If a sequence $\{c_n\}$ of complex numbers is bounded and $\{e_n\}$ is an orthogonal basis of separable Hilbert space $H$ and $x\in H$, then $$\left\|\sum_{n=1}^\infty c_n (x,e_n)e_n\right\|\leq \left\|c_n\right\|_\infty \left \|\sum_{n=1}^\infty(x,e_n)e_n\right\|.$$ It just doesn't "feel right", but I don't want to first use the triangle inequality on the whole sum, because then I can't get the RHS of the inequality.
2026-04-03 00:02:31.1775174551
Is this estimate correct?
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Using orthogonality, we have $$\left\lVert \sum_{n=1}^\infty c_n (x_n,e_n)e_n\right\rVert^2=\sum_{n=1}^\infty \left\lvert c_n\right\rvert^2 \left\lvert (x_n,e_n)\right\rvert^2 \leqslant \left\lVert \left(c_i\right)_{i\geqslant 1}\right\rVert^2 \sum_{n=1}^\infty \left\lvert (x_n,e_n)\right\rvert^2 $$ and we conclude using again orthognality.