Normed & Metric Spaces

Distance, generalized beyond the real line: the axioms and several norms.

The idea

A metric space is a set together with a distance function, and it is the setting in which the limit arguments of analysis — convergence, continuity, Cauchy sequences — can be carried out once and then reused. On the real line those arguments all run through the distance $|x - y|$; multivariable analysis needs them in $\mathbb{R}^{n}$, and later analysis needs them on spaces of functions. The definition isolates the three properties of $|x - y|$ that the arguments use.

Definition (Metric space).

A metric on a set $X$ is a function $d \colon X \times X \to \mathbb{R}$ such that, for all $x, y, z \in X$, $d(x,y) \ge 0, \qquad d(x,y) = d(y,x), \qquad d(x,z) \le d(x,y) + d(y,z),$ with $d(x,y) = 0$ if and only if $x = y$. The pair $(X, d)$ is a metric space.

The third condition is the triangle inequality: passing through an intermediate point never shortens a distance.

A norm $\lVert \cdot \rVert$ measures the size of a single vector, and every norm induces a metric by $d(x,y) = \lVert x - y \rVert$. One set can carry several norms, so a distance between two points is fixed only once the norm is chosen, and different choices give the same set different metric structures. On $\mathbb{R}^{2}$, for instance, the $L^{1}$ distance between two points $P$ and $Q$ sums the coordinate gaps $\Delta x$ and $\Delta y$, the $L^{2}$ distance measures the straight segment, and the $L^{\infty}$ distance keeps only the larger gap — and the unit balls of the three norms are a diamond, a disk, and a square.

Ways to work on it

• Walkthrough. The metric axioms and distance under the L^1 and L^2 norms.
• Practice. Compute a distance under the L^1, L^2, or L^∞ norm.
• Hardest. Spot a non-metric, then a supremum distance in ℝ^3.

Not sure where to start? Take the ten-question placement test.