# What is the difference between totally bounded and uniformly bounded?

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

• Please formulate your question as a question.
– J.R.
Commented Oct 16, 2012 at 19:40
• Could you give the definitions, and where are the functions defined? Commented Oct 16, 2012 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 $$x$$.

Uniformly bounded means each of the functions in the set is bounded with respect to the sup metric if we are talking about a metric space.

• What if $C$ does not depend on $x$ but does depend on $k$? Commented Nov 18, 2023 at 8:45
• Could the term uniformly bounded be used for a function $f:\mathbb{R}\rightarrow\mathbb{R}^n$, with $n$ playing the role of $k$ in your definition? In that case, this definition would be equivalent to $||f(x)||_\infty \le C$ for all $x \in \mathbb{R}$?
– Siva
Commented Jun 8 at 3:22