:: The Uniform Continuity of Functions on Normed Linear Spaces
:: by Takaya Nishiyama , Artur Korni{\l}owicz and Yasunari Shidama
::
:: Received April 6, 2004
:: Copyright (c) 2004-2012 Association of Mizar Users


begin

definition
let X be set ;
let S, T be RealNormSpace;
let f be PartFunc of S,T;
predf is_uniformly_continuous_on X means :Def1: :: NFCONT_2:def 1
( X c= dom f &amp; ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s &amp; ( for x1, x2 being Point of S st x1 in X &amp; x2 in X &amp; ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < r ) ) ) );
end;

:: deftheorem Def1 defines is_uniformly_continuous_on NFCONT_2:def_1_:_
 for X being set
for S, T being RealNormSpace
for f being PartFunc of S,T holds
( f is_uniformly_continuous_on X iff ( X c= dom f &amp; ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s &amp; ( for x1, x2 being Point of S st x1 in X &amp; x2 in X &amp; ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < r ) ) ) ) );

definition
let X be set ;
let S be RealNormSpace;
let f be PartFunc of the carrier of S,REAL;
predf is_uniformly_continuous_on X means :Def2: :: NFCONT_2:def 2
( X c= dom f &amp; ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s &amp; ( for x1, x2 being Point of S st x1 in X &amp; x2 in X &amp; ||.(x1 - x2).|| < s holds
abs ((f /. x1) - (f /. x2)) < r ) ) ) );
end;

:: deftheorem Def2 defines is_uniformly_continuous_on NFCONT_2:def_2_:_
 for X being set
for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL holds
( f is_uniformly_continuous_on X iff ( X c= dom f &amp; ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s &amp; ( for x1, x2 being Point of S st x1 in X &amp; x2 in X &amp; ||.(x1 - x2).|| < s holds
abs ((f /. x1) - (f /. x2)) < r ) ) ) ) );

theorem Th1: :: NFCONT_2:1
for X, X1 being set
for S, T being RealNormSpace
for f being PartFunc of S,T st f is_uniformly_continuous_on X &amp; X1 c= X holds
f is_uniformly_continuous_on X1
proofend;

theorem :: NFCONT_2:2
for X, X1 being set
for S, T being RealNormSpace
for f1, f2 being PartFunc of S,T st f1 is_uniformly_continuous_on X &amp; f2 is_uniformly_continuous_on X1 holds
f1 + f2 is_uniformly_continuous_on X /\ X1
proofend;

theorem :: NFCONT_2:3
for X, X1 being set
for S, T being RealNormSpace
for f1, f2 being PartFunc of S,T st f1 is_uniformly_continuous_on X &amp; f2 is_uniformly_continuous_on X1 holds
f1 - f2 is_uniformly_continuous_on X /\ X1
proofend;

theorem Th4: :: NFCONT_2:4
for X being set
for p being Real
for S, T being RealNormSpace
for f being PartFunc of S,T st f is_uniformly_continuous_on X holds
p (#) f is_uniformly_continuous_on X
proofend;

theorem :: NFCONT_2:5
for X being set
for S, T being RealNormSpace
for f being PartFunc of S,T st f is_uniformly_continuous_on X holds
- f is_uniformly_continuous_on X
proofend;

theorem :: NFCONT_2:6
for X being set
for S, T being RealNormSpace
for f being PartFunc of S,T st f is_uniformly_continuous_on X holds
||.f.|| is_uniformly_continuous_on X
proofend;

theorem Th7: :: NFCONT_2:7
for X being set
for S, T being RealNormSpace
for f being PartFunc of S,T st f is_uniformly_continuous_on X holds
f is_continuous_on X
proofend;

theorem Th8: :: NFCONT_2:8
for X being set
for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL st f is_uniformly_continuous_on X holds
f is_continuous_on X
proofend;

theorem Th9: :: NFCONT_2:9
for X being set
for S, T being RealNormSpace
for f being PartFunc of S,T st f is_Lipschitzian_on X holds
f is_uniformly_continuous_on X
proofend;

theorem :: NFCONT_2:10
for T, S being RealNormSpace
for f being PartFunc of S,T
for Y being Subset of S st Y is compact &amp; f is_continuous_on Y holds
f is_uniformly_continuous_on Y
proofend;

theorem :: NFCONT_2:11
for T, S being RealNormSpace
for f being PartFunc of S,T
for Y being Subset of S st Y c= dom f &amp; Y is compact &amp; f is_uniformly_continuous_on Y holds
f .: Y is compact by Th7, NFCONT_1:32;

theorem :: NFCONT_2:12
for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL
for Y being Subset of S st Y <> {} &amp; Y c= dom f &amp; Y is compact &amp; f is_uniformly_continuous_on Y holds
ex x1, x2 being Point of S st
( x1 in Y &amp; x2 in Y &amp; f /. x1 = upper_bound (f .: Y) &amp; f /. x2 = lower_bound (f .: Y) ) by Th8, NFCONT_1:37;

theorem :: NFCONT_2:13
for X being set
for S, T being RealNormSpace
for f being PartFunc of S,T st X c= dom f &amp; f | X is constant holds
f is_uniformly_continuous_on X by Th9, NFCONT_1:43;

begin

definition
let M be non empty NORMSTR ;
let f be Function of M,M;
attrf is contraction  means :Def3: :: NFCONT_2:def 3
 ex L being Real st
( 0 < L &amp; L < 1 &amp; ( for x, y being Point of M holds ||.((f . x) - (f . y)).|| <= L * ||.(x - y).|| ) );
end;

:: deftheorem Def3 defines contraction NFCONT_2:def_3_:_
 for M being non empty NORMSTR
for f being Function of M,M holds
( f is contraction iff ex L being Real st
( 0 < L &amp; L < 1 &amp; ( for x, y being Point of M holds ||.((f . x) - (f . y)).|| <= L * ||.(x - y).|| ) ) );

registration
let M be RealNormSpace;
cluster non empty Relation-like the carrier of M -defined the carrier of M -valued Function-like total quasi_total contraction for Element of K6(K7( the carrier of M, the carrier of M));
existence
 ex b1 being Function of M,M st b1 is contraction
proofend;
end;

definition
let M be RealNormSpace;
mode Contraction of M is contraction Function of M,M;
end;

Lm1:  ( ( for X being RealNormSpace
for x, y being Point of X holds
( ||.(x - y).|| = 0 iff x = y ) ) &amp; ( for X being RealNormSpace
for x, y being Point of X holds
( ||.(x - y).|| <> 0 iff x <> y ) ) &amp; ( for X being RealNormSpace
for x, y being Point of X holds
( ||.(x - y).|| > 0 iff x <> y ) ) &amp; ( for X being RealNormSpace
for x, y being Point of X holds ||.(x - y).|| = ||.(y - x).|| ) &amp; ( for X being RealNormSpace
for x, y, z being Point of X
for e being Real st e > 0 &amp; ||.(x - z).|| < e / 2 &amp; ||.(z - y).|| < e / 2 holds
||.(x - y).|| < e ) &amp; ( for X being RealNormSpace
for x, y, z being Point of X
for e being Real st e > 0 &amp; ||.(x - z).|| < e / 2 &amp; ||.(y - z).|| < e / 2 holds
||.(x - y).|| < e ) &amp; ( for X being RealNormSpace
for x being Point of X st ( for e being Real st e > 0 holds
||.x.|| < e ) holds
x = 0. X ) &amp; ( for X being RealNormSpace
for x, y being Point of X st ( for e being Real st e > 0 holds
||.(x - y).|| < e ) holds
x = y ) )
proofend;

Lm2:  for K, L, e being real number st 0 < K &amp; K < 1 &amp; 0 < e holds
ex n being Element of NAT st abs (L * (K to_power n)) < e
proofend;

theorem Th14: :: NFCONT_2:14
for X being RealBanachSpace
for f being Function of X,X st f is Contraction of X holds
ex xp being Point of X st
( f . xp = xp &amp; ( for x being Point of X st f . x = x holds
xp = x ) )
proofend;

theorem :: NFCONT_2:15
for X being RealBanachSpace
for f being Function of X,X st ex n0 being Element of NAT st iter (f,n0) is Contraction of X holds
ex xp being Point of X st
( f . xp = xp &amp; ( for x being Point of X st f . x = x holds
xp = x ) )
proofend;

theorem :: NFCONT_2:16
for K, L, e being real number st 0 < K &amp; K < 1 &amp; 0 < e holds
ex n being Element of NAT st abs (L * (K to_power n)) < e by Lm2;