What is the difference between totally bounded and uniformly bounded?

Solution 1:

To illustrate the concepts, I consider real functions in one real variable in the following. Of course this carries over to arbitrarly generalized contexts (domains in $\mathbb{R}^n$, metric spaces, Banach spaces, whatever).

A single function $f:\mathbb{R}\rightarrow\mathbb{R}$ is bounded, if there exists a constant $C\ge 0$ such that $|f(x)|\le C$ for all $x\in\mathbb{R}$.

The term uniformly bounded only makes sense if you are considering an object that depends on at least one additional parameter, e.g. a sequence of functions $(f_k)_k$ ($f_k(x)$ depends on the index $k$ and on $x$).

A sequence $(f_k:\mathbb{R}\rightarrow\mathbb{R})_k$ of functions is uniformly bounded if there exists a constant $C\ge 0$ s.t. for all $k$ we have $|f_k(x)|\le C$ for all $x\in\mathbb{R}$. The important thing here is that C does not depend on $x$. This is what the word uniformly means.

In contrast, such a sequence is (pointwise) bounded, if for all $x\in\mathbb{R}$ there exists a constant $C=C(x)\ge 0$ such that $|f_k(x)|\le C$ for all $k$. Here, $C$ depends on $x$.