I was reading Treves book and, on page 527, defined the following objects. A sequence $\tau = (\tau_{p})_{p\in \mathbb{Z}^{d}}$ is of slowly growing if there exists an integer $k \ge 0$ and a constant $C \ge 0$ such that: $$ (1+|p|)^{-k}|\tau_{p}| \le C $$ for every $p \in \mathbb{Z}^{d}$. The space of all slowly growing sequences is $s'$.
However, I came across a different definition of $s'$ which is the following: $s'$ is the space of all sequences $\tau = (\tau_{p})_{p\in \mathbb{Z}^{d}}$ satisfying: $$\sum_{p\in \mathbb{Z}^{d}}(1+|p|)^{k}|\tau_{p}| < +\infty$$ for some integer $k \ge 0$. I'm trying to prove that this spaces are the same. I already proved one implication, which is:
$(\Rightarrow)$ Suppose the second condition holds. Then, because $(1+|p|)^{-k}\le (1+|p|)^{k}$ for every $p \in \mathbb{Z}^{d}$, then: $$(1+|p|)^{-k}|\tau_{p}| \le (1+|p|)^{k}|\tau_{p}| \le \sum_{p\in \mathbb{Z}^{d}}(1+|p|)^{k}|\tau_{p}| < +\infty$$ and the first condition follows. But I'm really stuck proving the converse. Can anyone help me please?
NOTE: I tried to find some nice references on these sequences spaces (Treves book introduces it briefly) and its alternative representations. So, any good reference is also welcomed.
You meant $\sum_{p\in \mathbb{Z}^{d}}(1+|p|)^{-k}|\tau_{p}| < +\infty$ which implies that $(1+|p|)^{-k}|\tau_{p}|<C$, conversely if $(1+|p|)^{-k}|\tau_{p}|<C$ then $\sum_{p\in \mathbb{Z}^{d}}(1+|p|)^{-k-d-1}|\tau_{p}| < +\infty$. This space of polynomially bounded sequences $S'(\Bbb{Z}^d)$ is the dual of the space of fast decreasing sequence $S(\Bbb{Z}^d)$, ie. the map $x\to \sum_k x_k \tau_k$ is a well-defined linear functional continuous in the $S(\Bbb{Z}^d)$ topology.