:: The Differentiable Functions on Normed Linear Spaces
:: by Hiroshi Imura , Morishige Kimura and Yasunari Shidama
::
:: Received May 24, 2004
:: Copyright (c) 2004-2018 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, SUBSET_1, REAL_1, NORMSP_1, SEQ_1, NAT_1, PRE_TOPC,
RCOMP_1, TARSKI, CARD_1, ARYTM_3, ARYTM_1, XXREAL_0, RELAT_1, FUNCT_1,
SEQ_2, ORDINAL2, STRUCT_0, VALUED_0, SUPINF_2, RLVECT_1, VALUED_1,
COMPLEX1, XXREAL_2, FDIFF_1, ZFMISC_1, PARTFUN1, FUNCOP_1, FUNCT_2,
LOPBAN_1, XBOOLE_0, FCONT_1, RSSPACE, NFCONT_1, CFCONT_1, LOPBAN_2;
notations TARSKI, SUBSET_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, PRE_TOPC,
XCMPLX_0, XXREAL_0, XREAL_0, COMPLEX1, ORDINAL1, NUMBERS, REAL_1, NAT_1,
RLVECT_1, STRUCT_0, FUNCOP_1, VALUED_0, VALUED_1, SEQ_1, SEQ_2, VFUNCT_1,
NORMSP_0, NORMSP_1, RSSPACE, LOPBAN_1, LOPBAN_2, NFCONT_1, FDIFF_1,
RECDEF_1;
constructors REAL_1, SQUARE_1, COMPLEX1, SEQ_2, FDIFF_1, RSSPACE, VFUNCT_1,
LOPBAN_3, NFCONT_1, RECDEF_1, RELSET_1, BINOP_2, RVSUM_1, COMSEQ_2,
NUMBERS;
registrations FUNCT_1, ORDINAL1, RELSET_1, NUMBERS, XREAL_0, MEMBERED,
FDIFF_1, STRUCT_0, LOPBAN_1, LOPBAN_2, VALUED_0, VALUED_1, FUNCT_2,
FUNCOP_1, NAT_1;
requirements SUBSET, REAL, BOOLE, NUMERALS, ARITHM;
begin :: Differentiable Function on Normed Linear Spaces
reserve n,k for Element of NAT;
reserve x,y,X for set;
reserve g,r,p for Real;
reserve S for RealNormSpace;
reserve rseq for Real_Sequence;
reserve seq,seq1 for sequence of S;
reserve x0 for Point of S;
reserve Y for Subset of S;
:: Normed Linear Spaces varsions of RCOMP_1:37 -
:: RCOMP_1:37 --> NFCONT_1:4
theorem :: NDIFF_1:1
for x0 be Point of S for N1,N2 being Neighbourhood of x0 ex N
being Neighbourhood of x0 st N c= N1 & N c= N2;
theorem :: NDIFF_1:2
for X being Subset of S st X is open for r be Point of S st r in
X ex N being Neighbourhood of r st N c= X;
theorem :: NDIFF_1:3
for X being Subset of S st X is open for r be Point of S st r in X
holds ex g st 00.S;
theorem :: NDIFF_1:7
seq is non-zero iff for n being Nat holds seq.n<>0.S;
definition
let RNS be RealLinearSpace;
let S be sequence of RNS;
let a be Real_Sequence;
func a (#) S -> sequence of RNS means
:: NDIFF_1:def 2
for n being Nat holds it.n = a.n * S.n;
end;
definition
let RNS be RealLinearSpace;
let z be Point of RNS;
let a be Real_Sequence;
func a * z -> sequence of RNS means
:: NDIFF_1:def 3
for n being Nat holds it.n = a.n * z;
end;
theorem :: NDIFF_1:8
for rseq1,rseq2 be Real_Sequence holds (rseq1+rseq2)(#)seq=rseq1(#)seq
+rseq2(#)seq;
theorem :: NDIFF_1:9
for rseq be Real_Sequence for seq1, seq2 be sequence of S holds
rseq(#)(seq1+seq2)=rseq(#)seq1+rseq(#)seq2;
theorem :: NDIFF_1:10
for rseq be Real_Sequence holds r*(rseq(#)seq) =rseq(#)(r*seq);
theorem :: NDIFF_1:11
for rseq1,rseq2 be Real_Sequence holds (rseq1-rseq2)(#)seq=rseq1(#)seq
-rseq2(#)seq;
theorem :: NDIFF_1:12
for rseq be Real_Sequence for seq1, seq2 be sequence of S holds
rseq(#)(seq1-seq2)=rseq(#)seq1-rseq(#)seq2;
theorem :: NDIFF_1:13
rseq is convergent & seq is convergent implies rseq (#) seq is convergent;
theorem :: NDIFF_1:14
rseq is convergent & seq is convergent implies lim (rseq (#) seq
) = (lim rseq) * (lim seq);
theorem :: NDIFF_1:15
for k being Nat
holds (seq+seq1) ^\k=(seq ^\k) +(seq1 ^\k);
theorem :: NDIFF_1:16
(seq-seq1) ^\k=(seq ^\k) -(seq1 ^\k);
theorem :: NDIFF_1:17
seq is non-zero implies seq ^\k is non-zero;
definition
let S;
let x be Point of S;
let IT be sequence of S;
attr IT is x-convergent means
:: NDIFF_1:def 4
IT is convergent & lim IT = x;
end;
theorem :: NDIFF_1:18
for X be RealNormSpace for seq be sequence of X holds seq is
constant implies ( seq is convergent & for k be Element of NAT holds lim seq =
seq.k );
theorem :: NDIFF_1:19
for r be Real st 0 0.S
holds a*z is non-zero (0.S)-convergent;
theorem :: NDIFF_1:22
(for r be Point of S holds r in Y iff r in the carrier of S) iff Y=the
carrier of S;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1 for sequence of S;
reserve x0 for Point of S;
registration
let S;
cluster constant for sequence of S;
end;
:: Normed Linear Spaces versions of FDIFF_1
reserve h for (0.S)-convergent sequence of S;
reserve c for constant sequence of S;
definition
let S,T;
let IT be PartFunc of S,T;
attr IT is RestFunc-like means
:: NDIFF_1:def 5
IT is total & for h st h is non-zero holds (||.h.||")(#)(IT
/*h) is convergent & lim ((||.h.||")(#)(IT/*h)) = 0.T;
end;
registration
let S,T;
cluster RestFunc-like for PartFunc of S,T;
end;
definition
let S,T;
mode RestFunc of S,T is RestFunc-like PartFunc of S,T;
end;
theorem :: NDIFF_1:23
for R be PartFunc of S,T st R is total holds
R is RestFunc-like iff
for r be Real st r > 0 ex d be Real st d > 0 &
for z be Point of S st z <> 0.S &
||.z.|| < d holds ( ||.z.||"* ||. R/.z .||) < r;
theorem :: NDIFF_1:24
for R be RestFunc of S,T for s be (0.S)-convergent sequence of S
st s is non-zero
holds (R/*s) is convergent & lim (R/*s) = 0.T;
reserve R,R1,R2 for RestFunc of S,T;
reserve L,L1,L2 for Point of R_NormSpace_of_BoundedLinearOperators(S,T);
theorem :: NDIFF_1:25
for h1,h2 be PartFunc of S,T for seq be sequence of S holds h1
is total & h2 is total implies (h1+h2)/*seq = h1/*seq + h2/*seq & (h1-h2)/*seq
= h1/*seq - h2/*seq;
theorem :: NDIFF_1:26
for h be PartFunc of S,T for seq be sequence of S for r be Real
holds h is total implies (r(#)h)/*seq = r*(h/*seq);
theorem :: NDIFF_1:27
f is_continuous_in x0 iff x0 in dom f & for s1 be sequence of S
st rng s1 c= dom f & s1 is convergent & lim s1=x0 &
(for n being Nat holds s1.n<>x0)
holds f/*s1 is convergent & f/.x0=lim(f/*s1);
theorem :: NDIFF_1:28
for R1,R2 holds R1+R2 is RestFunc of S,T & R1-R2 is RestFunc of S,T;
theorem :: NDIFF_1:29
for r,R holds r(#)R is RestFunc of S,T;
definition
let S,T;
let f be PartFunc of S,T;
let x0 be Point of S;
pred f is_differentiable_in x0 means
:: NDIFF_1:def 6
ex N being Neighbourhood of x0
st N c= dom f & ex L,R st for x be Point of S st x in N holds f/.x - f/.x0 = L.
(x-x0) + R/.(x-x0);
end;
definition
let S,T;
let f be PartFunc of S,T;
let x0 be Point of S;
assume
f is_differentiable_in x0;
func diff(f,x0) -> Point of R_NormSpace_of_BoundedLinearOperators(S,T) means
:: NDIFF_1:def 7
ex N being Neighbourhood of x0 st N c= dom f & ex R st for x be Point of
S st x in N holds f/.x-f/.x0 = it.(x-x0) + R/.(x-x0);
end;
definition
let X;
let S,T;
let f be PartFunc of S,T;
pred f is_differentiable_on X means
:: NDIFF_1:def 8
X c=dom f & for x be Point of S st x in X holds f|X is_differentiable_in x;
end;
theorem :: NDIFF_1:30
for f be PartFunc of S,T holds (f is_differentiable_on X implies
X is Subset of S);
theorem :: NDIFF_1:31
for f be PartFunc of S,T for Z be Subset of S st Z is open holds
( f is_differentiable_on Z iff Z c=dom f & for x be Point of S st x in Z holds
f is_differentiable_in x );
theorem :: NDIFF_1:32
for f be PartFunc of S,T for Y be Subset of S holds f
is_differentiable_on Y implies Y is open;
definition
let S,T;
let f be PartFunc of S,T;
let X be set;
assume
f is_differentiable_on X;
func f`|X -> PartFunc of S, R_NormSpace_of_BoundedLinearOperators(S,T) means
:: NDIFF_1:def 9
dom it = X & for x be Point of S st x in X holds it/.x = diff(f,x);
end;
theorem :: NDIFF_1:33
for f be PartFunc of S,T for Z be Subset of S st Z is open & Z c= dom
f & ex r be Point of T st rng f = {r} holds f is_differentiable_on Z & for x be
Point of S st x in Z holds (f`|Z)/.x = 0.R_NormSpace_of_BoundedLinearOperators(
S,T);
registration
let S;
let h,n;
cluster h^\n -> (0.S)-convergent for sequence of S;
end;
registration
let S be non trivial RealNormSpace;
let h be (0.S)-convergent non-zero sequence of S ;
let n;
cluster h^\n -> (0.S)-convergent non-zero for sequence of S;
end;
theorem :: NDIFF_1:34
for x0 be Point of S for N being Neighbourhood of x0 st f
is_differentiable_in x0 & N c= dom f holds
for h be (0.S)-convergent sequence of S st h is non-zero
holds
for c 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;
theorem :: NDIFF_1:35
for f1,f2,x0 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);
theorem :: NDIFF_1:36
for f1,f2,x0 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);
theorem :: NDIFF_1:37
for r,f,x0 st f is_differentiable_in x0 holds r(#)f
is_differentiable_in x0 & diff((r(#)f),x0) = r*diff(f,x0);
theorem :: NDIFF_1:38
for f be PartFunc of S,S for Z be Subset of S st Z is open & Z c= dom
f & f|Z = id Z holds f is_differentiable_on Z & for x be Point of S st x in Z
holds (f`|Z)/.x = id (the carrier of S);
theorem :: NDIFF_1:39
for Z be Subset of S st Z is open for f1,f2 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 be Point of S st x in Z holds ((f1+f2)`|Z)/.x =
diff(f1,x) + diff(f2,x);
theorem :: NDIFF_1:40
for Z be Subset of S st Z is open for f1,f2 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 be Point of S st x in Z holds ((f1-f2)`|Z)/.x =
diff(f1,x) - diff(f2,x);
theorem :: NDIFF_1:41
for Z be Subset of S st Z is open for r,f st Z c= dom (r(#)f) & f
is_differentiable_on Z holds r(#)f is_differentiable_on Z & for x be Point of S
st x in Z holds ((r(#)f)`|Z)/.x =r*diff(f,x);
theorem :: NDIFF_1:42
for Z be Subset of S st Z is open holds (Z c= dom f & f|Z is constant
implies f is_differentiable_on Z & for x be Point of S st x in Z holds (f`|Z)/.
x = 0.R_NormSpace_of_BoundedLinearOperators(S,T));
theorem :: NDIFF_1:43
for f be PartFunc of S,S for r be Real for p be Point of S for Z be
Subset of S st Z is open holds ( Z c= dom f & (for x be Point of S st x in Z
holds f/.x = r*x + p) implies f is_differentiable_on Z & for x be Point of S st
x in Z holds (f`|Z)/.x = r * FuncUnit(S) );
theorem :: NDIFF_1:44
for x0 be Point of S holds f is_differentiable_in x0 implies f
is_continuous_in x0;
theorem :: NDIFF_1:45
f is_differentiable_on X implies f is_continuous_on X;
theorem :: NDIFF_1:46
for Z be Subset of S st Z is open holds ( f is_differentiable_on X & Z
c= X implies f is_differentiable_on Z );
theorem :: NDIFF_1:47
f is_differentiable_in x0 implies ex N being Neighbourhood of x0 st N
c= dom f & ex R st R/.0.S=0.T & R is_continuous_in 0.S & for x be Point of S st
x in N holds f/.x-f/.x0 = diff(f,x0).(x-x0) + R/.(x-x0);
definition
let S be non trivial RealNormSpace ,T;
let IT be PartFunc of S,T;
redefine attr IT is RestFunc-like means
:: NDIFF_1:def 10
IT is total & for h be
non-zero (0.S)-convergent sequence of S
holds (||.h.||")(#)(IT /*h) is convergent
& lim ((||.h.||")(#)(IT/*h)) = 0.T;
end;