# Dot Product is Inner Product

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## Contents

## 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

\(\displaystyle \mathbf u \cdot \mathbf v\) | \(=\) | \(\displaystyle \mathbf v \cdot \mathbf u\) | Dot Product Operator is Commutative | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \overline{\mathbf v \cdot \mathbf u}\) | 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.

$\blacksquare$