Inner product space

An inner product space is a vector space over $\mathbb{F}$ together with an inner product $\langle \cdot, \cdot \rangle$.

Example

Let $V = \mathbb{F}^n$, $u = (u_1, \ldots, u_n), v = (v_1, \ldots, v_n) \in \mathbb{F}^n$. Then $V$ with

$$ \langle u, v \rangle = \sum_{i=1}^n u_i \overline{v}_i. $$

For $\mathbb{F} = \mathbb{R}$, $\overline{v}_i=v_i$

$$ u \cdot v = u_1 v_1 + \cdots + u_n v_n. $$

Properties

• Cauchy-Schwarz inequality
• Triangle inequality
• Parallelogram law