Completeness and linear domain of the diagonal sequence model

The sequence is norm bounded, say $\mathrm{\parallel }{x}^{(n)}\mathrm{\parallel }\le M$. For each coordinate, $\mathrm{\mid }{x}_i^{(n)}-x_i^{(m)}\mathrm{\mid }\le \mathrm{\parallel }{x}^{(n)}-x^{(m)}\mathrm{\parallel }$, so [F2] gives a unique limit $x_i$. For every finite $F\subset I$, passing to the limit in its finite sum gives ${\sum }_{i\in F}\mathrm{\mid }{x}_i{\mathrm{\mid }}^2\le M^2$. Taking the finite-subset supremum proves $x\in {\mathrm{\ell }}^2(I)$. No selection of nonunique coordinate limits is involved.

If $x,y\in D(H_E)$ and $a,b\in \mathbb{C}$, then for every finite $F$, ${\sum }_{i\in F}\mathrm{\mid }{E}_i(a{x}_i+b{y}_i){\mathrm{\mid }}^2\le 2\mathrm{\mid }a{\mathrm{\mid }}^2{\sum }_{i\in F}\mathrm{\mid }{E}_i{x}_i{\mathrm{\mid }}^2+2\mathrm{\mid }b{\mathrm{\mid }}^2{\sum }_{i\in F}\mathrm{\mid }{E}_i{y}_i{\mathrm{\mid }}^2\mathrm{.}$ The right side is bounded independently of $F$, so $ax+by\in D(H_E)$; the zero vector is included. The coordinate action is linear and has square-summable output by the domain definition. Every finite-support vector belongs to this domain because each $E_i$ is finite.

Given $ϵ>0$, choose $N$ with $\mathrm{\parallel }{x}^{(n)}-x^{(m)}\mathrm{\parallel }\le ϵ$ for all $n,m\ge N$. For fixed $n\ge N$ and finite $F$, let $m\to \mathrm{\infty }$ to obtain ${\sum }_{i\in F}\mathrm{\mid }{x}_i^{(n)}-x_i{\mathrm{\mid }}^2\le {ϵ}^2$. Taking the supremum over $F$ gives $\mathrm{\parallel }{x}^{(n)}-x\mathrm{\parallel }\le ϵ$. Thus the space is complete and is Hilbert by [F3].

For any $x\in {\mathrm{\ell }}^2(I)$ and $ϵ>0$, the finite-subset supremum supplies a finite $F$ whose square sum differs from the total by less than ${ϵ}^2$. For every finite $G\subset I\setminus F$, its square sum is at most that difference, by applying the total bound to $F\cup G$. Hence the norm of the omitted tail is at most $ϵ$. Finite-coordinate truncations therefore approximate $x$ and lie in $D(H_E)$ by step 1.2, proving density. If $I$ is empty all spaces and maps are zero, and the same statements hold directly.