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Let $u \in L^\infty(\Omega)$ on a bounded domain $\Omega$. Let $w_j$ be the eigenfuntions of the Neumann Laplacian.

Is it true that $$a_n := \sum_{i=1}^n (u,w_j)w_j$$ is such that $\lVert a_n\rVert_{L^\infty(\Omega)} \leq C$ uniformly in $n$?

I've no clue why but I think answer is.

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