Dot Product is Inner Product

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The dot product is an inner product.


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.



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.



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.