:: The Differentiable Functions on Normed Linear Spaces
:: by Hiroshi Imura , Morishige Kimura and Yasunari Shidama
::
:: Received May 24, 2004
:: Copyright (c) 2004-2011 Association of Mizar Users

begin

theorem Th1: :: NDIFF_1:1

for S being RealNormSpace
for x0 being Point of S
for N1, N2 being Neighbourhood of x0 ex N being Neighbourhood of x0 st
( N c= N1 & N c= N2 )

proof end;

theorem Th2: :: NDIFF_1:2

for S being RealNormSpace
for X being Subset of S st X is open holds
for r being Point of S st r in X holds
ex N being Neighbourhood of r st N c= X

proof end;

theorem :: NDIFF_1:3

for S being RealNormSpace
for X being Subset of S st X is open holds
for r being Point of S st r in X holds
ex g being Real st
( 0 < g & { y where y is Point of S : ||.(y - r).|| < g } c= X )

proof end;

theorem Th4: :: NDIFF_1:4

for S being RealNormSpace
for X being Subset of S st ( for r being Point of S st r in X holds
ex N being Neighbourhood of r st N c= X ) holds
X is open

proof end;

theorem :: NDIFF_1:5

for S being RealNormSpace
for X being Subset of S holds
( ( for r being Point of S st r in X holds
ex N being Neighbourhood of r st N c= X ) iff X is open ) by Th2, Th4;

definition

let X be set ;
let S be ZeroStr ;
let f be Function of X,S;
redefine attr f is non-zero means :Def1: :: NDIFF_1:def 1
rng f c= NonZero S;
compatibility
( f is non-zero iff rng f c= NonZero S )

proof end;

end;

:: deftheorem Def1 defines non-zero NDIFF_1:def 1 :
for X being set
for S being ZeroStr
for f being Function of X,S holds
( f is non-zero iff rng f c= NonZero S );

theorem Th6: :: NDIFF_1:6

for S being RealNormSpace
for seq being sequence of S holds
( seq is non-zero iff for x being set st x in NAT holds
seq . x <> 0. S )

proof end;

theorem Th7: :: NDIFF_1:7

for S being RealNormSpace
for seq being sequence of S holds
( seq is non-zero iff for n being Element of NAT holds seq . n <> 0. S )

proof end;

definition

let RNS be RealLinearSpace;
let S be sequence of RNS;
let a be Real_Sequence;
func a (#) S -> sequence of RNS means :Def2: :: NDIFF_1:def 2
for n being Element of NAT holds it . n = (a . n) * (S . n);
existence
ex b₁ being sequence of RNS st
for n being Element of NAT holds b₁ . n = (a . n) * (S . n)

proof end;

uniqueness
for b₁, b₂ being sequence of RNS st ( for n being Element of NAT holds b₁ . n = (a . n) * (S . n) ) & ( for n being Element of NAT holds b₂ . n = (a . n) * (S . n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines (#) NDIFF_1:def 2 :
for RNS being RealLinearSpace
for S being sequence of RNS
for a being Real_Sequence
for b₄ being sequence of RNS holds
( b₄ = a (#) S iff for n being Element of NAT holds b₄ . n = (a . n) * (S . n) );

definition

let RNS be RealLinearSpace;
let z be Point of RNS;
let a be Real_Sequence;
func a * z -> sequence of RNS means :Def3: :: NDIFF_1:def 3
for n being Element of NAT holds it . n = (a . n) * z;
existence
ex b₁ being sequence of RNS st
for n being Element of NAT holds b₁ . n = (a . n) * z

proof end;

uniqueness
for b₁, b₂ being sequence of RNS st ( for n being Element of NAT holds b₁ . n = (a . n) * z ) & ( for n being Element of NAT holds b₂ . n = (a . n) * z ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def3 defines * NDIFF_1:def 3 :
for RNS being RealLinearSpace
for z being Point of RNS
for a being Real_Sequence
for b₄ being sequence of RNS holds
( b₄ = a * z iff for n being Element of NAT holds b₄ . n = (a . n) * z );

theorem :: NDIFF_1:8

for S being RealNormSpace
for seq being sequence of S
for rseq1, rseq2 being Real_Sequence holds (rseq1 + rseq2) (#) seq = (rseq1 (#) seq) + (rseq2 (#) seq)

proof end;

theorem Th9: :: NDIFF_1:9

for S being RealNormSpace
for rseq being Real_Sequence
for seq1, seq2 being sequence of S holds rseq (#) (seq1 + seq2) = (rseq (#) seq1) + (rseq (#) seq2)

proof end;

theorem Th10: :: NDIFF_1:10

for r being Real
for S being RealNormSpace
for seq being sequence of S
for rseq being Real_Sequence holds r * (rseq (#) seq) = rseq (#) (r * seq)

proof end;

theorem :: NDIFF_1:11

for S being RealNormSpace
for seq being sequence of S
for rseq1, rseq2 being Real_Sequence holds (rseq1 - rseq2) (#) seq = (rseq1 (#) seq) - (rseq2 (#) seq)

proof end;

theorem Th12: :: NDIFF_1:12

for S being RealNormSpace
for rseq being Real_Sequence
for seq1, seq2 being sequence of S holds rseq (#) (seq1 - seq2) = (rseq (#) seq1) - (rseq (#) seq2)

proof end;

theorem Th13: :: NDIFF_1:13

for S being RealNormSpace
for rseq being Real_Sequence
for seq being sequence of S st rseq is convergent & seq is convergent holds
rseq (#) seq is convergent

proof end;

theorem Th14: :: NDIFF_1:14

for S being RealNormSpace
for rseq being Real_Sequence
for seq being sequence of S st rseq is convergent & seq is convergent holds
lim (rseq (#) seq) = (lim rseq) * (lim seq)

proof end;

theorem Th15: :: NDIFF_1:15

for k being Element of NAT
for S being RealNormSpace
for seq, seq1 being sequence of S holds (seq + seq1) ^\ k = (seq ^\ k) + (seq1 ^\ k)

proof end;

theorem Th16: :: NDIFF_1:16

for k being Element of NAT
for S being RealNormSpace
for seq, seq1 being sequence of S holds (seq - seq1) ^\ k = (seq ^\ k) - (seq1 ^\ k)

proof end;

theorem Th17: :: NDIFF_1:17

for k being Element of NAT
for S being RealNormSpace
for seq being sequence of S st seq is non-zero holds
seq ^\ k is non-zero

proof end;

theorem :: NDIFF_1:18

canceled;

theorem :: NDIFF_1:19

canceled;

theorem :: NDIFF_1:20

canceled;

definition

let S be RealNormSpace;
let IT be sequence of S;
attr IT is convergent_to_0 means :Def4: :: NDIFF_1:def 4
( IT is non-zero & IT is convergent & lim IT = 0. S );

end;

:: deftheorem Def4 defines convergent_to_0 NDIFF_1:def 4 :
for S being RealNormSpace
for IT being sequence of S holds
( IT is convergent_to_0 iff ( IT is non-zero & IT is convergent & lim IT = 0. S ) );

theorem Th21: :: NDIFF_1:21

for X being RealNormSpace
for seq being sequence of X st seq is constant holds
( seq is convergent & ( for k being Element of NAT holds lim seq = seq . k ) )

proof end;

theorem Th22: :: NDIFF_1:22

for S being RealNormSpace
for seq being sequence of S
for x0 being Point of S
for r being Real st 0 < r & ( for n being Element of NAT holds seq . n = (1 / (n + r)) * x0 ) holds
seq is convergent

proof end;

theorem Th23: :: NDIFF_1:23

for S being RealNormSpace
for seq being sequence of S
for x0 being Point of S
for r being Real st 0 < r & ( for n being Element of NAT holds seq . n = (1 / (n + r)) * x0 ) holds
lim seq = 0. S

proof end;

theorem :: NDIFF_1:24

for S being RealNormSpace
for a being convergent_to_0 Real_Sequence
for z being Point of S st z <> 0. S holds
a * z is convergent_to_0

proof end;

theorem :: NDIFF_1:25

for S being RealNormSpace
for Y being Subset of S holds
( ( for r being Point of S holds
( r in Y iff r in the carrier of S ) ) iff Y = the carrier of S )

proof end;

registration

let S be non trivial RealNormSpace;
cluster V1() V4( NAT ) V5( the carrier of S) Function-like V11() total quasi_total convergent_to_0 Element of K21(K22(NAT, the carrier of S));
existence
ex b₁ being sequence of S st b₁ is convergent_to_0

proof end;

end;

registration

let S be non trivial RealNormSpace;
cluster V1() V4( NAT ) V5( the carrier of S) Function-like V8() V11() total quasi_total Element of K21(K22(NAT, the carrier of S));
existence
ex b₁ being sequence of S st b₁ is constant

proof end;

end;

definition

let S, T be non trivial RealNormSpace;
let IT be PartFunc of S,T;
attr IT is REST-like means :Def5: :: NDIFF_1:def 5
( IT is total & ( for h being convergent_to_0 sequence of S holds
( (||.h.|| ") (#) (IT /* h) is convergent & lim ((||.h.|| ") (#) (IT /* h)) = 0. T ) ) );

end;

:: deftheorem Def5 defines REST-like NDIFF_1:def 5 :
for S, T being non trivial RealNormSpace
for IT being PartFunc of S,T holds
( IT is REST-like iff ( IT is total & ( for h being convergent_to_0 sequence of S holds
( (||.h.|| ") (#) (IT /* h) is convergent & lim ((||.h.|| ") (#) (IT /* h)) = 0. T ) ) ) );

registration

let S, T be non trivial RealNormSpace;
cluster V1() V4( the carrier of S) V5( the carrier of T) Function-like REST-like Element of K21(K22( the carrier of S, the carrier of T));
existence
ex b₁ being PartFunc of S,T st b₁ is REST-like

proof end;

end;

definition

let S, T be non trivial RealNormSpace;
mode REST of S,T is REST-like PartFunc of S,T;

end;

theorem :: NDIFF_1:26

for S, T being non trivial RealNormSpace
for R being PartFunc of S,T st R is total holds
( R is REST-like iff for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Point of S st z <> 0. S & ||.z.|| < d holds
(||.z.|| ") * ||.(R /. z).|| < r ) ) )

proof end;

theorem Th27: :: NDIFF_1:27

for S, T being non trivial RealNormSpace
for R being REST of S,T
for s being convergent_to_0 sequence of S holds
( R /* s is convergent & lim (R /* s) = 0. T )

proof end;

theorem :: NDIFF_1:28

canceled;

theorem :: NDIFF_1:29

canceled;

theorem Th30: :: NDIFF_1:30

for S, T being non trivial RealNormSpace
for h1, h2 being PartFunc of S,T
for seq being sequence of S st h1 is total & h2 is total holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )

proof end;

theorem Th31: :: NDIFF_1:31

for S, T being non trivial RealNormSpace
for h being PartFunc of S,T
for seq being sequence of S
for r being Real st h is total holds
(r (#) h) /* seq = r * (h /* seq)

proof end;

theorem Th32: :: NDIFF_1:32

for T, S being non trivial RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for s1 being sequence of S st rng s1 c= dom f & s1 is convergent & lim s1 = x0 & ( for n being Element of NAT holds s1 . n <> x0 ) holds
( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) )

proof end;

theorem Th33: :: NDIFF_1:33

for T, S being non trivial RealNormSpace
for R1, R2 being REST of S,T holds
( R1 + R2 is REST of S,T & R1 - R2 is REST of S,T )

proof end;

theorem Th34: :: NDIFF_1:34

for T, S being non trivial RealNormSpace
for r being Real
for R being REST of S,T holds r (#) R is REST of S,T

proof end;

definition

let S, T be non trivial RealNormSpace;
let f be PartFunc of S,T;
let x0 be Point of S;
pred f is_differentiable_in x0 means :Def6: :: NDIFF_1:def 6
ex N being Neighbourhood of x0 st
( N c= dom f & ex L being Point of (R_NormSpace_of_BoundedLinearOperators (S,T)) ex R being REST of S,T st
for x being Point of S st x in N holds
(f /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) );

end;

:: deftheorem Def6 defines is_differentiable_in NDIFF_1:def 6 :
for S, T being non trivial RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S holds
( f is_differentiable_in x0 iff ex N being Neighbourhood of x0 st
( N c= dom f & ex L being Point of (R_NormSpace_of_BoundedLinearOperators (S,T)) ex R being REST of S,T st
for x being Point of S st x in N holds
(f /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) ) );

definition

let S, T be non trivial RealNormSpace;
let f be PartFunc of S,T;
let x0 be Point of S;
assume A1: f is_differentiable_in x0 ;
func diff (f,x0) -> Point of (R_NormSpace_of_BoundedLinearOperators (S,T)) means :Def7: :: NDIFF_1:def 7
ex N being Neighbourhood of x0 st
( N c= dom f & ex R being REST of S,T st
for x being Point of S st x in N holds
(f /. x) - (f /. x0) = (it . (x - x0)) + (R /. (x - x0)) );
existence
ex b₁ being Point of (R_NormSpace_of_BoundedLinearOperators (S,T)) ex N being Neighbourhood of x0 st
( N c= dom f & ex R being REST of S,T st
for x being Point of S st x in N holds
(f /. x) - (f /. x0) = (b₁ . (x - x0)) + (R /. (x - x0)) )

proof end;

uniqueness
for b₁, b₂ being Point of (R_NormSpace_of_BoundedLinearOperators (S,T)) st ex N being Neighbourhood of x0 st
( N c= dom f & ex R being REST of S,T st
for x being Point of S st x in N holds
(f /. x) - (f /. x0) = (b₁ . (x - x0)) + (R /. (x - x0)) ) & ex N being Neighbourhood of x0 st
( N c= dom f & ex R being REST of S,T st
for x being Point of S st x in N holds
(f /. x) - (f /. x0) = (b₂ . (x - x0)) + (R /. (x - x0)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def7 defines diff NDIFF_1:def 7 :
for S, T being non trivial RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S st f is_differentiable_in x0 holds
for b₅ being Point of (R_NormSpace_of_BoundedLinearOperators (S,T)) holds
( b₅ = diff (f,x0) iff ex N being Neighbourhood of x0 st
( N c= dom f & ex R being REST of S,T st
for x being Point of S st x in N holds
(f /. x) - (f /. x0) = (b₅ . (x - x0)) + (R /. (x - x0)) ) );

definition

let X be set ;
let S, T be non trivial RealNormSpace;
let f be PartFunc of S,T;
pred f is_differentiable_on X means :Def8: :: NDIFF_1:def 8
( X c= dom f & ( for x being Point of S st x in X holds
f | X is_differentiable_in x ) );

end;

:: deftheorem Def8 defines is_differentiable_on NDIFF_1:def 8 :
for X being set
for S, T being non trivial RealNormSpace
for f being PartFunc of S,T holds
( f is_differentiable_on X iff ( X c= dom f & ( for x being Point of S st x in X holds
f | X is_differentiable_in x ) ) );

theorem Th35: :: NDIFF_1:35

for X being set
for S, T being non trivial RealNormSpace
for f being PartFunc of S,T st f is_differentiable_on X holds
X is Subset of S

proof end;

theorem Th36: :: NDIFF_1:36

for S, T being non trivial RealNormSpace
for f being PartFunc of S,T
for Z being Subset of S st Z is open holds
( f is_differentiable_on Z iff ( Z c= dom f & ( for x being Point of S st x in Z holds
f is_differentiable_in x ) ) )

proof end;

theorem :: NDIFF_1:37

for S, T being non trivial RealNormSpace
for f being PartFunc of S,T
for Y being Subset of S st f is_differentiable_on Y holds
Y is open

proof end;

definition

let S, T be non trivial RealNormSpace;
let f be PartFunc of S,T;
let X be set ;
assume A1: f is_differentiable_on X ;
func f `| X -> PartFunc of S,(R_NormSpace_of_BoundedLinearOperators (S,T)) means :Def9: :: NDIFF_1:def 9
( dom it = X & ( for x being Point of S st x in X holds
it /. x = diff (f,x) ) );
existence
ex b₁ being PartFunc of S,(R_NormSpace_of_BoundedLinearOperators (S,T)) st
( dom b₁ = X & ( for x being Point of S st x in X holds
b₁ /. x = diff (f,x) ) )

proof end;

uniqueness
for b₁, b₂ being PartFunc of S,(R_NormSpace_of_BoundedLinearOperators (S,T)) st dom b₁ = X & ( for x being Point of S st x in X holds
b₁ /. x = diff (f,x) ) & dom b₂ = X & ( for x being Point of S st x in X holds
b₂ /. x = diff (f,x) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def9 defines `| NDIFF_1:def 9 :
for S, T being non trivial RealNormSpace
for f being PartFunc of S,T
for X being set st f is_differentiable_on X holds
for b<sub>5</sub> being PartFunc of S,(R_NormSpace_of_BoundedLinearOperators (S,T)) holds
( b<sub>5</sub> = f `| X iff ( dom b₅ = X & ( for x being Point of S st x in X holds
b₅ /. x = diff (f,x) ) ) );

theorem :: NDIFF_1:38

for S, T being non trivial RealNormSpace
for f being PartFunc of S,T
for Z being Subset of S st Z is open & Z c= dom f & ex r being Point of T st rng f = {r} holds
( f is_differentiable_on Z & ( for x being Point of S st x in Z holds
(f `| Z) /. x = 0. (R_NormSpace_of_BoundedLinearOperators (S,T)) ) )

proof end;

registration

let S be non trivial RealNormSpace;
let h be convergent_to_0 sequence of S;
let n be Element of NAT ;
cluster h ^\ n -> convergent_to_0 ;
coherence
h ^\ n is convergent_to_0

proof end;

end;

theorem Th39: :: NDIFF_1:39

for S, T being non trivial RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S
for N being Neighbourhood of x0 st f is_differentiable_in x0 & N c= dom f holds
for h being convergent_to_0 sequence of S
for c being V8() sequence of S st rng c = {x0} & rng (h + c) c= N holds
( (f /* (h + c)) - (f /* c) is convergent & lim ((f /* (h + c)) - (f /* c)) = 0. T )

proof end;

theorem Th40: :: NDIFF_1:40

for T, S being non trivial RealNormSpace
for f1, f2 being PartFunc of S,T
for x0 being Point of S st f1 is_differentiable_in x0 & f2 is_differentiable_in x0 holds
( f1 + f2 is_differentiable_in x0 & diff ((f1 + f2),x0) = (diff (f1,x0)) + (diff (f2,x0)) )

proof end;

theorem Th41: :: NDIFF_1:41

for T, S being non trivial RealNormSpace
for f1, f2 being PartFunc of S,T
for x0 being Point of S st f1 is_differentiable_in x0 & f2 is_differentiable_in x0 holds
( f1 - f2 is_differentiable_in x0 & diff ((f1 - f2),x0) = (diff (f1,x0)) - (diff (f2,x0)) )

proof end;

theorem Th42: :: NDIFF_1:42

for T, S being non trivial RealNormSpace
for r being Real
for f being PartFunc of S,T
for x0 being Point of S st f is_differentiable_in x0 holds
( r (#) f is_differentiable_in x0 & diff ((r (#) f),x0) = r * (diff (f,x0)) )

proof end;

theorem :: NDIFF_1:43

for S being non trivial RealNormSpace
for f being PartFunc of S,S
for Z being Subset of S st Z is open & Z c= dom f & f | Z = id Z holds
( f is_differentiable_on Z & ( for x being Point of S st x in Z holds
(f `| Z) /. x = id the carrier of S ) )

proof end;

theorem :: NDIFF_1:44

for S, T being non trivial RealNormSpace
for Z being Subset of S st Z is open holds
for f1, f2 being PartFunc of S,T st Z c= dom (f1 + f2) & f1 is_differentiable_on Z & f2 is_differentiable_on Z holds
( f1 + f2 is_differentiable_on Z & ( for x being Point of S st x in Z holds
((f1 + f2) `| Z) /. x = (diff (f1,x)) + (diff (f2,x)) ) )

proof end;

theorem :: NDIFF_1:45

for S, T being non trivial RealNormSpace
for Z being Subset of S st Z is open holds
for f1, f2 being PartFunc of S,T st Z c= dom (f1 - f2) & f1 is_differentiable_on Z & f2 is_differentiable_on Z holds
( f1 - f2 is_differentiable_on Z & ( for x being Point of S st x in Z holds
((f1 - f2) `| Z) /. x = (diff (f1,x)) - (diff (f2,x)) ) )

proof end;

theorem :: NDIFF_1:46

for S, T being non trivial RealNormSpace
for Z being Subset of S st Z is open holds
for r being Real
for f being PartFunc of S,T st Z c= dom (r (#) f) & f is_differentiable_on Z holds
( r (#) f is_differentiable_on Z & ( for x being Point of S st x in Z holds
((r (#) f) `| Z) /. x = r * (diff (f,x)) ) )

proof end;

theorem :: NDIFF_1:47

for S, T being non trivial RealNormSpace
for f being PartFunc of S,T
for Z being Subset of S st Z is open & Z c= dom f & f | Z is constant holds
( f is_differentiable_on Z & ( for x being Point of S st x in Z holds
(f `| Z) /. x = 0. (R_NormSpace_of_BoundedLinearOperators (S,T)) ) )

proof end;

theorem :: NDIFF_1:48

for S being non trivial RealNormSpace
for f being PartFunc of S,S
for r being Real
for p being Point of S
for Z being Subset of S st Z is open & Z c= dom f & ( for x being Point of S st x in Z holds
f /. x = (r * x) + p ) holds
( f is_differentiable_on Z & ( for x being Point of S st x in Z holds
(f `| Z) /. x = r * (FuncUnit S) ) )

proof end;

theorem Th49: :: NDIFF_1:49

for T, S being non trivial RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S st f is_differentiable_in x0 holds
f is_continuous_in x0

proof end;

theorem :: NDIFF_1:50

for X being set
for S, T being non trivial RealNormSpace
for f being PartFunc of S,T st f is_differentiable_on X holds
f is_continuous_on X

proof end;

theorem :: NDIFF_1:51

for X being set
for T, S being non trivial RealNormSpace
for f being PartFunc of S,T
for Z being Subset of S st Z is open & f is_differentiable_on X & Z c= X holds
f is_differentiable_on Z

proof end;

theorem :: NDIFF_1:52

for T, S being non trivial RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S st f is_differentiable_in x0 holds
ex N being Neighbourhood of x0 st
( N c= dom f & ex R being REST of S,T st
( R /. (0. S) = 0. T & R is_continuous_in 0. S & ( for x being Point of S st x in N holds
(f /. x) - (f /. x0) = ((diff (f,x0)) . (x - x0)) + (R /. (x - x0)) ) ) )

proof end;