## Closure of $ell^2$ in the space of real sequences

Using the product topology on $overline{mathbb{R}}^omega$, is $ell^2$ (the space of real square summable sequences) a dense subset of $overline{mathbb{R}}^omega$ ?

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Let $A : D(A) subset H rightarrow H$ be unbounded and $B$ be a bounded operator, both of them are self-adjoint, then $(AB)^* = B^*A^*$ and $(BA)^* = A^*B^*$, right? I just wanted to be sure that …

The following is a question I came up with when I was studying the same problem in dimension 1 (for which also I have the questions that follows) but I put in generality. Let $U_1, U_2 subset …