  We study $L_2$-approximation problems $\text{APP}_d$ in the worst case
setting in the weighted Korobov spaces $H_{d,\a,{\bm \ga}}$ with parameter
sequences ${\bm \ga}=\{\ga_j\}$ and $\a=\{\az_j\}$ of positive real numbers
$1\ge \ga_1\ge \ga_2\ge \cdots\ge 0$ and $\frac1 2<\az_1\le \az_2\le \cdots$.
We consider the minimal worst case error $e(n,\text{APP}_d)$ of algorithms that
use $n$ arbitrary continuous linear functionals with $d$ variables. We study
polynomial convergence of the minimal worst case error, which means that
$e(n,\text{APP}_d)$ converges to zero polynomially fast with increasing $n$. We
recall the notions of polynomial, strongly polynomial, weak and
$(t_1,t_2)$-weak tractability. In particular, polynomial tractability means
that we need a polynomial number of arbitrary continuous linear functionals in
$d$ and $\va^{-1}$ with the accuracy $\va$ of the approximation. We obtain that
the matching necessary and sufficient condition on the sequences ${\bm \ga}$
and $\a$ for strongly polynomial tractability or polynomial tractability is
$$\dz:=\liminf_{j\to\infty}\frac{\ln \ga_j^{-1}}{\ln j}>0,$$ and the exponent
of strongly polynomial tractability is $$p^{\text{str}}=2\max\big\{\frac 1 \dz,
\frac 1 {2\az_1}\big\}.$$
