let X be set ; for RNS being RealNormSpace
for CNS being ComplexNormSpace
for f being PartFunc of RNS,CNS 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 RNS,CNS st f is_uniformly_continuous_on X holds
||.f.|| is_uniformly_continuous_on X
let CNS be ComplexNormSpace; for f being PartFunc of RNS,CNS st f is_uniformly_continuous_on X holds
||.f.|| is_uniformly_continuous_on X
let f be PartFunc of RNS,CNS; ( 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
;
then A2:
X c= dom ||.f.||
by NORMSP_0:def 3;
for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r ) )
proof
let r be
Real;
( 0 < r implies ex s being Real st
( 0 < s & ( for x1, x2 being Point of RNS 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 RNS 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
RNS st
x1 in X &
x2 in X &
||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < r
by A1;
reconsider s =
s as
Real ;
take
s
;
( 0 < s & ( for x1, x2 being Point of RNS 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 RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r
let x1,
x2 be
Point of
RNS;
( 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 CLVECT_1:110;
||.((f /. x1) - (f /. x2)).|| < r
by A4, A5, A6, A7;
hence
|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r
by A8, XXREAL_0:2;
verum
end;
hence
||.f.|| is_uniformly_continuous_on X
by A2, NFCONT_2:def 2; verum