Dot Product is Inner Product
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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