Bounded: |f(x)|\le M for all x\in \mathbb E.

Uniformly bounded: |f_n(x)|<M for all x\in\mathbb E, n=1, 2, 3,...

Pointwise bounded: |f_n(x)<\phi(x)| where \phi is a finite valued function, and x\in\mathbb E, n=1, 2, 3,...

K is compact, f_n is pointwise bounded and equitcontinuous \Rightarrow f_n is uniformly bounded, and f_n contains a uniformly convergent subsequence.


Continuous: f: \mathbb E\subset X\to Y, p\in\mathbb E, \forall \varepsilon>0, \exists \delta>0 s.t. \forall x\in\mathbb E, d_X(x, p)<\delta\Rightarrow d_Y(f(x), f(p))<\varepsilon.

Uniformly continuous: \forall \varepsilon>0, \exists \delta>0 s.t. d_X(p, q)<\delta\Rightarrow d_Y(f(p), f(q))<\varepsilon.

Equicontinuous: \forall \varepsilon>0, \exists \delta>0 s.t. d(x, y)<\delta\Rightarrow |f(x)-f(y)|<\varepsilon.

Equicontinuous \Rightarrow uniformly continuous.

Compact set, uniformly convergence \Rightarrow equicontinuous.


Convergent: A sequence \{p_n\} in a metric space X, \exists p\in X s.t. \forall \varepsilon>0, \exists N s.t. n\ge N\Rightarrow d(p_n, p)<\varepsilon.

Uniformly convergent (definition 1): a sequence of functions \{f_n\}, n=1, 2, 3, ... converges uniformly on \mathbb E to a function f if \forall \varepsilon>0, \exists N s.t. n\ge N\Rightarrow |f_n(x)-f(x)|\le \varepsilon for all x\in\mathbb E.

Uniformly convergent (definition 2): \varepsilon>0, \exists N s.t. m\ge N, n\ge N, x\in\mathbb E\Rightarrow |f_n(x)-f_m(x)|\le \varepsilon.

Pointwise convergent: f(x)=\lim_{n\to\infty} f(x) (x\in\mathbb E).

If \{f_n\} is uniformly convergent, then \lim_{t\to x}\lim_{n\to\infty}f_n(t)=\lim_{n\to\infty}\lim_{t\to x}f_n(x).

K is compact, \{f_n\} is continuous and pointwise convergent, f_n\ge f_{n+1} \Rightarrow f_n\to f uniformly.

f_n\to f uniformly, where f_n\in\mathcal R Rightarrow \int_a^b f d_\alpha=\lim_{n\to\infty}\int_a^b f_n d\alpha.

\{f_n\} is differentiable, \{f_n'\} converge uniformly on [a, b]\Rightarrow f_n converges uniformly, and $f'(x)=\lim_{n\to\infty}f_n'(x) (a\le x\le b)$.

\{f_n\} is differentiable, f' is uniformly bounded \Rightarrow f_n is equicontinuous.