Unconditional convergence for $L^1_{[-T,T]}$ Norm

Question

I have $f \in L^1$, $T > 0$ and $P_T f$ with $P_T f(x) = \sum_{k \in \mathbb{Z}} f(x+2kT)$

I want to show that this series converges unconditionally regarding the norm in $L^1[-T,T]$.

Since I know that absolute convergence implies unconditional convergence for $\mathbb{R}$ and a countable set $I$ I have to show that $\sum_{k \in \mathbb{Z}}||f(x+2kT)||_{L^1_{[-T,T]}}$ converges. But since $f \in L^1$ $||f(x+2kT)||_{L^1_{[-T,T]}} < \infty$ and $\sum_{k \in \mathbb{Z}}||f(x+2kT)||_{L^1_{[-T,T]}} < \infty$.

Is it that simple? I'm not sure if I understand the problem correctly.