Bounded: for all
.
Uniformly bounded: for all
,
Pointwise bounded: where
is a finite valued function, and
,
is compact,
is pointwise bounded and equitcontinuous
is uniformly bounded, and
contains a uniformly convergent subsequence.
Continuous: s.t.
.
Uniformly continuous: s.t.
.
Equicontinuous: s.t.
.
Equicontinuous uniformly continuous.
Compact set, uniformly convergence equicontinuous.
Convergent: A sequence in a metric space
s.t.
s.t.
.
Uniformly convergent (definition 1): a sequence of functions converges uniformly on
to a function
if
s.t.
for all
.
Uniformly convergent (definition 2): s.t.
.
Pointwise convergent: .
If is uniformly convergent, then
.
is compact,
is continuous and pointwise convergent,
uniformly.
uniformly, where
.
is differentiable,
converge uniformly on
converges uniformly, and $f'(x)=\lim_{n\to\infty}f_n'(x) (a\le x\le b)$.
is differentiable,
is uniformly bounded
is equicontinuous.