In finite element books, we have estimates for $$||u-u_h||$$ and also estimates for $$||u - I_h(u)||$$ where $I_h(u): V \mapsto V_h$ projects a function from the infinite dimensional space to the finite dimensional one. For instance, $V = H^1$ and $V_h \subset V$ is the set of piecewise poly of degree $r$.
What is the difference between $u_h$ and $I_h(u)$ ?
Here's my attempt:
Usually the interpolant is defined as (whenever point evaluation makes sense) $$I_h(u)(x) = \sum_{i=1}^N u(x_i)\phi_i(x) $$
Also, we know that $u_h \in V_h$, so there must exists a set of coefficients $\{U_i\}_i$ (called DoFs) such that $$u_h(x)= \sum_{i=1}^N U_i \phi_i(x)$$
So it seems to me that $u_h$ and $I_h(u)$ are really similar, but the only difference is in the coefficients: if $U_i$ coincides with $u(x_i)$ for every $i$, then the interpolant coincides with the discrete solution $u_h$. In practice, we obtain $\{ U_i\}_i$ by solving linear systems, so they will not be equal to $u(x_i)$