:: by Takaya Nishiyama , Artur Korni{\l}owicz and Yasunari Shidama

::

:: Received April 6, 2004

:: Copyright (c) 2004-2021 Association of Mizar Users

:: deftheorem 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 & ( for r being Real st 0 < r holds

ex s being Real st

( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

||.((f /. x1) - (f /. x2)).|| < r ) ) ) ) );

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 & ( for r being Real st 0 < r holds

ex s being Real st

( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

||.((f /. x1) - (f /. x2)).|| < r ) ) ) ) );

:: deftheorem 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 & ( for r being Real st 0 < r holds

ex s being Real st

( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

|.((f /. x1) - (f /. x2)).| < r ) ) ) ) );

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 & ( for r being Real st 0 < r holds

ex s being Real st

( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

|.((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 & X1 c= X holds

f is_uniformly_continuous_on X1

for S, T being RealNormSpace

for f being PartFunc of S,T st f is_uniformly_continuous_on X & X1 c= X holds

f is_uniformly_continuous_on X1

proof end;

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 & f2 is_uniformly_continuous_on X1 holds

f1 + f2 is_uniformly_continuous_on X /\ X1

for S, T being RealNormSpace

for f1, f2 being PartFunc of S,T st f1 is_uniformly_continuous_on X & f2 is_uniformly_continuous_on X1 holds

f1 + f2 is_uniformly_continuous_on X /\ X1

proof end;

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 & f2 is_uniformly_continuous_on X1 holds

f1 - f2 is_uniformly_continuous_on X /\ X1

for S, T being RealNormSpace

for f1, f2 being PartFunc of S,T st f1 is_uniformly_continuous_on X & f2 is_uniformly_continuous_on X1 holds

f1 - f2 is_uniformly_continuous_on X /\ X1

proof end;

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

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

proof end;

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

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

proof end;

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

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

proof end;

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

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

proof end;

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

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

proof end;

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

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

proof end;

theorem :: NFCONT_2:10

for S, T being RealNormSpace

for f being PartFunc of S,T

for Y being Subset of S st Y is compact & f is_continuous_on Y holds

f is_uniformly_continuous_on Y

for f being PartFunc of S,T

for Y being Subset of S st Y is compact & f is_continuous_on Y holds

f is_uniformly_continuous_on Y

proof end;

theorem :: NFCONT_2:11

for S, T being RealNormSpace

for f being PartFunc of S,T

for Y being Subset of S st Y c= dom f & Y is compact & f is_uniformly_continuous_on Y holds

f .: Y is compact by Th7, NFCONT_1:32;

for f being PartFunc of S,T

for Y being Subset of S st Y c= dom f & Y is compact & 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 <> {} & Y c= dom f & Y is compact & f is_uniformly_continuous_on Y holds

ex x1, x2 being Point of S st

( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) ) by Th8, NFCONT_1:37;

for f being PartFunc of the carrier of S,REAL

for Y being Subset of S st Y <> {} & Y c= dom f & Y is compact & f is_uniformly_continuous_on Y holds

ex x1, x2 being Point of S st

( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & 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 & f | X is constant holds

f is_uniformly_continuous_on X by Th9, NFCONT_1:43;

for S, T being RealNormSpace

for f being PartFunc of S,T st X c= dom f & f | X is constant holds

f is_uniformly_continuous_on X by Th9, NFCONT_1:43;

:: 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 & L < 1 & ( for x, y being Point of M holds ||.((f . x) - (f . y)).|| <= L * ||.(x - y).|| ) ) );

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 & L < 1 & ( for x, y being Point of M holds ||.((f . x) - (f . y)).|| <= L * ||.(x - y).|| ) ) );

registration

let M be RealNormSpace;

ex b_{1} being Function of M,M st b_{1} is contraction

end;
cluster Relation-like the carrier of M -defined the carrier of M -valued non empty Function-like total quasi_total contraction for Element of K16(K17( the carrier of M, the carrier of M));

existence ex b

proof end;

Lm1: ( ( for X being RealNormSpace

for x, y being Point of X holds

( ||.(x - y).|| = 0 iff x = y ) ) & ( for X being RealNormSpace

for x, y being Point of X holds

( ||.(x - y).|| <> 0 iff x <> y ) ) & ( for X being RealNormSpace

for x, y being Point of X holds

( ||.(x - y).|| > 0 iff x <> y ) ) & ( for X being RealNormSpace

for x, y being Point of X holds ||.(x - y).|| = ||.(y - x).|| ) & ( for X being RealNormSpace

for x, y, z being Point of X

for e being Real st e > 0 & ||.(x - z).|| < e / 2 & ||.(z - y).|| < e / 2 holds

||.(x - y).|| < e ) & ( for X being RealNormSpace

for x, y, z being Point of X

for e being Real st e > 0 & ||.(x - z).|| < e / 2 & ||.(y - z).|| < e / 2 holds

||.(x - y).|| < e ) & ( 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 ) & ( 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 ) )

proof end;

Lm2: for K, L, e being Real st 0 < K & K < 1 & 0 < e holds

ex n being Nat st |.(L * (K to_power n)).| < e

proof end;

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 & ( for x being Point of X st f . x = x holds

xp = x ) )

for f being Function of X,X st f is Contraction of X holds

ex xp being Point of X st

( f . xp = xp & ( for x being Point of X st f . x = x holds

xp = x ) )

proof end;

theorem :: NFCONT_2:15

for X being RealBanachSpace

for f being Function of X,X st ex n0 being Nat st iter (f,n0) is Contraction of X holds

ex xp being Point of X st

( f . xp = xp & ( for x being Point of X st f . x = x holds

xp = x ) )

for f being Function of X,X st ex n0 being Nat st iter (f,n0) is Contraction of X holds

ex xp being Point of X st

( f . xp = xp & ( for x being Point of X st f . x = x holds

xp = x ) )

proof end;

theorem :: NFCONT_2:16