I have the Hilbert space of square integrable functions on $[a, b]$, and what I would like to have is to discretize this space, i.e., find a sequence of finite-dimensional Hilbert spaces $H_K$ of dimension $K$ such that
$|\langle u, v \rangle - \langle u^{(K)}, v^{(K)} \rangle| \rightarrow 0$
as $K \rightarrow \infty$, where $v^{(K)}$ is a 'discretization' of $v$, defined appropriately. My idea was to define discretized vectors in such a way that $\langle u^{(K)}, v^{(K)} \rangle$ is the $K$-th Riemann sum for the integral that I would get for $\langle u, v \rangle$ (say, by equally partitioning $[a, b]$ and taking the points of this partition to represent a basis for the finite spaces).
But I know that I need a Lebesgue limit rather than a Riemann limit. Is such discretization possible?
