Let $B$ be a finite dimensional manifold, $H$ be a complete Hilbert space and $V$ a finite dimensional subspace of $H$.
We set $\Gamma:=\Gamma(B;H)$, i.e. the smooth maps $B\to H$ and $W:=\{s\in \Gamma\mid \exists b\in B, \mbox{ s.t. }s(b)\perp V\}$.
If we regard $\Gamma$ as the infinite dimensional space and $W$ as its sub-space.
Q: Can we say that the codimension of $W$ is $\max\{\dim V- \dim B,0\}$?