Compact does not imply Hilbert Schmidt

Given: Countable Choice, the space ${\mathrm{\ell }}^2$ with its standard basis $(u_n)$, and the diagonal operator with $d_0=0$, $d_n=n^{-1\mathrm{/}2}$ for $n\ge 1$.

Diagonal criteria. For a diagonal operator with bounded sequence $d$, boundedness, compactness, the Hilbert–Schmidt criterion ${\sum }_n\mathrm{\mid }{d}_n{\mathrm{\mid }}^2<+\mathrm{\infty }$ relative to the standard basis, and the trace-class criterion ${\sum }_n\mathrm{\mid }{d}_n\mathrm{\mid }<+\mathrm{\infty }$ hold as in the diagonal example; the Hilbert–Schmidt property and norm are basis-independent, and the standard basis is orthonormal with $\mathrm{\parallel }{u}_n{\mathrm{\parallel }}_2=1$ (Diagonal Schatten class criteria on ell two, Hilbert–Schmidt operator and Hilbert–Schmidt norm, The Hilbert–Schmidt norm is basis independent, Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$).

Divergence and convergence of $p$-series. For rational $p>1$ the series ${\sum }_{k\ge 1}{k}^{-p}$ converges, while at $p=1$ the harmonic series ${\sum }_{k\ge 1}1\mathrm{/}k$ diverges; in particular ${\sum }_{n\ge 1}{n}^{-1\mathrm{/}2}$ is not summable because its terms dominate the harmonic terms for $n\ge 1$ (For rational $p>0$, $\sum 1\mathrm{/}{k}^p$ converges iff $p>1$).

Countable Choice is the standing hypothesis (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).