dot product equal for all $y$, then the two functions are equal

Question

Let's say we have two functions : $f, g : A \rightarrow B$, where $A, B$ are finite vector spaces with scalar product $ \langle , \rangle$ (so $A, B$ are euclidian spaces), and such that $\forall x$ : $$\langle f(x), y \rangle = \langle g(x), y \rangle$$

Then do we have : $f(x) = g(x)$, $\forall x \in A$ ?

According to me, there is a link with Riesz theorem but I am not sure, also if $f, g$ are not continuous I don't think this is in true.

On the other hand, if instead of $\forall x$, if we have $\forall x, y \in A$, then it might be true, but I am not sure. Any ideas ?

Answer 1

This is equivalent to asking if $f \equiv 0$ whenever ­$\langle\ f(x),y \rangle=0$, $\forall x \in A$. This is false if, for example, $y$ is orthogonal to the subspace spanned by $\{f(x) \ : \ x \in X\}$.

Now if you include the condition $\forall y \in B$, then it is true. Just take $y=f(x)$ to conclude that $\langle\ f(x),f(x) \rangle=0 \, \ \forall x $ and thus $f \equiv 0$.