Let V be a Hilbert space, $V_0 \subset V$ a subspace (finite dimensional) and $f \in V_0$ and $g \in V$. I've got the minimization problem $min \{ \frac{1}{2} ||v||^2 - <f, v> \}$, where $v \in V_g$ for $V_g = g + V_0$ (which is an affine space).
Can you explain me, using a basis, how you can compute the basis-presentation of the solution?
Making $$ v={\displaystyle \sum_{k}\alpha_{k}v_{k}+g} $$
$$ f=\sum_{j}\beta_{j}v_{j} $$
and assuming $\left\{ v_{k}\right\} $ as an orthonormal set, we have
$$ \frac{1}{2}\left\Vert v\right\Vert ^{2}-\left\langle f,v\right\rangle =\frac{1}{2}\left\Vert \sum_{k}\alpha_{k}v_{k}+g\right\Vert ^{2}-\left\langle \sum_{j}\beta_{j}v_{j},\sum_{k}\alpha_{k}v_{k}+g\right\rangle =\frac{1}{2}\left\Vert \alpha_{k}^{2}\right\Vert +\frac{1}{2}\left\Vert g\right\Vert ^{2}+\sum_{k}\alpha_{k}\left\langle g,v_{k}\right\rangle -\sum_{k}\alpha_{k}\beta_{k}+\sum_{k}\beta_{k}\left\langle g,v_{k}\right\rangle $$
so the condition for stationary points is
$$ \alpha_{k}+\left\langle g,v_{k}\right\rangle -\beta_{k}=0 $$