Let $V$ be an inner product space, and let $u \in V$ be a vector that satisfies $\langle u, v\rangle = 0$ for all $v ∈ V$. Prove or disprove that $u = 0_V$.
I'm studying linear algebra and I encountered some difficulties with this task. I would love to get some hints or an answer for this one.
Since the condition holds for every $u, v\in V$ it holds for $u=v$, so we have $\langle u, u\rangle=0$, but the properties of the inner product implies that $u=0$, since a inneer product is a Positive definite bilinear form.