# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 8: Inner Product Spaces

An inner product space is a vector space $V$ over a field $F$ along with the additional structure of a mapping called an inner product, a generalization of the dot product.

The inner product $\langle \cdot , \cdot \rangle : V \times V \rightarrow F$ must satisfy the following properties:

linearity in the first argument

$\forall a \in F, x, y \in V$:

$\langle a x, y \rangle = a \langle x, y \rangle$

$\langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle$

positive-definiteness

$\langle x, x \rangle \geq 0$

$\langle x, x \rangle = 0 \Longleftrightarrow x = \boldsymbol{0}$

conjugate symmetry

$\langle x, y \rangle = \overline{ \langle y, x \rangle}$