Diagonal multiplication is self-adjoint

By [F5] the ambient space is Hilbert and the operator domain and action are linear. Every finitely supported sequence belongs to $D(H_E)$. For $x\in {\mathrm{\ell }}^2(I)$, the tails of its square-sum can be made arbitrarily small, so finite-coordinate truncations converge in norm to $x$; finite-support vectors are therefore dense in ${\mathrm{\ell }}^2(I)$.

For $x,y\in D(H_E)$, Cauchy–Schwarz [F4] makes ${\sum }_i\mathrm{\mid }{E}_i{x}_i\overline{y_i}\mathrm{\mid }$ finite, and reality of $E_i$ gives $⟨H_{E}x,y⟩={\sum }_i{E}_i{x}_i\overline{y_i}=⟨x,H_{E}y⟩$. Thus $H_E$ is symmetric.

If $y\in D(H_E^{\ast })$, there is $z\in {\mathrm{\ell }}^2(I)$ such that $⟨H_{E}x,y⟩=⟨x,z⟩$ for every $x\in D(H_E)$. Taking $x=u_i$ gives $E_i\overline{y_i}=\overline{z_i}$, hence $z_i=E_i{y}_i$ for every $i$; since $z\in {\mathrm{\ell }}^2(I)$, $y\in D(H_E)$ and $H_E^{\ast }y=H_{E}y$. Conversely symmetry in step 1.2 shows every $y\in D(H_E)$ is in $D(H_E^{\ast })$ with that value. Therefore $D(H_E^{\ast })=D(H_E)$ and $H_E^{\ast }=H_E$.