LAIP1

From Exampleproblems

Define an inner product.

For $x, y \in S$ , the inner product $\langle x, y\rangle$ is a function $\langle\cdot ,\cdot\rangle : S \times S \longrightarrow \mathbb{C}$ (or $\mathbb{R}$ if $S\,$ is a real vector space) with the properties:

$\langle x, y\rangle = \overline{\langle y, x\rangle}$
$\langle \alpha x, y\rangle = \alpha\langle x, y\rangle$
$\langle x + y, z\rangle = \langle x, z\rangle + \langle y, z\rangle$
$\langle x, x\rangle > 0$ if $x \ne 0\,$ , and $\langle x, x\rangle = 0 \Longleftrightarrow x = 0$