Infinity norm
High school
Consider a vector $x = (x_1, \dots, x_n)$. The infinity norm of $x$ is the largest absolute value of its entries:
\[\norm{x}_{\infty} = \max_i \abs{x_i}.\]Examples
Worked examples
- The vector $x = (1, 1) \in \reals^2$ : $\norm{x}_{\infty} = \max(\abs{1}, \abs{1}) = 1$ High school
- The vector $x = (3, -4) \in \reals^2$ : $\norm{x}_{\infty} = \max(\abs{3}, \abs{-4}) = 4$ High school
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The vector $x = (5, 0) \in \reals^2$
: $\norm{x}_{\infty} = \max(\abs{5}, \abs{0}) = 5$
High school
All three norms of $(5, 0)$ equal $5$, because the vector lies along an axis: the norms differ only for vectors that point between the axes.
Properties
- The infinity norm is the limit of the $p$-norms $\norm{x}_p = (\abs{x_1}^p + \dots + \abs{x_n}^p)^{1/p}$ as $p \to \infty$, which explains its name.
- For functions on an interval, the analogous norm is the sup norm $\sup_t \abs{f(t)}$, which measures the largest deviation of $f$ from $0$.
Distinctions involving this term
- Types of norms: A norm measures the size of a vector, and $\reals^n$ has several common norms that measure size in different ways. The Euclidean norm is the straight-line length, the 1-norm adds the sizes of the entries, and the infinity norm takes the largest entry. Each is a norm, so each satisfies the triangle inequality, and for every vector, $\norm{x}_{\infty} \leq \norm{x}_2 \leq \norm{x}_1$.