Dot Product is Inner Product

Theorem
The dot product is an inner product.

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

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

Bilinearity
It follows from Dot Product Operator is Bilinear.

Non-Negative Definiteness
It follows from Dot Product with Self is Non-Negative.

Positiveness
It follows from Dot Product with Self is Zero iff Zero Vector.

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