Consider the problem on the picture. I am struggling with part (b) of the excercise. I have managed to show that we have $L^1$ convergence, but I am unable to show $L^2$ convergence. Does anyone have a slight hint sending me in the right direction? Thanks a lot!
2026-03-25 17:35:41.1774460141
L2 convergence for a simple function approximation
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Hint: If $g \in L^{2}$ and $g$ is orthogonal to $I_{j,n}$ for all $n$ and $j$ then $g=0$ a.e.. By Hilbert space Theory this implies that linear combinations of the functions $I_{j,n}$ form a dense subspace of $L^{2}$. When $f$ is such a linear combination the result follows trivially (with $E_n=f$ for $n$ sufficiently large). The general case follows by approximating $f$ by a finite linear combination of the functions $I_{j,n}$ since $\|E_nf-E_ng\|\leq \|f-g\|$.