Is there some connection between the Wronskian determinant and Sobolev spaces, e.g. $H^1$?
I know that we often seek solutions to differential equations in Sobolev spaces because these spaces place additional regularity requirements on potential solutions. So, for example, $\|f\|_{H^1}$ depends on both $f$ and its derivative $f'$.
But the Wronskian determinant also depends on functions and their derivatives. In fact, by analogy: If the usual determinant of matrix $M \in \mathbb{R}^{n \times n}$ is often thought of as an $n$-linear form where columns of $M$ represent vectors of $\mathbb{R}^{n \times n}$, then might it be possible to think of the columns of the Wronskian
$$W = \left| \begin{bmatrix} f(x) & g(x) \\ f'(x) & g'(x) \end{bmatrix} \right|$$
as vectors in some space underlying space $V$, and the Wronskian as being an $n$-linear form on $V$, i.e. $W(u,v)$ where $u = (f(x), f'(x))$ and $v = (g(x),g'(x))$?
I guess what I'm driving at is this: So many concepts in undergrad math are taught as like, "Just follow these instructions..." when actually there is a rich and deep theory that is being glossed over. Is the Wronskian one of those concepts? Is there a connection here that is worth exploring?