormSpace holds FuncUnit X = id the carrier of X;

theorem Th4: :: LOPBAN_2:4  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being RealNormSpace
for f, g, h being bounded 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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being RealNormSpace
for f, g, h being bounded LinearOperator of X,X holds f * (g * h) = (f * g) * h
proof end;

theorem Th6: :: LOPBAN_2:6  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being RealNormSpace
for f being bounded 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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being RealNormSpace
for f, g, h being Element of BoundedLinearOperators X,X holds f * (g * h) = (f * g) * h
proof end;

theorem Th8: :: LOPBAN_2:8  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being RealNormSpace
for f being Element of BoundedLinearOperators X,X holds
( f * (FuncUnit X) = f & (FuncUnit X) * f = f )
proof end;

theorem Th9: :: LOPBAN_2:9  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being RealNormSpace
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being RealNormSpace
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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)
proof end;

theorem Th12: :: LOPBAN_2:12  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being RealNormSpace
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;
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
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;

:: 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)) #);

registration
let X be RealNormSpace;
cluster Ring_of_BoundedLinearOperators X -> non empty strict ;
coherence
( not Ring_of_BoundedLinearOperators X is empty & Ring_of_BoundedLinearOperators X is strict )
proof end;
end;

Lm1: now
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 ) )
assume A1: e = FuncUnit X ; :: thesis: ( x * e = x & e * x = x )
reconsider f = x as Element of BoundedLinearOperators X,X ;
thus x * e = the mult of (Ring_of_BoundedLinearOperators X) . x,e
.= (FuncMult X) . f,(FuncUnit X) by A1
.= f * (FuncUnit X) by Def4
.= x by Th8 ; :: thesis: e * x = x
thus e * x = the mult of (Ring_of_BoundedLinearOperators X) . e,x
.= (FuncMult X) . (FuncUnit X),f by A1
.= (FuncUnit X) * f by Def4
.= x by Th8 ; :: thesis: verum
end;

registration
let X be RealNormSpace;
cluster Ring_of_BoundedLinearOperators X -> non empty unital strict ;
coherence
Ring_of_BoundedLinearOperators X is unital
proof end;
end;

Lm2: now
let X be RealNormSpace; :: thesis: 1. (Ring_of_BoundedLinearOperators X) = FuncUnit X
set F = Ring_of_BoundedLinearOperators X;
reconsider e = FuncUnit X as Element of (Ring_of_BoundedLinearOperators X) ;
for x being Element of (Ring_of_BoundedLinearOperators X) holds
( x * e = x & e * x = x ) by Lm1;
hence 1. (Ring_of_BoundedLinearOperators X) = FuncUnit X by GROUP_1:def 5; :: thesis: verum
end;

theorem Th13: :: LOPBAN_2:13  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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) )
proof end;

theorem Th14: :: LOPBAN_2:14  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being RealNormSpace holds Ring_of_BoundedLinearOperators X is Ring
proof end;

registration
let X be RealNormSpace;
cluster Ring_of_BoundedLinearOperators X -> non empty Abelian add-associative right_zeroed right_complementable unital associative strict right_unital distributive left_unital ;
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 ) by Th14;
end;

definition
let X be RealNormSpace;
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
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;

:: 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)) #);

registration
let X be RealNormSpace;
cluster R_Algebra_of_BoundedLinearOperators X -> non empty strict ;
coherence
( not R_Algebra_of_BoundedLinearOperators X is empty & R_Algebra_of_BoundedLinearOperators X is strict )
proof end;
end;

Lm3: now
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 ) )
assume A1: e = FuncUnit X ; :: thesis: ( x * e = x & e * x = x )
reconsider f = x as Element of BoundedLinearOperators X,X ;
thus x * e = the mult of (R_Algebra_of_BoundedLinearOperators X) . x,e
.= (FuncMult X) . f,(FuncUnit X) by A1
.= f * (FuncUnit X) by Def4
.= x by Th8 ; :: thesis: e * x = x
thus e * x = the mult of (R_Algebra_of_BoundedLinearOperators X) . e,x
.= (FuncMult X) . (FuncUnit X),f by A1
.= (FuncUnit X) * f by Def4
.= x by Th8 ; :: thesis: verum
end;

registration
let X be RealNormSpace;
cluster R_Algebra_of_BoundedLinearOperators X -> non empty unital strict ;
coherence
R_Algebra_of_BoundedLinearOperators X is unital
proof end;
end;

Lm4: now
let X be RealNormSpace; :: thesis: 1. (R_Algebra_of_BoundedLinearOperators X) = FuncUnit X
set F = R_Algebra_of_BoundedLinearOperators X;
reconsider e = FuncUnit X as Element of (R_Algebra_of_BoundedLinearOperators X) ;
for x being Element of (R_Algebra_of_BoundedLinearOperators X) holds
( x * e = x & e * x = x ) by Lm3;
hence 1. (R_Algebra_of_BoundedLinearOperators X) = FuncUnit X by GROUP_1:def 5; :: thesis: verum
end;

theorem Th15: :: LOPBAN_2:15  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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) )
proof end;

definition
mode BLAlgebra is non empty Abelian add-associative right_zeroed right_complementable associative Algebra-like AlgebraStr ;
end;

theorem :: LOPBAN_2:16  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being RealNormSpace holds R_Algebra_of_BoundedLinearOperators X is BLAlgebra
proof end;

registration
cluster l1_Space -> complete ;
coherence
l1_Space is complete
proof end;
end;

registration
cluster l1_Space -> non trivial complete ;
coherence
not l1_Space is trivial
proof end;
end;

registration
cluster non trivial NORMSTR ;
existence
not for b1 being RealBanachSpace holds b1 is trivial
proof end;
end;

theorem Th17: :: LOPBAN_2:17  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non trivial RealNormSpace ex w being VECTOR of X st ||.w.|| = 1
proof end;

theorem Th18: :: LOPBAN_2:18  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non trivial RealNormSpace holds (BoundedLinearOperatorsNorm X,X) . (id the carrier of X) = 1
proof end;

definition
attr c1 is strict;
struct Normed_AlgebraStr -> AlgebraStr , NORMSTR ;
aggr Normed_AlgebraStr(# carrier, mult, add, Mult, unity, Zero, norm #) -> Normed_AlgebraStr ;
end;

registration
cluster non empty Normed_AlgebraStr ;
existence
not for b1 being Normed_AlgebraStr holds b1 is empty
proof end;
end;

definition
let X be RealNormSpace;
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
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;

:: 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) #);

registration
let X be RealNormSpace;
cluster R_Normed_Algebra_of_BoundedLinearOperators X -> non empty strict ;
coherence
( not R_Normed_Algebra_of_BoundedLinearOperators X is empty & R_Normed_Algebra_of_BoundedLinearOperators X is strict )
proof end;
end;

Lm5: now
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 ) )
assume A1: e = FuncUnit X ; :: thesis: ( x * e = x & e * x = x )
reconsider f = x as Element of BoundedLinearOperators X,X ;
thus x * e = the mult of (R_Normed_Algebra_of_BoundedLinearOperators X) . x,e
.= (FuncMult X) . f,(FuncUnit X) by A1
.= f * (FuncUnit X) by Def4
.= x by Th8 ; :: thesis: e * x = x
thus e * x = the mult of (R_Normed_Algebra_of_BoundedLinearOperators X) . e,x
.= (FuncMult X) . (FuncUnit X),f by A1
.= (FuncUnit X) * f by Def4
.= x by Th8 ; :: thesis: verum
end;

registration
let X be RealNormSpace;
cluster R_Normed_Algebra_of_BoundedLinearOperators X -> non empty unital strict ;
coherence
R_Normed_Algebra_of_BoundedLinearOperators X is unital
proof end;
end;

Lm6: now
let X be RealNormSpace; :: thesis: 1. (R_Normed_Algebra_of_BoundedLinearOperators X) = FuncUnit X
set F = R_Normed_Algebra_of_BoundedLinearOperators X;
reconsider e = FuncUnit X as Element of (R_Normed_Algebra_of_BoundedLinearOperators X) ;
for x being Element of (R_Normed_Algebra_of_BoundedLinearOperators X) holds
( x * e = x & e * x = x ) by Lm5;
hence 1. (R_Normed_Algebra_of_BoundedLinearOperators X) = FuncUnit X by GROUP_1:def 5; :: thesis: verum
end;

theorem Th19: :: LOPBAN_2:19  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 )
proof end;

theorem Th20: :: LOPBAN_2:20  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being RealNormSpace holds
( 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 Algebra-like & R_Normed_Algebra_of_BoundedLinearOperators X is RealLinearSpace-like )
proof end;

registration
cluster non empty Abelian add-associative right_zeroed right_complementable RealLinearSpace-like associative Algebra-like RealNormSpace-like strict Normed_AlgebraStr ;
existence
ex b1 being non empty Normed_AlgebraStr st
( b1 is RealNormSpace-like & b1 is Abelian & b1 is add-associative & b1 is right_zeroed & b1 is right_complementable & b1 is associative & b1 is Algebra-like & b1 is RealLinearSpace-like & b1 is strict )
proof end;
end;

definition
mode Normed_Algebra is non empty Abelian add-associative right_zeroed right_complementable RealLinearSpace-like associative Algebra-like RealNormSpace-like Normed_AlgebraStr ;
end;

registration
let X be RealNormSpace;
cluster R_Normed_Algebra_of_BoundedLinearOperators X -> non empty Abelian add-associative right_zeroed right_complementable RealLinearSpace-like unital associative Algebra-like RealNormSpace-like strict ;
correctness
coherence
( 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 Algebra-like & R_Normed_Algebra_of_BoundedLinearOperators X is RealLinearSpace-like );
by Th20;
end;

definition
let X be non empty Normed_AlgebraStr ;
attr X is Banach_Algebra-like_1 means :: LOPBAN_2:def 9
for x, y being Element of X holds ||.(x * y).|| <= ||.x.|| * ||.y.||;
attr X is Banach_Algebra-like_2 means :: LOPBAN_2:def 10
||.(1. X).|| = 1;
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);
end;

:: 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.|| );

:: 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 );

:: 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) );

definition
let X be Normed_Algebra;
attr X is Banach_Algebra-like means :Def12: :: 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 );
end;

:: deftheorem Def12 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 ) );

registration
cluster Banach_Algebra-like -> left-distributive left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Normed_AlgebraStr ;
coherence
for b1 being Normed_Algebra st b1 is Banach_Algebra-like holds
( b1 is Banach_Algebra-like_1 & b1 is Banach_Algebra-like_2 & b1 is Banach_Algebra-like_3 & b1 is left-distributive & b1 is left_unital & b1 is complete ) by Def12;
cluster left-distributive left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 -> Banach_Algebra-like Normed_AlgebraStr ;
coherence
for b1 being Normed_Algebra st b1 is Banach_Algebra-like_1 & b1 is Banach_Algebra-like_2 & b1 is Banach_Algebra-like_3 & b1 is left-distributive & b1 is left_unital & b1 is complete holds
b1 is Banach_Algebra-like by Def12;
end;

registration
let X be non trivial RealBanachSpace;
cluster R_Normed_Algebra_of_BoundedLinearOperators X -> non empty Abelian add-associative right_zeroed right_complementable RealLinearSpace-like unital associative left-distributive Algebra-like RealNormSpace-like complete strict Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ;
coherence
R_Normed_Algebra_of_BoundedLinearOperators X is Banach_Algebra-like
proof end;
end;

registration
cluster left-distributive left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like Normed_AlgebraStr ;
existence
ex b1 being Normed_Algebra st b1 is Banach_Algebra-like
proof end;
end;

definition
mode Banach_Algebra is Banach_Algebra-like Normed_Algebra;
end;

theorem :: LOPBAN_2:21  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being RealNormSpace holds 1. (Ring_of_BoundedLinearOperators X) = FuncUnit X by Lm2;

theorem :: LOPBAN_2:22  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being RealNormSpace holds 1. (R_Algebra_of_BoundedLinearOperators X) = FuncUnit X by Lm4;

theorem :: LOPBAN_2:23  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being RealNormSpace holds 1. (R_Normed_Algebra_of_BoundedLinearOperators X) = FuncUnit X by Lm6;