Norm induced by an inner product

Definition

Let $V$ be an inner product space over $\R$ or $\C$, with inner product $\langle \cdot, \cdot \rangle$. The norm induced by the inner product is the map $$\norm{x} = \sqrt{\langle x, x \rangle} \qquad (x \in V).$$

Remarks

It is a norm; the triangle inequality follows from the Cauchy–Schwarz inequality. For the dot product on $\R^n$, it is the Euclidean length $\sqrt{x_1^2 + \cdots + x_n^2}$. It satisfies the parallelogram law, and the polarisation identity recovers the inner product from it.

Explanations

L1 human

The length of a vector is the square root of its inner product with itself. In the plane, this is the Pythagorean formula $\sqrt{x^2 + y^2}$. So every inner product gives a way to measure lengths.