:: Complex {B}anach Space of Bounded Linear Operators :: by Noboru Endou :: :: Received February 24, 2004 :: Copyright (c) 2004-2012 Association of Mizar Users begin definition let X be set ; let Y be non empty set ; let F be Function of [:COMPLEX,Y:],Y; let c be complex number ; let f be Function of X,Y; :: original:[;] redefine funcF [;] (c,f) -> Element of Funcs (X,Y); coherence F [;] (c,f) is Element of Funcs (X,Y) proofend; 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) proofend; 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 proofend; 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 proofend; theorem Th2: :: CLOPBAN1:2 for X being non empty set for Y being ComplexLinearSpace for h, f 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) ) proofend; 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) proofend; 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) proofend; 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 proofend; 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)) . [(- 1r),f])) = FuncZero (X,Y) proofend; 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 proofend; 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] proofend; 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] proofend; 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))] proofend; 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 proofend; 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; cluster ComplexVectSpace (X,Y) -> constituted-Functions ; coherence ComplexVectSpace (X,Y) is constituted-Functions 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 funcf . x -> VECTOR of Y; coherence f . x is VECTOR of Y proofend; 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) ) proofend; begin definition let X, Y be non empty CLSStruct ; let IT be Function of X,Y; attrIT 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 ) proofend; end; definition let X, Y be ComplexLinearSpace; mode LinearOperator of X,Y is additive homogeneous Function of X,Y; 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 ) proofend; 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 proofend; 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 ) proofend; end; theorem Th13: :: CLOPBAN1:13 for X, Y being ComplexLinearSpace holds LinearOperators (X,Y) is linearly-closed proofend; 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 Th13, CSSPACE:11; 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 Th13, CSSPACE:11; 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; cluster C_VectorSpace_of_LinearOperators (X,Y) -> strict ; coherence C_VectorSpace_of_LinearOperators (X,Y) is strict ; end; registration let X, Y be ComplexLinearSpace; cluster C_VectorSpace_of_LinearOperators (X,Y) -> constituted-Functions ; coherence C_VectorSpace_of_LinearOperators (X,Y) is constituted-Functions by MONOID_0:80; end; definition let X, Y be ComplexLinearSpace; let f be Element of (C_VectorSpace_of_LinearOperators (X,Y)); let v be VECTOR of X; :: original:. redefine funcf . v -> VECTOR of Y; coherence f . v is VECTOR of Y proofend; end; theorem Th15: :: CLOPBAN1:15 for X, Y being ComplexLinearSpace for f, g, h being VECTOR of (C_VectorSpace_of_LinearOperators (X,Y)) holds ( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) ) proofend; theorem Th16: :: CLOPBAN1:16 for X, Y being ComplexLinearSpace for f, h being VECTOR of (C_VectorSpace_of_LinearOperators (X,Y)) for c being Complex holds ( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) ) proofend; theorem Th17: :: CLOPBAN1:17 for X, Y being ComplexLinearSpace holds 0. (C_VectorSpace_of_LinearOperators (X,Y)) = the carrier of X --> (0. Y) proofend; theorem Th18: :: CLOPBAN1:18 for X, Y being ComplexLinearSpace holds the carrier of X --> (0. Y) is LinearOperator of X,Y proofend; begin 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.|| = ||.g.|| ) proofend; definition let X, Y be ComplexNormSpace; let IT be LinearOperator of X,Y; attrIT is Lipschitzian means :Def6: :: CLOPBAN1:def 6 ex K being Real st ( 0 <= K & ( for x being VECTOR of X holds ||.(IT . x).|| <= K * ||.x.|| ) ); 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 * ||.x.|| ) ) ); 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 proofend; 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 proofend; end; definition let X, Y be ComplexNormSpace; func BoundedLinearOperators (X,Y) -> Subset of (C_VectorSpace_of_LinearOperators (X,Y)) 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 (C_VectorSpace_of_LinearOperators (X,Y)) st for x being set holds ( x in b1 iff x is Lipschitzian LinearOperator of X,Y ) proofend; uniqueness for b1, b2 being Subset of (C_VectorSpace_of_LinearOperators (X,Y)) 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 proofend; end; :: deftheorem Def7 defines BoundedLinearOperators CLOPBAN1:def_7_:_ for X, Y being ComplexNormSpace for b3 being Subset of (C_VectorSpace_of_LinearOperators (X,Y)) 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 proofend; end; theorem Th21: :: CLOPBAN1:21 for X, Y being ComplexNormSpace holds BoundedLinearOperators (X,Y) is linearly-closed proofend; theorem :: CLOPBAN1:22 for X, Y being ComplexNormSpace holds CLSStruct(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) #) is Subspace of C_VectorSpace_of_LinearOperators (X,Y) by Th21, CSSPACE:11; registration let X, Y be ComplexNormSpace; cluster CLSStruct(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ; coherence ( CLSStruct(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) #) is Abelian & CLSStruct(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) #) is add-associative & CLSStruct(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) #) is right_zeroed & CLSStruct(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) #) is right_complementable & CLSStruct(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) #) is vector-distributive & CLSStruct(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) #) is scalar-distributive & CLSStruct(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) #) is scalar-associative & CLSStruct(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) #) is scalar-unital ) by Th21, CSSPACE:11; end; definition let X, Y be ComplexNormSpace; func C_VectorSpace_of_BoundedLinearOperators (X,Y) -> ComplexLinearSpace equals :: CLOPBAN1:def 8 CLSStruct(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) #); coherence CLSStruct(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) #) 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(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) #); registration let X, Y be ComplexNormSpace; cluster C_VectorSpace_of_BoundedLinearOperators (X,Y) -> strict ; coherence C_VectorSpace_of_BoundedLinearOperators (X,Y) is strict ; end; registration let X, Y be ComplexNormSpace; cluster -> Relation-like Function-like for Element of the carrier of (C_VectorSpace_of_BoundedLinearOperators (X,Y)); coherence for b1 being Element of (C_VectorSpace_of_BoundedLinearOperators (X,Y)) holds ( b1 is Function-like & b1 is Relation-like ) ; end; definition let X, Y be ComplexNormSpace; let f be Element of (C_VectorSpace_of_BoundedLinearOperators (X,Y)); let v be VECTOR of X; :: original:. redefine funcf . v -> VECTOR of Y; coherence f . v is VECTOR of Y proofend; end; theorem Th23: :: CLOPBAN1:23 for X, Y being ComplexNormSpace for f, g, h being VECTOR of (C_VectorSpace_of_BoundedLinearOperators (X,Y)) holds ( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) ) proofend; theorem Th24: :: CLOPBAN1:24 for X, Y being ComplexNormSpace for f, h being VECTOR of (C_VectorSpace_of_BoundedLinearOperators (X,Y)) for c being Complex holds ( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) ) proofend; theorem Th25: :: CLOPBAN1:25 for X, Y being ComplexNormSpace holds 0. (C_VectorSpace_of_BoundedLinearOperators (X,Y)) = the carrier of X --> (0. Y) proofend; definition let X, Y be ComplexNormSpace; let f be set ; 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 A1, Def7; end; :: deftheorem Def9 defines modetrans CLOPBAN1:def_9_:_ for X, Y being ComplexNormSpace for f being set 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 : ||.t.|| <= 1 } ; coherence { ||.(u . t).|| where t is VECTOR of X : ||.t.|| <= 1 } is non empty Subset of REAL proofend; 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 : ||.t.|| <= 1 } ; theorem Th26: :: CLOPBAN1:26 for X, Y being ComplexNormSpace for g being Lipschitzian LinearOperator of X,Y holds PreNorms g is bounded_above proofend; 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 ) proofend; definition let X, Y be ComplexNormSpace; func BoundedLinearOperatorsNorm (X,Y) -> Function of (BoundedLinearOperators (X,Y)),REAL means :Def11: :: CLOPBAN1:def 11 for x being set st x in BoundedLinearOperators (X,Y) holds it . x = upper_bound (PreNorms (modetrans (x,X,Y))); existence ex b1 being Function of (BoundedLinearOperators (X,Y)),REAL st for x being set st x in BoundedLinearOperators (X,Y) holds b1 . x = upper_bound (PreNorms (modetrans (x,X,Y))) proofend; uniqueness for b1, b2 being Function of (BoundedLinearOperators (X,Y)),REAL st ( for x being set st x in BoundedLinearOperators (X,Y) holds b1 . x = upper_bound (PreNorms (modetrans (x,X,Y))) ) & ( for x being set st x in BoundedLinearOperators (X,Y) holds b2 . x = upper_bound (PreNorms (modetrans (x,X,Y))) ) holds b1 = b2 proofend; end; :: deftheorem Def11 defines BoundedLinearOperatorsNorm CLOPBAN1:def_11_:_ for X, Y being ComplexNormSpace for b3 being Function of (BoundedLinearOperators (X,Y)),REAL holds ( b3 = BoundedLinearOperatorsNorm (X,Y) iff for x being set 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 proofend; theorem Th29: :: CLOPBAN1:29 for X, Y being ComplexNormSpace for f being Lipschitzian LinearOperator of X,Y holds (BoundedLinearOperatorsNorm (X,Y)) . f = upper_bound (PreNorms f) proofend; definition let X, Y be ComplexNormSpace; func C_NormSpace_of_BoundedLinearOperators (X,Y) -> non empty CNORMSTR equals :: CLOPBAN1:def 12 CNORMSTR(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(BoundedLinearOperatorsNorm (X,Y)) #); coherence CNORMSTR(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(BoundedLinearOperatorsNorm (X,Y)) #) 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(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))),(BoundedLinearOperatorsNorm (X,Y)) #); theorem Th30: :: CLOPBAN1:30 for X, Y being ComplexNormSpace holds the carrier of X --> (0. Y) = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) proofend; theorem Th31: :: CLOPBAN1:31 for X, Y being ComplexNormSpace for f being Point of (C_NormSpace_of_BoundedLinearOperators (X,Y)) for g being Lipschitzian LinearOperator of X,Y st g = f holds for t being VECTOR of X holds ||.(g . t).|| <= ||.f.|| * ||.t.|| proofend; theorem Th32: :: CLOPBAN1:32 for X, Y being ComplexNormSpace for f being Point of (C_NormSpace_of_BoundedLinearOperators (X,Y)) holds 0 <= ||.f.|| proofend; theorem Th33: :: CLOPBAN1:33 for X, Y being ComplexNormSpace for f being Point of (C_NormSpace_of_BoundedLinearOperators (X,Y)) st f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) holds 0 = ||.f.|| proofend; registration let X, Y be ComplexNormSpace; cluster -> Relation-like Function-like for Element of the carrier of (C_NormSpace_of_BoundedLinearOperators (X,Y)); coherence for b1 being Element of (C_NormSpace_of_BoundedLinearOperators (X,Y)) holds ( b1 is Function-like & b1 is Relation-like ) ; end; definition let X, Y be ComplexNormSpace; let f be Element of (C_NormSpace_of_BoundedLinearOperators (X,Y)); let v be VECTOR of X; :: original:. redefine funcf . v -> VECTOR of Y; coherence f . v is VECTOR of Y proofend; end; theorem Th34: :: CLOPBAN1:34 for X, Y being ComplexNormSpace for f, g, h being Point of (C_NormSpace_of_BoundedLinearOperators (X,Y)) holds ( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) ) proofend; theorem Th35: :: CLOPBAN1:35 for X, Y being ComplexNormSpace for f, h being Point of (C_NormSpace_of_BoundedLinearOperators (X,Y)) for c being Complex holds ( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) ) proofend; theorem Th36: :: CLOPBAN1:36 for X, Y being ComplexNormSpace for f, g being Point of (C_NormSpace_of_BoundedLinearOperators (X,Y)) for c being Complex holds ( ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) ) & ( f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) implies ||.f.|| = 0 ) & ||.(c * f).|| = |.c.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| ) proofend; 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 ) proofend; theorem Th38: :: CLOPBAN1:38 for X, Y being ComplexNormSpace holds C_NormSpace_of_BoundedLinearOperators (X,Y) is ComplexNormSpace proofend; registration let X, Y be ComplexNormSpace; cluster C_NormSpace_of_BoundedLinearOperators (X,Y) -> non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ; coherence ( 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 & C_NormSpace_of_BoundedLinearOperators (X,Y) is vector-distributive & C_NormSpace_of_BoundedLinearOperators (X,Y) is scalar-distributive & C_NormSpace_of_BoundedLinearOperators (X,Y) is scalar-associative & C_NormSpace_of_BoundedLinearOperators (X,Y) is scalar-unital & C_NormSpace_of_BoundedLinearOperators (X,Y) is Abelian & C_NormSpace_of_BoundedLinearOperators (X,Y) is add-associative & C_NormSpace_of_BoundedLinearOperators (X,Y) is right_zeroed & C_NormSpace_of_BoundedLinearOperators (X,Y) is right_complementable ) by Th38; end; theorem Th39: :: CLOPBAN1:39 for X, Y being ComplexNormSpace for f, g, h being Point of (C_NormSpace_of_BoundedLinearOperators (X,Y)) holds ( h = f - g iff for x being VECTOR of X holds h . x = (f . x) - (g . x) ) proofend; begin definition let X be ComplexNormSpace; attrX 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 cluster non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete for CNORMSTR ; existence ex b1 being ComplexNormSpace st b1 is complete proofend; end; definition mode ComplexBanachSpace is complete ComplexNormSpace; end; Lm2: for e being Real for seq being Real_Sequence st seq is convergent & ex k being Element of NAT st for i being Element of NAT st k <= i holds seq . i <= e holds lim seq <= e proofend; 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).|| ) proofend; theorem Th41: :: CLOPBAN1:41 for X, Y being ComplexNormSpace st Y is complete holds for seq being sequence of (C_NormSpace_of_BoundedLinearOperators (X,Y)) st seq is Cauchy_sequence_by_Norm holds seq is convergent proofend; theorem Th42: :: CLOPBAN1:42 for X being ComplexNormSpace for Y being ComplexBanachSpace holds C_NormSpace_of_BoundedLinearOperators (X,Y) is ComplexBanachSpace proofend; registration let X be ComplexNormSpace; let Y be ComplexBanachSpace; cluster C_NormSpace_of_BoundedLinearOperators (X,Y) -> non empty complete ; coherence C_NormSpace_of_BoundedLinearOperators (X,Y) is complete by Th42; end;