# Dot Product is Inner Product

## Theorem

The dot product is an inner product.

## Proof

Let $\mathbf u, \mathbf v \in \R^n$.

We will check the four defining properties of an inner product in turn.

### Conjugate Symmetry

 $\displaystyle \mathbf u \cdot \mathbf v$ $=$ $\displaystyle \mathbf v \cdot \mathbf u$ Dot Product Operator is Commutative $\displaystyle$ $=$ $\displaystyle \overline{\mathbf v \cdot \mathbf u}$ Complex Number equals Conjugate iff Wholly Real

Thus the dot product possesses conjugate symmetry.

$\Box$

### Bilinearity

From Dot Product Operator is Bilinear, the dot product possesses bilinearity.

$\Box$

### Non-Negative Definiteness

From Dot Product with Self is Non-Negative, the dot product possesses non-negative definiteness.

$\Box$

### Positiveness

From Dot Product with Self is Zero iff Zero Vector, the dot product possesses positiveness.

$\Box$

Hence the dot product satisfies the definition of an inner product.

$\blacksquare$