The Supremum Norm

From Error Bounds to a Norm

In “Examples of Nonuniform Convergence,” we examined the error \(|f_n(x)-f(x)|\) across an entire domain. The supremum of that error records its least upper bound, whether or not any input actually realizes it. This tutorial develops that idea into a standard way to measure the size of a function.

A norm is a numerical measure of size that respects addition and scalar multiplication. For functions, the supremum norm measures the largest absolute value taken over the domain. Its importance in the study of sequences of functions is direct: when the difference \(f_n-f\) is bounded, its supremum norm is exactly the supremum error used in the Supremum Criterion for Uniform Convergence, established earlier in the course.

Definition of the Supremum Norm

Fix a nonempty set \(E\), and consider functions \(f:E\to\mathbb{R}\). A function is bounded on \(E\) if there is a finite constant \(M\geq0\) such that \(|f(x)|\leq M\) for every \(x\in E\). The collection of all such functions will be denoted by \(B(E)\). It is nonempty and closed under addition and real scalar multiplication: sums and scalar multiples of bounded functions are bounded.

The subscript \(\infty\) is standard notation for this norm. It does not mean that the norm is infinite: for every \(f\in B(E)\), the value \(\|f\|_\infty\) is a finite nonnegative real number. If a function is unbounded, this supremum is infinite, so it does not have a finite supremum norm and is not an element of \(B(E)\).

The supremum need not be a value of \(|f|\) at any point. When it is attained, it is also the maximum of \(|f|\); otherwise, values of \(|f|\) can approach the supremum arbitrarily closely without equaling it. The norm is defined using the supremum in either situation.

The Supremum Norm Satisfies the Norm Axioms

A norm on a real vector space assigns a nonnegative real number to each element and satisfies three conditions: only the zero element has norm zero, multiplying by a scalar multiplies the norm by its absolute value, and the norm of a sum is no greater than the sum of the norms. The supremum norm has all three properties.

Proof. Since \(|f(x)|\geq0\) for every \(x\), its supremum is nonnegative. If \(f\) is the zero function, then \(|f(x)|=0\) at every input, so \(\|f\|_\infty=0\). Conversely, suppose \(\|f\|_\infty=0\). For each \(x\in E\),

It follows that \(|f(x)|=0\), and hence \(f(x)=0\), for every \(x\in E\). This proves the definiteness condition.

For scalar multiplication, if \(c=0\), then \(\|cf\|_\infty=0=|c|\|f\|_\infty\). If \(c\neq0\), then for every \(x\in E\), \(|cf(x)|=|c||f(x)|\). Since multiplication by the positive constant \(|c|\) preserves order, the least upper bound of these values is \(|c|\) times the least upper bound of \(|f(x)|\). Therefore \(\|cf\|_\infty=|c|\|f\|_\infty\).

For the triangle inequality, the pointwise triangle inequality gives

for every \(x\in E\). Thus \(\|f\|_\infty+\|g\|_\infty\) is an upper bound for all values \(|f(x)+g(x)|\), and their supremum cannot exceed that upper bound. Hence \(\|f+g\|_\infty\leq\|f\|_\infty+\|g\|_\infty\). All the norm axioms hold. \(\square\)

The nonempty-domain hypothesis matters for definiteness: it ensures there are inputs at which a function can be tested. The boundedness hypothesis matters because a norm, as defined here, takes finite values. Together these conditions make \(B(E)\) a vector space on which the supremum norm is well-defined.

The Norm Defines a Distance Between Functions

A norm can also compare two elements by measuring the size of their difference. For bounded functions, this gives a distance that tracks the largest pointwise separation over the domain. In particular, the distance is small only when the two functions are close at every input.

Proof. Nonnegativity follows from the nonnegativity of a norm. By the definiteness condition of the norm, \(d_\infty(f,g)=\|f-g\|_\infty=0\) exactly when \(f-g\) is the zero function, which is exactly when \(f=g\). Also,

Finally, the triangle inequality for the norm, applied to \(f-h=(f-g)+(g-h)\), yields

These are the metric properties, so the supremum distance is a metric on \(B(E)\). \(\square\)

A Useful Consequence: The Reverse Triangle Inequality

The triangle inequality also gives a bound on how much the norm can change when a function is altered. This estimate is useful whenever a function is replaced by an approximation: a small supremum distance guarantees that the two functions have nearly the same norm.

Proof. Since \(f=(f-g)+g\), the triangle inequality gives

so \(\|f\|_\infty-\|g\|_\infty\leq\|f-g\|_\infty\). Interchanging \(f\) and \(g\) gives \(\|g\|_\infty-\|f\|_\infty\leq\|g-f\|_\infty\). By homogeneity, \(\|g-f\|_\infty=\|f-g\|_\infty\). The two inequalities together imply

which is equivalent to the stated absolute-value inequality. \(\square\)

Why the Supremum, Rather Than a Pointwise Maximum?

The supremum norm works on domains where a maximum may fail to exist. The domain might be unbounded, or the values might approach their least upper bound only near a missing endpoint. The definition does not require a function to attain its largest absolute value, and restricting it to functions with an attained maximum would exclude many useful bounded functions.

A second important point is that the domain is part of the norm. The same formula can have different supremum norms on different domains because the collection of inputs changes. For example, the function \(q(x)=x\) has supremum norm \(1\) on \([0,1)\), but supremum norm \(2\) on \([0,2]\). On the first domain, \(1\) is approached but not attained; on the second, the value \(2\) is attained. Norm statements should therefore always specify the domain.

For sequences of functions, the definition connects size and error: when \(f_n\) and \(f\) belong to \(B(E)\), the norm \(\|f_n-f\|_\infty\) is the supremum error across \(E\). The Supremum Criterion for Uniform Convergence from earlier in the course expresses uniform convergence by asking that these errors tend to zero. The next tutorial develops this connection further. A common pitfall is to substitute a maximum for a supremum without checking attainment; the boundary-layer example in “Examples of Nonuniform Convergence” illustrates why that distinction can affect the calculation.