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

\(\ds \mathbf u \cdot \mathbf v\)

\(=\)

\(\ds \mathbf v \cdot \mathbf u\)

Dot Product Operator is Commutative

\(\ds \)

\(=\)

\(\ds \overline{\mathbf v \cdot \mathbf u}\)

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.

$\blacksquare$

Sources

• 1965: Michael Spivak: Calculus on Manifolds ... (previous) ... (next): 1. Functions on Euclidean Space: Norm and Inner Product