Properties of $\bigcap_{p > 1} \ell_p$ Consider the following space of sequences $$\left\{a=(a_n)_{n\in\mathbb{N}}:a\in\bigcap_{p>1}\ell_p, a_n\in\mathbb{R}\right\}$$ What are some of its properties? What is its relation to $\ell_1$ and $\ell_\infty$?

Obviously, $\ell_\infty\supset\ell_p\supset\ell_1$ ($p>1$), and $\bigcap_{p>1}\ell_p\supset\ell_1$, but $\bigcap_{p>1}\ell_p\ne\ell_1$, because the sequence $x_k=\frac{1}{k}$ belongs to each $\ell_p$ ($p>1$), but not to $\ell_1$.
