Uniform Norm¶

Definition¶

On a nonempty set $X$ , the uniform norm $\|f\|_\infty$ of a bounded function $f\colon X\to\mathbb{R}$ is defined as:

\|f\|_\infty = \sup_{x \in X} |f(x)|.

Interpretation¶

The uniform norm measures the size of a bounded function by its worst case: the supremum of the absolute value $|f(x)|$ over the whole set $X$ . Applied to a difference, $\|f-g\|_\infty$ is the largest vertical gap between the graphs of $f$ and $g$ , and it serves as the distance between the two functions.

Two functions f and g with horizontal lines at the maximum absolute value of f, and a vertical arrow at the place where the gap between f and g is largest.

The dashed lines in the figure are at $\pm\|f\|_\infty$ , and the dot marks where $|f|$ reaches its supremum. The green arrow marks where the vertical gap between $f$ and $g$ is largest; its length is $\|f-g\|_\infty$ $g ∥_{\infty}$ . That gap is longer than $\|f\|_\infty$ itself, because $g$ is negative where $f$ is large, so the norm of a difference cannot be read off from the norms of $f$ and $g$ alone.

The red segment marks the largest gap between $f_n$ and $f$ : that is $\|f_n-f\|_\infty$ $f ∥_{\infty}$ . Uniform convergence means it tends to $0$ .

Boundedness of $f$ makes the supremum finite. The supremum need not be attained: for $f(x)=\mathrm{e}^{-x}$ on $(0,+\infty)$ in the Examples, $\|f\|_\infty=1$ while $|f(x)|<1$ everywhere. It behaves like a length on the bounded functions: $\|f\|_\infty=0$ only for the zero function, $\|\lambda f\|_\infty=|\lambda|\,\|f\|_\infty$ $∣ λ ∣ ∥ f ∥_{\infty}$ , and the triangle inequality $\|f+g\|_\infty\le\|f\|_\infty+\|g\|_\infty$ holds because $|f(x)+g(x)|\le\|f\|_\infty+\|g\|_\infty$ for every $x$ .

The norm turns uniform convergence into convergence of numbers, as in the Key Results. It also appears in normal convergence, in the Weierstrass $M$ -test and in the Weierstrass approximation theorem.

A common slip is to confuse $\|f_n-f\|_\infty\to0$ with $f_n(x)\to f(x)$ at every $x$ . On $[0,1)$ the functions $x^n$ tend to $0$ at every point, yet $\|x^n\|_\infty=1$ for every $n$ .

These three properties return later as the definition of a norm; for vectors in $\mathbb{R}^n$ the same recipe gives the maximum norm.

Key Results¶

A sequence of functions $(f_n)_{n=1}^\infty$ converges uniformly to a function $f$ on $X$ if and only if:

\lim_{n \to \infty} \|f_n - f\|_\infty = 0.

For possibly unbounded functions, the same supremum of the error $|f_n-f|$ may be $+\infty$ ; it is an extended error measure rather than a norm on the space of all functions. Uniform convergence means these errors eventually become finite and tend to zero.

Examples¶

For the function $f\colon (0,+\infty) \to \mathbb{R}$ defined by $f(x) = \mathrm{e}^{-x}$ $e^{- x}$ , we have $\|f\|_\infty = 1.$
For the function $g\colon \mathbb{R} \to \mathbb{R}$ defined by $g(x) = \sin x$ $sin x$ , we have $\|g\|_\infty = 1.$

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 2

Aliases: sup norm, supremum norm Keywords: infinity norm, Chebyshev norm Symbols: element of, infinity, lambda, real numbers

Last updated: 20 October 2026 03:02