Bounded Sequences of Functions

Definition (Pointwise Bounded)

A sequence of functions $f_n:E\to \mathbb{C}$ is pointwise bounded if $\forall x\in E,\{f_n{\}}_{n\ge 1}$ is a bounded sequence where $\exists M_x$ such that $|f_n(x)|\le M_x$.

This is dependent on $x$. Meaning, there needs to be a particular $M$ for each $x\in E$.

Definition (Uniformly Bounded)

A sequence of functions $f_n:E\to \mathbb{C}$ is uniformly bounded if $\exists M$ such that $|f_n(x)|\le M$ for any $x\in E$.

This $M$ must be true for all $x\in E$.

Corollary (Uniformly Bounded is Pointwise Bounded)

Uniformly bounded implies pointwise bounded, but not necessarily the converse.

Example 1

For example, let

$$f_n(x)=xE=\mathbb{R}$$

The sequence is pointwise bounded by $x+1$ for each $n$, but not uniformly bounded.

Example 2

Let $f_n(x)=n$ over $E=[0,1]$. The sequence is not pointwise bounded, and so it is not uniformly bounded, but every $f_n$ is bounded.

Example 3

Let $f_n(x)=\sin (nx)$ over $E=[0,1]$. This sequence is uniformly bounded by $1$ and thus pointwise bounded by $1$.