Conditions for proving infinite codimension?

Question

I am working with moduli spaces of curves and I am not used to work with infinite dimensional spaces. Then I was wondering if there are sufficient explicit conditions to stablish that certain subspace $W$ has infinite co-dimension within a largen infinite-dimensional space $U$. For example, working with finite-dimensional $\mathbb{K}$-vector spaces, proving that $W$ is the kernel of certain function $f:U\to\mathbb{K}$ (assumming $0$ is a regular value of $f$) shows that $W$ has co-dimension $1$ in $U$. Is there any analogous result in the infinite dimension setting for proving infinite codimension? Or, any other radically different result (maybe topological...) that states conditions under which $W$ has infinite-codimension? Thanks a lot in advance.