let X be set ; for RNS being RealNormSpace
for CNS being ComplexNormSpace
for f being PartFunc of CNS,RNS st f is_uniformly_continuous_on X holds
||.f.|| is_uniformly_continuous_on X
let RNS be RealNormSpace; for CNS being ComplexNormSpace
for f being PartFunc of CNS,RNS st f is_uniformly_continuous_on X holds
||.f.|| is_uniformly_continuous_on X
let CNS be ComplexNormSpace; for f being PartFunc of CNS,RNS st f is_uniformly_continuous_on X holds
||.f.|| is_uniformly_continuous_on X
let f be PartFunc of CNS,RNS; ( f is_uniformly_continuous_on X implies ||.f.|| is_uniformly_continuous_on X )
assume A1:
f is_uniformly_continuous_on X
; ||.f.|| is_uniformly_continuous_on X
then
X c= dom f
;
hence A2:
X c= dom ||.f.||
by NORMSP_0:def 3; NCFCONT2:def 5 for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r ) )
let r be Real; ( 0 < r implies ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r ) ) )
assume
0 < r
; ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r ) )
then consider s being Real such that
A3:
0 < s
and
A4:
for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < r
by A1;
take
s
; ( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r ) )
thus
0 < s
by A3; for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r
let x1, x2 be Point of CNS; ( x1 in X & x2 in X & ||.(x1 - x2).|| < s implies |.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r )
assume that
A5:
x1 in X
and
A6:
x2 in X
and
A7:
||.(x1 - x2).|| < s
; |.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r
|.((||.f.|| /. x1) - (||.f.|| /. x2)).| =
|.((||.f.|| . x1) - (||.f.|| /. x2)).|
by A2, A5, PARTFUN1:def 6
.=
|.((||.f.|| . x1) - (||.f.|| . x2)).|
by A2, A6, PARTFUN1:def 6
.=
|.(||.(f /. x1).|| - (||.f.|| . x2)).|
by A2, A5, NORMSP_0:def 3
.=
|.(||.(f /. x1).|| - ||.(f /. x2).||).|
by A2, A6, NORMSP_0:def 3
;
then A8:
|.((||.f.|| /. x1) - (||.f.|| /. x2)).| <= ||.((f /. x1) - (f /. x2)).||
by NORMSP_1:9;
||.((f /. x1) - (f /. x2)).|| < r
by A4, A5, A6, A7;
hence
|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r
by A8, XXREAL_0:2; verum