Let $X$ be the completed space with your norm. We collect several facts about $X$:
$x\in X$ if and only if $P_Nx \in X$ for all $N$ and $|P_N x|$ is uniformly bounded, where $P_Nx$ is the finite sequence that equals $x$ in the first $N$ slots and has $0$'s afterwards.
$\ell^1$ is dense in $X$ using the $|\cdot|$ norm.
Finite sequences are also dense in $X$, since the set of finite sequences are dense in $\ell^1$ with respect to the $|\cdot|_1$ norm, which dominates $|\cdot|$.
$X$ embeds into $\ell^2$, because the norm dominates the $\ell^2$ norm on sequences, because $|x|_p \leq |x|_q$ for $p \geq q$.
$X\subseteq \ell^p$ for all $p > 1$; if we had $x\in \ell^2 \setminus \ell^p$ for any $p > 1$, then $|P_N x|_q$ for all $q \in [1,p]$ would not be uniformly bounded,
Jochen's formulation holds, i.e. $x\in X$ if and only if $x\in \cap_{1 < p \leq 2} \ell^p$ and $|x| < \infty$.
$X$ embeds into every $\ell^p$ continuously, by the monotonicity of the $\ell^p$ norms and $$ (p-1)|x|_p = \int_1^p |x|_p ~dq \leq \int_1^p |x|_q ~dq \leq |x| $$
We also collect facts about $X^*$:
$\ell^q \subset X^*$ for all finite $q$.
$X^* \subseteq \ell^\infty$, by the embedding of $\ell^1$ into $X$
$\cup_{q < \infty} \ell^q \subsetneq X^*$, since we can find $\sum y_k$ in $X^*-\cup_{q < \infty} \ell^q*$:
- Let $y_k \in \ell^{2^k + 1}\setminus \ell^{2^{k-1} + 1}$ for $k \in \mathbb{N}_0$, be such that $|y_k|_{2^k + 1} \leq 2^{-k}$.
- $\sum_k y_k$ converges to some sequence in $\ell^\infty$.
- $\sum y_k \in X^*$ because $|\langle x, \sum y_k\rangle| \leq \sum |x|_{1 + 2^{-k}} |y_k|_{1 + 2^k} \leq \sum 2^{-k} |x|_{1 + 2^{-k}} \approx |x|$.
I am hopeful that this leads to a characterization of $X^*$, but am not sure how to prove it, nor what would be a nicer norm for it.
