Interpolation constraint in Hilbert space The paper "Geodesic Interpolating Splines" made the following comment on an interpolation problem:
Interpolation problem:
Let $\mathcal{H}$ be a Hilbert space, let $f_1, \dots, f_N \in \mathcal{H}$, and $c_1,\dots,c_N \in \mathbb{R}$ be given. 
Find $h \in \mathcal{H}$ such that $\|h\|$ is minimum subject to the constraints $\langle f_i, h \rangle = c_i$ for $i=1,\dots,N$.
Comment:
It is indeed clear that the constraints are not affected if $h$ is replaced by $h + v$ where $v$ is orthogonal to all the $f_i$, so that the solution much be in fact be searched in the linear space spanned by $f_1, \dots, f_N$ and express the unknown $h$ as a linear combination $h = \sum_{i=1}^{N} \alpha_i f_i$.
I understand that the interpolation constraint will remain the same if we replace $h$ by $h + v$. But, how this property implies that the solution should be searched in linear space? 
Is it possible to prove their claim with the elementary level knowledge of Hilbert space?

If you can find minimizer $h$ outside of the $\text{span}\{f_1,\cdots,f_N\}$, then since $\langle f_i,h\rangle=0=c_i$, we have $c_1=\cdots=c_N=0$, which may be contradiction (if there exists $c_i\neq0$ given) or such $h$ does not exists (because if $\text{span}\{f_1,\cdots,f_N\}^\perp\neq\emptyset$, then take $a\in\text{span}\{f_1,\cdots,f_N\}^\perp$, and $||a/n||\rightarrow 0\in\text{span}\{f_1,\cdots,f_N\}$ satisfies the given conditions for all $n\in\mathbb{N}$), which is again a contradiction. 
