Inner Product with Zero Vector

Theorem
Let $V$ be an inner product space with inner product $\innerprod \cdot \cdot$.

Then:


 * $\innerprod 0 x = \innerprod x 0 = 0$

for all $x \in V$, where $0$ is the zero vector of $V$.

Proof
We have:

so:


 * $\innerprod 0 x = 0$

From conjugate symmetry, we have:


 * $\innerprod x 0 = \overline {\innerprod 0 x}$

so:


 * $\innerprod x 0 = \overline 0 = 0$