In finite-dimensional spaces like $mathbb{R}^n$, the concept of the sup-norm simplifies to the max-norm. Visualize balls in such a space under the max-norm; they take the shape of cubes perfectly aligned with the coordinate axes.
In practical applications, consider a set $A$ of experiments, each yielding a numerical value alongside an associated error. These sets of experiments and their errors can be represented as vectors $x, e in mathbb{R}^A$. Often, we need to determine an overall error for the entire set based on individual errors. The definition of error can vary depending on the application. A frequently used method is the uniform error, quantified by the max-norm, denoted as $||e||_infty$.
To formally define the Uniform Norm, let’s first establish some foundational concepts. Consider a non-empty set $X$ ($X neq emptyset$). For a function $f: X rightarrow mathbb{R}$, real numbers $B, C in mathbb{R}$ are defined as a lower and an upper bound of $f$ respectively if:
$$ forall{x in X} quad B le f(x) $$
and
$$ forall{x in X} quad f(x) le C $$
It’s important to note that one or both of these bounds might be undefined (infinite).
Next, we define the function $|f|: X rightarrow mathbb{R}$ as:
$$ forall_{x in X} quad |f|(x) := |f(x)| $$
A function $f: X rightarrow mathbb{R}$ is considered bounded if it satisfies any of the following 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 $
Therefore, boundedness of $f$ is equivalent to $f$ having both a lower and an upper bound. Finally, for a function $f: X rightarrow mathbb{R}$, we define $sup(f)$ as the least upper bound of $f$ (if it exists). This means $sup(f) = s in mathbb{R}$ such that:
$$ forall{x in X} quad f(x) le s $$
and
$$ forall{t < s} quad exists_{x in X} quad t < f(x) $$
In simpler terms, for a finite set $X$, $sup(f)$ is simply the maximum value of $f$, i.e., $sup(f) = max(f)$.
Let $mathcal{B}(X)$ represent the set of all bounded functions $f: X rightarrow mathbb{R}$. This set is often referred to as a space due to its favorable mathematical properties, particularly as an algebra over $mathbb{R}$ and a superspace for many well-known spaces. We define a norm $||f||_infty$, known as the sup-norm, uniform norm, or $L^infty$-norm, for $f in mathcal{B}(X)$ as:
$$ ||f||_infty := sup(|f|) $$
This uniform norm provides a way to measure the “size” of a bounded function, focusing on the largest absolute value it attains over its domain. It’s a crucial concept in various areas of analysis, particularly in functional analysis and approximation theory, offering a robust measure of function magnitude.
