Inner Product with Zero Vector

Theorem
Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.

Let $0_V$ be the zero vector of $V$.

Then:


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

for all $x \in V$.

Proof
We have:

so:


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

From conjugate symmetry, we have:


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

so:


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