A closed-form convergence criterion for the weak greedy algorithm

In 2002, V. N. Temlyakov established a criterion for the convergence of the weak greedy algorithm in a Hilbert space for a given weakness sequence $\tau =\{t_1,t_2,\dots \}$. The criterion requires verifying a certain limiting relation for every nonnegative square-summable sequence. We give an equivalent closed-form criterion: the weak greedy algorithm converges if and only if ${\sum }_{n=1}^{\mathrm{\infty }}(1+n{\sum }_{k=1}^n{t}_k^2{)}^{-1\mathrm{/}2}{t}_n^2=+\mathrm{\infty }$.