Cauchy's Inequality/Vector Form

Theorem
Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $V$.

Then:
 * $\paren {\mathbf a \cdot \mathbf b}^2 \le \paren {\mathbf a \cdot \mathbf a} \paren {\mathbf b \cdot \mathbf b}$

where $\cdot$ denotes dot product.

Proof
Let us express $\mathbf a$ and $\mathbf b$ in component form:

where the ordered $n$-tuple $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is the standard ordered basis of $V$.

Then we have: