Absolute convergence of a sum of Fourier coefficients in a Hilbert Space

Question

Let $(\varphi_k)$ be an orthonormal basis of a Hilbert space $\mathbf{H}$, and consider the sum $\sum_{k}\left < x, \varphi_k\right > \overline{\left < y, \varphi_k \right >}$ for some $x,y\in \mathbf{H}$. It converges to $\left < x, y\right >$. Does it necessarily absolutbly converge? I'm pretty sure it does in $\ell_2$ and even in $L_2[a,b]$. But does it hold in the general case, and how does one prove it?

Answer 1

Since $$\sum_k | \langle x, \phi_k \rangle |^2 = \sum_k \langle x,\phi_k \rangle \overline{ \langle x,\phi_k \rangle} = \langle x,x \rangle = \|x\|^2$$ you can use e.g. Cauchy-Schwarz to find that $$ \sum_k | \langle x,\phi_k \rangle \overline{ \langle y,\phi_k \rangle}| = \sum_k | \langle x,\phi_k \rangle | \cdot |\overline{ \langle y,\phi_k \rangle}| \le \|x\| \|y\|$$ so yes, it converges absolutely.

Answer 2

Yes, it converges absolutely because $$ 2|\langle x,\varphi_k\rangle\langle \varphi_k,y\rangle| \le |\langle x,\varphi_k\rangle|^2+|\langle y,\varphi_k\rangle|^2, $$ and because the sums of the terms on the right converge by Bessel's inequality.