Inner Product with Zero Vector

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

Then:


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

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

Proof
We have:

so:


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

From conjugate symmetry, we have:


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

so:


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