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i'm having trouble with this apparently simple request. I need to say if the subset of $H=L^2(-2,2)$ characterized by $X=\{u\in H : u(x)=a+bx^2\}$ with $a,b \in \mathbb{R}$ is close or not.

I was thinking to proceed in the following way: consider a sequence $u_n=a_n+b_nx^2 \in X$ which converges with the norm of $L^2$ to a certain function $u\in H$. I want to prove that $u\in X$. The convergence in the 2-norm implies the convergence almos everywhere of a subsequence $u_{n_k}$ but i don't know how to go further and conclude that $u=a+bx^2$ with $a_n \rightarrow a$ and $b_n \rightarrow b$.

Michele Mascherpa
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This follows from the fact that your $X$ is finite dimensional, all finite dimensional subspaces of a normed space are closed.

See How to proof that a finite-dimensional linear subspace is a closed set or https://math.berkeley.edu/~sarason/Class_Webpages/solutions_202B_assign10.pdf

The proof of your claim is basically the proof of the theorem.