I want to convert the following continuous notation $$ \min_u \int_\Omega \lambda(x) (u(x) - f(x))^2 \, dx \tag{1} $$
to a discrete version. Let me assume that $\lambda,u,f$ are vectorized with length $N$.
The following is not equivalent to (1), right? $$ \min_u \langle \lambda, (u-f)^2 \rangle \tag{2} $$
(3) is equivalent to (1), I think. $$ \min_u \sum_{i=1}^N \lambda_i (u_i-f_i)^2 \tag{3} $$
However, I want to denote the discrete version with vector form. How to elegantly denote?
I also want to convert the following: $$ \int_\Omega \lambda(x) |\nabla u(x)| dx $$ where $u:\Omega\to\mathbb R^2$ and $|\nabla u(x)|=\sqrt{u_x^2 + u_y^2}$. It seems very complicated.