Proving the uniqueness of a vector

Question

I recieved the following question: Let $W$ be an inner product sapce, $dim\space W=n$, and let $\varphi: W \to \mathbb R$ be linear. Then there exists a unique vector $w_0 \in W$ such that $\varphi(w)= (w,w_0)$ for all $w \in W$. So i was able to prove that such vector exists by setting $w_0= \sum_{i=1}^n \varphi(u_i)u_i$ where $\lbrace u_1,...,u_n \rbrace $ is an orthonormal basis for $W$. I am having issues proving the uniqueness of such vector. Any assistance will be welcomed.

Answer 1

If $h$ is another such vector, $\forall w \in W$ $0 = \varphi(w) - \varphi(w) = (w, w_0 - h)$, so $h = w_0$ because the inner product is not degenerate.