I have a question regarding the projection onto a finite-dimensional subspace of a semi-normed vector space:
Let $V$ be a real vector space (either finite or infinite-dimensional) and let $\langle\cdot,\cdot\rangle:V\times V \to \mathbf{R}$ be a symmetric positive semi-definite bilinearform. Denote the corresponding semi-norm by $\|\cdot\|=\sqrt{\langle\cdot,\cdot\rangle}$ and let further $U\subseteq V$ be a finite-dimensional subspace of $V$.
Question: Under which conditions is the projection $P_U:V\to V, v\mapsto argmin_{u\in U} \|v-u\|$ well-defined in the sense that the argmin exists and is unique?
If I am not mistaken, then it suffices that the restriction of $\langle\cdot,\cdot\rangle$ to $U$ is an inner product. Or do I need completeness or definiteness of $\langle\cdot,\cdot\rangle$ on $V$ for $P_U$ to be well-defined?
Edit: Here is my approach: Let $\{u_1,\dots,u_d\}$ be a basis of $U$. Let $v\in V$ be fixed. Then for any $u=\sum_{j=1}^d u_j\beta_j$ it holds:
\begin{align*} \|v-u\|^2&=\|v\|^2-2\langle v,u\rangle + \|u\|^2 \\&=\|v\|^2-2\sum_{j=1}^d \beta_j \langle u_j,v\rangle+\sum_{j,k=1}^d \beta_j\beta_k \langle u_j,u_k\rangle\\&=:\|v\|^2-2\beta^T \gamma(v) + \beta^T G\beta. \end{align*}
Differentiating with respect to $\beta$ and equating with $0$ shows that the coefficients of the minimizer must satisfy the equations $G\beta=\gamma(v)$. If the restriction of $\langle \cdot,\cdot \rangle$ to $U$ is an inner product, then $G$ is positive definite and the unique minimizer has coefficients $G^{-1}\gamma(v)$.