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Let $H$ be a complex Hilbert space with inner product $⟨⋅, ⋅⟩$.

In the rest of this question, we write $T f$ for the sequence of Fourier coefficients of a complex-valued, integrable, $2 π$-periodic function $f$, i.e. $T f(n)=\frac{1}{2 π} ∫_{-π}^π f(x) e^{-\mathrm{i} n x} \mathrm{~d} x$. Throughout the rest of this question, you may take for granted that the system $\{\frac{1}{\sqrt{2 π}} e^{\mathrm{i} n x}\}_{n ∈ ℤ}$ is an orthonormal basis of $L^2(-π, π)$.