Let $(H_n)$ a collection enumerable of Hilbert spaces. Consider $$ H=\bigoplus_{n=1}^\infty H_n=\left\{(x_n):\sum_{n=1}^\infty\|x_n\|^2<\infty\right\}. $$ If each $H_n$ is separable, then $H$ also is separable?
If $H_n=\mathbb{C}$ is immediately to check: $H=\ell_2(\mathbb{N})$. But, the problem is for arbitrary Hilbert space $H_n$.
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Maybe one way to resolve is using what Sambo commented. So, if I consider $$ K=\bigoplus^{\operatorname{alg}}H_n=\left\{(x_n)\in H:x_n\neq0\mbox{ for finite choices of }x_n\right\}=\bigcup_{n=1}^\infty \times_{k=1}^n H_k\times_{k=n+1}^\infty\{0\}, $$ we have that $\overline{K}=H$. It's clear that $K$ is separable, since each $\times_{k=1}^n H_k\times_{k=n+1}^\infty\{0\}$ is separable, so the reunion must be separable too. Hence, $H$ is separable for being closure of $K$.