Inner product

Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $V$ be a vector space over $\mathbb{K}$. An inner product on $V$ is a map $\langle \cdot, \cdot \rangle \colon V \times V \to \mathbb{K}$ such that, for all $x, y, z \in V$ and $a, b \in \mathbb{K}$:

1. $\langle ax + by, z \rangle = a \langle x, z \rangle + b \langle y, z \rangle$.
2. $\langle y, x \rangle = \overline{\langle x, y \rangle}$.
3. $\langle x, x \rangle > 0$ if $x \ne 0$.

Remarks

Over $\R$, the conjugation does nothing, so an inner product is a symmetric positive definite bilinear form. Over $\C$, conditions 1 and 2 give $\langle x, by \rangle = \bar b \langle x, y \rangle$, so the map is conjugate linear in the second argument. Many physicists, and Mathlib, use the opposite convention: linear in the second argument. The dot product on $\R^n$ and $\langle x, y \rangle = \sum_i x_i \overline{y_i}$ on $\C^n$ are inner products.

Explanations

L1 human

An inner product generalises the dot product to abstract vector spaces. It lets us talk about lengths, angles and perpendicular vectors. On the space of continuous functions on $[0, 1]$, the integral $\int_0^1 f(t) g(t)\, dt$ is an inner product.