Let's say I have a vector $$f(x) = \begin{bmatrix} f_1(x) \\ f_2(x) \\ \vdots \\ f_n(x) \end{bmatrix} \in \mathbb{R}^n $$ consisting of continuous functions $f_i, i = 1,..., n$ and $x \in [0, 1]$. If you look at the space $$\mbox{span} \{ f(x) | x \in [0,1] \}$$ then this has dimension $n$ except if the ratio between 2 different $f_i(x) $stays the same or if there are $f_i(x)$ which are zero.
Discretize now the interval $[0,1]$ uniformly into $x_i, i = 1, ..., m$ with $m$ big enough and define now $F := [f(x_1), ..., f(x_m)]$, If you calculate a singular value decomposition of this $F$, this allows you to check if $F$ can be well approximated by a lower-dimensional subspace (by checking the singular values). If $m$ is big enough, this can tell you if $\mbox{span} \{ f(x) | x \in [0,1] \}$ can be well approximated by a low dimensional subspace.
I was wondering if given the function $f$ there is a method to immediately check if this space can be approximated by a low-dimensional subspace, without doing the discretisation and calculating an svd.
many thanks