LOPBAN_1: Banach Space of Bounded Linear Operators

:: Banach Space of Bounded Linear Operators
:: by Yasunari Shidama
::
:: Received December 22, 2003
:: Copyright (c) 2003-2011 Association of Mizar Users

begin

definition

let X be set ;
let Y be non empty set ;
let F be Function of [:REAL,Y:],Y;
let a be real number ;
let f be Function of X,Y;
:: original: [;]
redefine func F [;] (a,f) -> Element of Funcs (X,Y);
coherence

proof end;

end;

theorem Th1: :: LOPBAN_1:1

for X being non empty set
for Y being non empty addLoopStr ex ADD being BinOp of (Funcs (X, the carrier of Y)) st
for f, g being Element of Funcs (X, the carrier of Y) holds ADD . (f,g) = the addF of Y .: (f,g)

proof end;

theorem Th2: :: LOPBAN_1:2

for X being non empty set
for Y being RealLinearSpace ex MULT being Function of [:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)) st
for r being Real
for f being Element of Funcs (X, the carrier of Y)
for s being Element of X holds (MULT . [r,f]) . s = r * (f . s)

proof end;

definition

let X be non empty set ;
let Y be non empty addLoopStr ;
func FuncAdd (X,Y) -> BinOp of (Funcs (X, the carrier of Y)) means :Def1: :: LOPBAN_1:def 1
for f, g being Element of Funcs (X, the carrier of Y) holds it . (f,g) = the addF of Y .: (f,g);
existence by Th1;
uniqueness

proof end;

end;

:: deftheorem Def1 defines FuncAdd LOPBAN_1:def 1 :

definition

let X be non empty set ;
let Y be RealLinearSpace;
func FuncExtMult (X,Y) -> Function of [:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)) means :Def2: :: LOPBAN_1:def 2
for a being Real
for f being Element of Funcs (X, the carrier of Y)
for x being Element of X holds (it . [a,f]) . x = a * (f . x);
existence by Th2;
uniqueness

proof end;

end;

:: deftheorem Def2 defines FuncExtMult LOPBAN_1:def 2 :

definition

let X be set ;
let Y be non empty ZeroStr ;
func FuncZero (X,Y) -> Element of Funcs (X, the carrier of Y) equals :: LOPBAN_1:def 3
X --> (0. Y);
coherence by FUNCT_2:11;

end;

:: deftheorem defines FuncZero LOPBAN_1:def 3 :

Lm1: for A, B being non empty set
for x being Element of A
for f being Function of A,B holds x in dom f

proof end;

theorem Th3: :: LOPBAN_1:3

for X being non empty set
for Y being non empty addLoopStr
for f, g, h being Element of Funcs (X, the carrier of Y) holds
( h = (FuncAdd (X,Y)) . (f,g) iff for x being Element of X holds h . x = (f . x) + (g . x) )

proof end;

theorem :: LOPBAN_1:4

canceled;

theorem Th5: :: LOPBAN_1:5

for X being non empty set
for Y being RealLinearSpace
for h, f being Element of Funcs (X, the carrier of Y)
for a being Real holds
( h = (FuncExtMult (X,Y)) . [a,f] iff for x being Element of X holds h . x = a * (f . x) )

proof end;

theorem Th6: :: LOPBAN_1:6

for X being non empty set
for Y being RealLinearSpace
for f, g being Element of Funcs (X, the carrier of Y) holds (FuncAdd (X,Y)) . (f,g) = (FuncAdd (X,Y)) . (g,f)

proof end;

theorem Th7: :: LOPBAN_1:7

for X being non empty set
for Y being RealLinearSpace
for f, g, h being Element of Funcs (X, the carrier of Y) holds (FuncAdd (X,Y)) . (f,((FuncAdd (X,Y)) . (g,h))) = (FuncAdd (X,Y)) . (((FuncAdd (X,Y)) . (f,g)),h)

proof end;

theorem Th8: :: LOPBAN_1:8

for X being non empty set
for Y being RealLinearSpace
for f being Element of Funcs (X, the carrier of Y) holds (FuncAdd (X,Y)) . ((FuncZero (X,Y)),f) = f

proof end;

theorem Th9: :: LOPBAN_1:9

for X being non empty set
for Y being RealLinearSpace
for f being Element of Funcs (X, the carrier of Y) holds (FuncAdd (X,Y)) . (f,((FuncExtMult (X,Y)) . [(- 1),f])) = FuncZero (X,Y)

proof end;

theorem Th10: :: LOPBAN_1:10

for X being non empty set
for Y being RealLinearSpace
for f being Element of Funcs (X, the carrier of Y) holds (FuncExtMult (X,Y)) . [1,f] = f

proof end;

theorem Th11: :: LOPBAN_1:11

for X being non empty set
for Y being RealLinearSpace
for f being Element of Funcs (X, the carrier of Y)
for a, b being Real holds (FuncExtMult (X,Y)) . [a,((FuncExtMult (X,Y)) . [b,f])] = (FuncExtMult (X,Y)) . [(a * b),f]

proof end;

theorem Th12: :: LOPBAN_1:12

for X being non empty set
for Y being RealLinearSpace
for f being Element of Funcs (X, the carrier of Y)
for a, b being Real holds (FuncAdd (X,Y)) . (((FuncExtMult (X,Y)) . [a,f]),((FuncExtMult (X,Y)) . [b,f])) = (FuncExtMult (X,Y)) . [(a + b),f]

proof end;

Lm2: for X being non empty set
for Y being RealLinearSpace
for f, g being Element of Funcs (X, the carrier of Y)
for a being Real holds (FuncAdd (X,Y)) . (((FuncExtMult (X,Y)) . [a,f]),((FuncExtMult (X,Y)) . [a,g])) = (FuncExtMult (X,Y)) . [a,((FuncAdd (X,Y)) . (f,g))]

proof end;

theorem Th13: :: LOPBAN_1:13

for X being non empty set
for Y being RealLinearSpace holds RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) is RealLinearSpace

proof end;

definition

let X be non empty set ;
let Y be RealLinearSpace;
func RealVectSpace (X,Y) -> RealLinearSpace equals :: LOPBAN_1:def 4
RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #);
coherence by Th13;

end;

:: deftheorem defines RealVectSpace LOPBAN_1:def 4 :

registration

let X be non empty set ;
let Y be RealLinearSpace;
cluster RealVectSpace (X,Y) -> strict ;
coherence ;

end;

registration

let X be non empty set ;
let Y be RealLinearSpace;
cluster RealVectSpace (X,Y) -> constituted-Functions ;
coherence by MONOID_0:89;

end;

definition

let X be non empty set ;
let Y be RealLinearSpace;
let f be VECTOR of (RealVectSpace (X,Y));
let x be Element of X;
:: original: .
redefine func f . x -> VECTOR of Y;
coherence

proof end;

end;

theorem :: LOPBAN_1:14

for X being non empty set
for Y being RealLinearSpace
for f, g, h being VECTOR of (RealVectSpace (X,Y)) holds
( h = f + g iff for x being Element of X holds h . x = (f . x) + (g . x) ) by Th3;

theorem :: LOPBAN_1:15

for X being non empty set
for Y being RealLinearSpace
for f, h being VECTOR of (RealVectSpace (X,Y))
for a being Real holds
( h = a * f iff for x being Element of X holds h . x = a * (f . x) ) by Th5;

theorem :: LOPBAN_1:16

for X being non empty set
for Y being RealLinearSpace holds 0. (RealVectSpace (X,Y)) = X --> (0. Y) ;

begin

definition

let X, Y be non empty RLSStruct ;
let IT be Function of X,Y;
canceled;
attr IT is homogeneous means :Def6: :: LOPBAN_1:def 6
for x being VECTOR of X
for r being Real holds IT . (r * x) = r * (IT . x);

end;

:: deftheorem LOPBAN_1:def 5 :

:: deftheorem Def6 defines homogeneous LOPBAN_1:def 6 :

registration

let X be non empty RLSStruct ;
let Y be RealLinearSpace;
cluster non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive homogeneous Element of bool [: the carrier of X, the carrier of Y:];
existence

proof end;

end;

definition

let X, Y be RealLinearSpace;
mode LinearOperator of X,Y is additive homogeneous Function of X,Y;

end;

definition

let X, Y be RealLinearSpace;
func LinearOperators (X,Y) -> Subset of (RealVectSpace ( the carrier of X,Y)) means :Def7: :: LOPBAN_1:def 7
for x being set holds
( x in it iff x is LinearOperator of X,Y );
existence

proof end;

proof end;

end;

:: deftheorem Def7 defines LinearOperators LOPBAN_1:def 7 :

registration

let X, Y be RealLinearSpace;
cluster LinearOperators (X,Y) -> non empty functional ;
coherence

proof end;

end;

theorem Th17: :: LOPBAN_1:17

for X, Y being RealLinearSpace holds LinearOperators (X,Y) is linearly-closed

proof end;

theorem :: LOPBAN_1:18

for X, Y being RealLinearSpace holds RLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(RealVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(RealVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(RealVectSpace ( the carrier of X,Y)))) #) is Subspace of RealVectSpace ( the carrier of X,Y) by Th17, RSSPACE:13;

registration

let X, Y be RealLinearSpace;
cluster RLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(RealVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(RealVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(RealVectSpace ( the carrier of X,Y)))) #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
coherence by Th17, RSSPACE:13;

end;

definition

let X, Y be RealLinearSpace;
func R_VectorSpace_of_LinearOperators (X,Y) -> RealLinearSpace equals :: LOPBAN_1:def 8
RLSStruct(# (LinearOperators (X,Y)),(Zero_ ((LinearOperators (X,Y)),(RealVectSpace ( the carrier of X,Y)))),(Add_ ((LinearOperators (X,Y)),(RealVectSpace ( the carrier of X,Y)))),(Mult_ ((LinearOperators (X,Y)),(RealVectSpace ( the carrier of X,Y)))) #);
coherence ;

end;

:: deftheorem defines R_VectorSpace_of_LinearOperators LOPBAN_1:def 8 :

registration

let X, Y be RealLinearSpace;
cluster R_VectorSpace_of_LinearOperators (X,Y) -> strict ;
coherence ;

end;

registration

let X, Y be RealLinearSpace;
cluster R_VectorSpace_of_LinearOperators (X,Y) -> constituted-Functions ;
coherence by MONOID_0:89;

end;

definition

let X, Y be RealLinearSpace;
let f be Element of (R_VectorSpace_of_LinearOperators (X,Y));
let v be VECTOR of X;
:: original: .
redefine func f . v -> VECTOR of Y;
coherence

proof end;

end;

theorem :: LOPBAN_1:19

canceled;

theorem Th20: :: LOPBAN_1:20

for X, Y being RealLinearSpace
for f, g, h being VECTOR of (R_VectorSpace_of_LinearOperators (X,Y)) holds
( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )

proof end;

theorem Th21: :: LOPBAN_1:21

for X, Y being RealLinearSpace
for f, h being VECTOR of (R_VectorSpace_of_LinearOperators (X,Y))
for a being Real holds
( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) )

proof end;

theorem Th22: :: LOPBAN_1:22

for X, Y being RealLinearSpace holds 0. (R_VectorSpace_of_LinearOperators (X,Y)) = the carrier of X --> (0. Y)

proof end;

theorem Th23: :: LOPBAN_1:23

for X, Y being RealLinearSpace holds the carrier of X --> (0. Y) is LinearOperator of X,Y

proof end;

begin

theorem Th24: :: LOPBAN_1:24

for X being RealNormSpace
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.|| )

proof end;

definition

let X, Y be RealNormSpace;
let IT be LinearOperator of X,Y;
attr IT is bounded means :Def9: :: LOPBAN_1:def 9
ex K being Real st
( 0 <= K & ( for x being VECTOR of X holds ||.(IT . x).|| <= K * ||.x.|| ) );

end;

:: deftheorem Def9 defines bounded LOPBAN_1:def 9 :

theorem Th25: :: LOPBAN_1:25

for X, Y being RealNormSpace
for f being LinearOperator of X,Y st ( for x being VECTOR of X holds f . x = 0. Y ) holds
f is bounded

proof end;

registration

let X, Y be RealNormSpace;
cluster non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive homogeneous bounded Element of bool [: the carrier of X, the carrier of Y:];
existence

proof end;

end;

definition

let X, Y be RealNormSpace;
func BoundedLinearOperators (X,Y) -> Subset of (R_VectorSpace_of_LinearOperators (X,Y)) means :Def10: :: LOPBAN_1:def 10
for x being set holds
( x in it iff x is bounded LinearOperator of X,Y );
existence

proof end;

proof end;

end;

:: deftheorem Def10 defines BoundedLinearOperators LOPBAN_1:def 10 :

registration

let X, Y be RealNormSpace;
cluster BoundedLinearOperators (X,Y) -> non empty ;
coherence

proof end;

end;

theorem Th26: :: LOPBAN_1:26

for X, Y being RealNormSpace holds BoundedLinearOperators (X,Y) is linearly-closed

proof end;

theorem :: LOPBAN_1:27

for X, Y being RealNormSpace holds RLSStruct(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(R_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(R_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(R_VectorSpace_of_LinearOperators (X,Y)))) #) is Subspace of R_VectorSpace_of_LinearOperators (X,Y) by Th26, RSSPACE:13;

registration

let X, Y be RealNormSpace;
cluster RLSStruct(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(R_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(R_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(R_VectorSpace_of_LinearOperators (X,Y)))) #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
coherence by Th26, RSSPACE:13;

end;

definition

let X, Y be RealNormSpace;
func R_VectorSpace_of_BoundedLinearOperators (X,Y) -> RealLinearSpace equals :: LOPBAN_1:def 11
RLSStruct(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(R_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(R_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(R_VectorSpace_of_LinearOperators (X,Y)))) #);
coherence ;

end;

:: deftheorem defines R_VectorSpace_of_BoundedLinearOperators LOPBAN_1:def 11 :

registration

let X, Y be RealNormSpace;
cluster R_VectorSpace_of_BoundedLinearOperators (X,Y) -> strict ;
coherence ;

end;

registration

let X, Y be RealNormSpace;
cluster -> Relation-like Function-like Element of the carrier of (R_VectorSpace_of_BoundedLinearOperators (X,Y));
coherence ;

end;

definition

let X, Y be RealNormSpace;
let f be Element of (R_VectorSpace_of_BoundedLinearOperators (X,Y));
let v be VECTOR of X;
:: original: .
redefine func f . v -> VECTOR of Y;
coherence

proof end;

end;

theorem :: LOPBAN_1:28

canceled;

theorem Th29: :: LOPBAN_1:29

for X, Y being RealNormSpace
for f, g, h being VECTOR of (R_VectorSpace_of_BoundedLinearOperators (X,Y)) holds
( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )

proof end;

theorem Th30: :: LOPBAN_1:30

for X, Y being RealNormSpace
for f, h being VECTOR of (R_VectorSpace_of_BoundedLinearOperators (X,Y))
for a being Real holds
( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) )

proof end;

theorem Th31: :: LOPBAN_1:31

for X, Y being RealNormSpace holds 0. (R_VectorSpace_of_BoundedLinearOperators (X,Y)) = the carrier of X --> (0. Y)

proof end;

definition

let X, Y be RealNormSpace;
let f be set ;
assume A1: f in BoundedLinearOperators (X,Y) ;
func modetrans (f,X,Y) -> bounded LinearOperator of X,Y equals :Def12: :: LOPBAN_1:def 12
f;
coherence by A1, Def10;

end;

:: deftheorem Def12 defines modetrans LOPBAN_1:def 12 :

definition

let X, Y be RealNormSpace;
let u be LinearOperator of X,Y;
func PreNorms u -> non empty Subset of REAL equals :: LOPBAN_1:def 13
{ ||.(u . t).|| where t is VECTOR of X : ||.t.|| <= 1 } ;
coherence

proof end;

end;

:: deftheorem defines PreNorms LOPBAN_1:def 13 :

theorem Th32: :: LOPBAN_1:32

for X, Y being RealNormSpace
for g being bounded LinearOperator of X,Y holds PreNorms g is bounded_above

proof end;

theorem :: LOPBAN_1:33

for X, Y being RealNormSpace
for g being LinearOperator of X,Y holds
( g is bounded iff PreNorms g is bounded_above )

proof end;

theorem Th34: :: LOPBAN_1:34

for X, Y being RealNormSpace ex NORM being Function of (BoundedLinearOperators (X,Y)),REAL st
for f being set st f in BoundedLinearOperators (X,Y) holds
NORM . f = upper_bound (PreNorms (modetrans (f,X,Y)))

proof end;

definition

let X, Y be RealNormSpace;
func BoundedLinearOperatorsNorm (X,Y) -> Function of (BoundedLinearOperators (X,Y)),REAL means :Def14: :: LOPBAN_1:def 14
for x being set st x in BoundedLinearOperators (X,Y) holds
it . x = upper_bound (PreNorms (modetrans (x,X,Y)));
existence by Th34;
uniqueness

proof end;

end;

:: deftheorem Def14 defines BoundedLinearOperatorsNorm LOPBAN_1:def 14 :

theorem Th35: :: LOPBAN_1:35

for X, Y being RealNormSpace
for f being bounded LinearOperator of X,Y holds modetrans (f,X,Y) = f

proof end;

theorem Th36: :: LOPBAN_1:36

for X, Y being RealNormSpace
for f being bounded LinearOperator of X,Y holds (BoundedLinearOperatorsNorm (X,Y)) . f = upper_bound (PreNorms f)

proof end;

definition

let X, Y be RealNormSpace;
func R_NormSpace_of_BoundedLinearOperators (X,Y) -> non empty NORMSTR equals :: LOPBAN_1:def 15
NORMSTR(# (BoundedLinearOperators (X,Y)),(Zero_ ((BoundedLinearOperators (X,Y)),(R_VectorSpace_of_LinearOperators (X,Y)))),(Add_ ((BoundedLinearOperators (X,Y)),(R_VectorSpace_of_LinearOperators (X,Y)))),(Mult_ ((BoundedLinearOperators (X,Y)),(R_VectorSpace_of_LinearOperators (X,Y)))),(BoundedLinearOperatorsNorm (X,Y)) #);
coherence ;

end;

:: deftheorem defines R_NormSpace_of_BoundedLinearOperators LOPBAN_1:def 15 :

theorem Th37: :: LOPBAN_1:37

for X, Y being RealNormSpace holds the carrier of X --> (0. Y) = 0. (R_NormSpace_of_BoundedLinearOperators (X,Y))

proof end;

theorem Th38: :: LOPBAN_1:38

for X, Y being RealNormSpace
for f being Point of (R_NormSpace_of_BoundedLinearOperators (X,Y))
for g being bounded LinearOperator of X,Y st g = f holds
for t being VECTOR of X holds ||.(g . t).|| <= ||.f.|| * ||.t.||

proof end;

theorem Th39: :: LOPBAN_1:39

for X, Y being RealNormSpace
for f being Point of (R_NormSpace_of_BoundedLinearOperators (X,Y)) holds 0 <= ||.f.||

proof end;

theorem Th40: :: LOPBAN_1:40

for X, Y being RealNormSpace
for f being Point of (R_NormSpace_of_BoundedLinearOperators (X,Y)) st f = 0. (R_NormSpace_of_BoundedLinearOperators (X,Y)) holds
0 = ||.f.||

proof end;

registration

let X, Y be RealNormSpace;
cluster -> Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (X,Y));
coherence ;

end;

definition

let X, Y be RealNormSpace;
let f be Element of (R_NormSpace_of_BoundedLinearOperators (X,Y));
let v be VECTOR of X;
:: original: .
redefine func f . v -> VECTOR of Y;
coherence

proof end;

end;

theorem Th41: :: LOPBAN_1:41

for X, Y being RealNormSpace
for f, g, h being Point of (R_NormSpace_of_BoundedLinearOperators (X,Y)) holds
( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )

proof end;

theorem Th42: :: LOPBAN_1:42

for X, Y being RealNormSpace
for f, h being Point of (R_NormSpace_of_BoundedLinearOperators (X,Y))
for a being Real holds
( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) )

proof end;

theorem Th43: :: LOPBAN_1:43

for X, Y being RealNormSpace
for f, g being Point of (R_NormSpace_of_BoundedLinearOperators (X,Y))
for a being Real holds
( ( ||.f.|| = 0 implies f = 0. (R_NormSpace_of_BoundedLinearOperators (X,Y)) ) & ( f = 0. (R_NormSpace_of_BoundedLinearOperators (X,Y)) implies ||.f.|| = 0 ) & ||.(a * f).|| = (abs a) * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )

proof end;

theorem Th44: :: LOPBAN_1:44

for X, Y being RealNormSpace holds
( R_NormSpace_of_BoundedLinearOperators (X,Y) is reflexive & R_NormSpace_of_BoundedLinearOperators (X,Y) is discerning & R_NormSpace_of_BoundedLinearOperators (X,Y) is RealNormSpace-like )

proof end;

theorem Th45: :: LOPBAN_1:45

for X, Y being RealNormSpace holds R_NormSpace_of_BoundedLinearOperators (X,Y) is RealNormSpace

proof end;

registration

let X, Y be RealNormSpace;
cluster R_NormSpace_of_BoundedLinearOperators (X,Y) -> non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ;
coherence by Th45;

end;

theorem Th46: :: LOPBAN_1:46

for X, Y being RealNormSpace
for f, g, h being Point of (R_NormSpace_of_BoundedLinearOperators (X,Y)) holds
( h = f - g iff for x being VECTOR of X holds h . x = (f . x) - (g . x) )

proof end;

begin

definition

let X be RealNormSpace;
attr X is complete means :Def16: :: LOPBAN_1:def 16
for seq being sequence of X st seq is CCauchy holds
seq is convergent ;

end;

:: deftheorem Def16 defines complete LOPBAN_1:def 16 :

registration end;

definition

mode RealBanachSpace is complete RealNormSpace;

end;

Lm3: 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

proof end;

theorem Th47: :: LOPBAN_1:47

for X being RealNormSpace
for seq being sequence of X st seq is convergent holds
( ||.seq.|| is convergent & lim ||.seq.|| = ||.(lim seq).|| )

proof end;

theorem Th48: :: LOPBAN_1:48

for X, Y being RealNormSpace st Y is complete holds
for seq being sequence of (R_NormSpace_of_BoundedLinearOperators (X,Y)) st seq is CCauchy holds
seq is convergent

proof end;

theorem Th49: :: LOPBAN_1:49

for X being RealNormSpace
for Y being RealBanachSpace holds R_NormSpace_of_BoundedLinearOperators (X,Y) is RealBanachSpace

proof end;

registration

let X be RealNormSpace;
let Y be RealBanachSpace;
cluster R_NormSpace_of_BoundedLinearOperators (X,Y) -> non empty complete ;
coherence by Th49;

end;