Let $(W,\langle\cdot,\cdot\rangle_W)$ be an inner product space and let $V$ be a linear subspace of $W$. Let $f: V \rightarrow W$ be an injective linear map.
If we fix $v' \in V$ and $w' \in W'$, does there exists an inner product on $V$ such that
$$\langle v',v\rangle_V = \langle w',f(v)\rangle_W, \qquad \forall v \in V \quad?$$
EDIT: After the comment of aschepler, this doesn't work if $\langle w',f(v')\rangle\leq 0$.
So let's add the assumption that $\langle w',f(v')\rangle \geq 0$.