Forming a manifold from curves

Question

If we have $\{\frac{1}{\sqrt{\pi}}f_0(x),\sqrt{\frac{2}{\pi}}f_{2n-1}(x),\sqrt{\frac{2}{\pi}}f_{2n}(x)\}_{n=0}^\infty $..(1), which represents an orthonormal basis of $L^2(\mathbb{R})$ with respect to the weight function $\omega(x)=\frac{1}{1+x^2}$. If I want to make a manifold whose basis consists of functions from (1), what would be the first step? Should I take let's say only finite number of $f_n$, and then take their derivatives to get tangent vectors? Does infinite number of functions change anything?