Can somebody please explain me what the difference is between totally bounded and uniformly bounded functions?

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    $\begingroup$ Please formulate your question as a question. $\endgroup$ – J.R. Oct 16 '12 at 19:40
  • $\begingroup$ Could you give the definitions, and where are the functions defined? $\endgroup$ – Davide Giraudo Oct 16 '12 at 19:42

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 $x$.


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