An inner product on a real vector space $V$ is a function $\langle\cdot, \cdot\rangle: V \times V \rightarrow \mathbb{R}$ satisfying

(i) $\langle\mathbf{x}, \mathbf{x}\rangle \geq 0$, with equality if and only if $\mathbf{x}=\mathbf{0}$

(ii) $\langle\alpha \mathbf{x}+\beta \mathbf{y}, \mathbf{z}\rangle=\alpha\langle\mathbf{x}, \mathbf{z}\rangle+\beta\langle\mathbf{y}, \mathbf{z}\rangle$

(iii) $\langle\mathrm{x}, \mathbf{y}\rangle=\langle\mathbf{y}, \mathbf{x}\rangle$

for all $\mathrm{x}, \mathrm{y}, \mathrm{z} \in V$ and all $\alpha, \beta \in \mathbb{R}$. A vector space endowed with an inner product is called an inner product space.

Note that any inner product on $V$ induces a norm on $V$ :

$$

\|\mathrm{x}\|=\sqrt{\langle\mathrm{x}, \mathrm{x}\rangle}

$$

One can verify that the axioms for norms are satisfied under this definition and follow (almost) directly from the axioms for inner products. Therefore any inner product space is also a normed space (and hence also a metric space). ${ }^{4}$

Two vectors $\mathrm{x}$ and $\mathrm{y}$ are said to be orthogonal if $\langle\mathrm{x}, \mathrm{y}\rangle=0$; we write $\mathrm{x} \perp \mathrm{y}$ for shorthand. Orthogonality generalizes the notion of perpendicularity from Euclidean space. If two orthogonal vectors $\mathrm{x}$ and $\mathrm{y}$ additionally have unit length (i.e. $\|\mathrm{x}\|=\|\mathrm{y}\|=1$, then they are described as orthonormal.

The standard inner product on $\mathbb{R}^{n}$ is given by

$$

\langle\mathbf{x}, \mathbf{y}\rangle=\sum_{i=1}^{n} x_{i} y_{i}=\mathbf{x}^{\top} \mathbf{y}

$$

The matrix notation on the righthand side (see the Transposition section if it's unfamiliar) arises because this inner product is a special case of matrix multiplication where we regard the resulting $1 \times 1$ matrix as a scalar. The inner product on $\mathbb{R}^{n}$ is also often written $\mathrm{x} \cdot \mathrm{y}$ (hence the alternate name dot product). The reader can verify that the two-norm $\|\cdot\|_{2}$ on $\mathbb{R}^{n}$ is induced by thişinnerte $V$ product.