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
Thus the dot product possesses conjugate symmetry.

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

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

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

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