Let $X$ be the completed space with your norm.

Using the general fact that $|x|_p \leq |x|_q$ if $p \geq q$, your norm dominates the $\ell^2$ norm on sequences, and hence the completion of your space $X$ embeds into $\ell^2$.

You have $\ell^1$ is dense in $X$ (defined as the completion) using the $|\cdot|$ norm. Since the set of finite sequences are dense in $\ell^1$ with respect to the $|\cdot|_1$ norm, which dominates $|\cdot|$, you have that finite sequences are also dense in $X$.

Given a sequence $x$, write $P_Nx$ to be the finite sequence that equals $x$ in the first $N$ slots and $0$ afterward. That $x\in X$ if and only if $P_Nx \in X$ for all $N$ and $|P_N x|$ is uniformly bounded.

However, if $x\in \ell^2 \setminus \ell^p$ for any $p > 1$, then necessarily $|P_N x|_q$ for all $q \in [1,p]$ is not uniformly bounded, this shows that $x\in X$ implies that $x\in \ell^p$ for all $p > 1$.

And so Jochen's formulation holds, that

• $x\in X$ if and only if $x\in \cap_{1 < p \leq 2} \ell^p$ and $|x| < \infty$.

The general fact on the monotonicity of the $\ell^p$ norms also shows that

$$ (p-1)|x|_p = \int_1^p |x|_p ~dq \leq \int_1^p |x|_q ~dq \leq |x| $$

so that $X$ embeds into every $\ell^p$ continuously. Hence $X^*$ contains all $\ell^q$ for $q < \infty$. On the other hand, the embedding of $\ell^1$ into $X$ shows that $X^*$ is contained in $\ell^\infty$.

We can show that $X^*$ is strictly larger than $\cup_{q < \infty} \ell^q$:

• Let $y_k \in \ell^{2^k + 1}\setminus \ell^{2^{k-1} + 1}$ for $k \in \mathbb{N}_0$, be such that $|y_k|_{2^k + 1} \leq 2^{-k}$. Then $\sum_k y_k$ converges to some sequence in $\ell^\infty$.
• The pairing $|\langle x, \sum y_k\rangle| \leq \sum |x|_{1 + 2^{-k}} |y_k|_{1 + 2^k} \leq \sum 2^{-k} |x|_{1 + 2^{-k}} \approx |x|$ and so we see that $\sum y_k \in X^*$.

(I am hopeful that this example actually leads to a characterization of $X^*$, but am not sure how to prove it, nor what would be a nicer norm for it.)

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.