Here is a question from Axler's Linear Algebra Done Right (Chapter 6.B problem 11).
Suppose $\langle \cdot, \cdot \rangle_1$ and $\langle \cdot, \cdot \rangle_2$ are inner products on $V$ such that $\langle v,w\rangle_1 = 0$ if and only if $\langle v,w \rangle_2 = 0$. Prove that there is a positive number $c$ such that $\langle v,w\rangle_1 = c\langle v,w\rangle_2$ for every $v,w \in V$.
It doesn't say that $V$ is finite dimensional, but I tried showing the result assuming this. If $\{e_1,\dots,e_n\}$ is an orthonormal basis with respect to $\langle \cdot,\cdot \rangle_1$, then $\langle e_i,e_j \rangle_1 = 0$ for all $i \neq j$. This implies $\langle e_i,e_j \rangle_2 = 0$ for all $i \neq j$. So we can use the equivalence they give to show that a set of vectors is orthogonal with respect to one inner product if and only if they are orthogonal with respect to the other. But I'm not sure how to get the result from here.
Another idea I had was to represent the linear functional $\langle \cdot,w \rangle_1$ using Riesz Representation. There exists some $u \in V$ such that $\langle \cdot,w\rangle_1 = \langle \cdot, u \rangle_2$. That is, for any $v\in V$, we have \begin{align*} \langle v,w \rangle_1 = \langle v,u \rangle_2 \end{align*} If we want $\langle v,u \rangle_2 = c\langle v,w\rangle_1$ for some positive $c$, then we would have $c = \frac{\langle v,u \rangle_2}{\langle v,w\rangle_1}$, but I'm not sure how to show this is a positive constant.