We are working on functions that are in $L^2([-1,1])$, is the space of functions that are linear $f(x)=ax+b$, closed?
I am not entirely sure of how to prove this. I mean, if I have a sequence of functions, that converges to another function $\{g_n\}_ \rightarrow g$, in the $L^2$-norm, and all the $g_n$ are linear, I must then show that g is linear, or atleast linear a. e..
I am not even sure that it holds if we have that a sequence of linear functions converges pointwise to a function, that function has to be linear?
Easiest way: finite dimensional subspaces are always closed.
But as an exercise, you should try to show (for the pointwise case) that if $a_n x +b_n \to f(x)$, then $(a_n)_n$ and $(b_n)$ are convergent. Then it is not hard to see that $f$ is linear.
Now, convince yourself with a variant of the above argument that this still holds if we have convergence only almost everywhere. In fact, convergence at two points suffices.
Finally, for the $L^2$ case, use that if $f_n \to f$ in $L^2$, then $f_n \to f$ almost everywhere for a subsequence.