Hilbert spaces — the Hilbert cube inside $\ell^2$

Let $\ell^2$ denote the vector space of all square summable sequences $\{a_n\}_{n\ge1}$ of real numbers with the inner product given by $\langle\{a_n\},\{b_n\}\rangle=\sum_{n=1}^{\infty}a_nb_n$. Let

$$H=\left\{\{a_n\}\in\ell^2:|a_n|\le\frac1n \text{for all positive integers }n\right\}.$$

Which of the following statements are true?