Triangle Inequality (Norms)

x + y ≤ x + y — the norm axiom behind absolute value, Euclidean length, and distance.

The idea

Theorem (Triangle inequality).

For any norm $\lVert \cdot \rVert$ on a vector space and any vectors $x$ and $y$, $\lVert x + y \rVert \;\leq\; \lVert x \rVert + \lVert y \rVert.$

A norm assigns a length to each vector subject to three axioms: lengths are nonnegative and vanish only for the zero vector, scaling a vector scales its length, and this inequality. The triangle inequality is therefore not a fact discovered about lengths but part of the definition of length: a function that violates it is not a norm.

Read as geometry, the statement says that the straight step to $x + y$ is never longer than going out along $x$ and then along $y$, as the figure shows. For the Euclidean norm the two routes have equal length only in the degenerate case where the legs point the same way — $y = \lambda x$ for some $\lambda \ge 0$, or one of them zero.

Because the statement holds for an arbitrary norm, it covers every length measure at once, and applying it to differences of vectors gives the form used for distances: no route through an intermediate point is shorter than the direct one.

Ways to work on it

• Walkthrough. The norm axiom and its absolute-value, Euclidean, and metric-space instances.
• Practice. Derive the reverse triangle inequality.
• Hardest. Chain the inequality across a sum and recover the metric axiom.

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