TITLE: Simple explanation of uniform norm / sup-norm?
QUESTION [10 upvotes]: I've read the Uniform Norm Wikipedia page, but my most of it went over my head.

What is the sup-norm in simple and / or intuitive terms?
Are there any good examples which illustrate it?

REPLY [2 votes]: In the case of a finite-dimensional space $\ \mathbb R^n\ $ the sup-norm is simply the max-norm. The balls of such a space with the max-norm are cubes which have edges parallel to coordinate axes.

In applications, we often make a bunch $\ A\ $ of experiments, each has a numeric value, and an associated error. The said bunch and errors result in the respective vectors $\ x\ e\in\mathbb R^A.\ $ We often need to decide, based on the individual errors, what is the overall error for the whole given bunch. Depending on the nature of applications, errors are defined differently. One of the common ways is the uniform error, it is the max-norm $\ ||e||_\infty.$

Let $\ X\ $ be a non-empty set $\ (X\ne\emptyset).\ $ Given a function $\ f:X\rightarrow\mathbb R,\ $ we say that real numbers
$\ B\,\ C\in\mathbb R\ $ are a lower and an upper bound of $\ f
\,\ \Leftarrow:\Rightarrow$
$$ \forall_{x\in X}\quad B \le f(x) $$
and
$$ \forall_{x\in X}\quad f(x) \le C $$
One or both bounds may be not well-defined (infinite).
Next, we define function
$\ |f|:X\rightarrow\mathbb R\ $ as follows:
$$ \forall_{x\in X}\quad |f|(x)\ :=\ |f(x)| $$
A function $\ f:X\rightarrow\mathbb R\ $ is bounded
$\ \Leftarrow:\Rightarrow\ f\,\ $ satisfies any of the following four equivalent conditions:

$\ f\ $ admits both an upper and a lower bound;
$\ |f|\ $ admits an upper bound;
$\ \exists_{C\in\mathbb R}\quad |f|\le C $
$\ \exists_{C\in\mathbb R}\forall_{x\in X}\quad |f|(x)\le C\ $

Thus $\ f\ $ is bounded $\ \Leftrightarrow\ f\ $ admits both a lower and an upper bound. Finally, the final elementary definition here: given
$\ f:X\rightarrow\mathbb R,\ $ we define $\ \sup (f)\ $ as the least upper bound of $\ f\ $ (when it exists), i.e. as a real number $\ s\in\mathbb R\ $ such that:
$$ \forall_{x\in X}\quad f(x)\le s $$
and
$$ \forall_{t<s}\exists_{x\in X}\quad f(x)>t $$
(In the case of a finite $\ X,\ $ we have $\ \sup(f) = \max(f)).$

Let $\ \mathcal B(X)\ $ be the set of all bounded functions $\ f:X\rightarrow\mathbb R.\ $ This set deserves to be called space since it has many nice properties, in particular, it is an algebra over $\mathbb R,\ $ and it is a superspace for several more famous spaces. We define a norm $\ ||f||_\infty\ $ called the sup-norm or the uniform norm or $L^\infty$-norm of $\ f\in\mathcal B(X),\ $ as follows
$$ ||f||_\infty\ :=\ \sup(|f|) $$