Real-order Bessel-potential completion $H^s$

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). For $n\ge 1$ and $s\in \mathbb{R}$, define $H^s({\mathbb{R}}^n)$ to be the normed-space completion of $\mathcal{S}({\mathbb{R}}^n)$ with the positive-definite norm $q_s(u)=\mathrm{\parallel }⟨\xi {⟩}^s\widehat{u}{\mathrm{\parallel }}_2$ from Weighted Fourier candidate norm on Schwartz space and The weighted Fourier seminorm separates Schwartz functions.

Concretely, its elements are equivalence classes $[u_j]$ of norm-Cauchy sequences $(u_j{)}_{j\in \mathbb{N}}$ in Schwartz space, where $(u_j)\sim (v_j)$ exactly when ${\lim }_{j\to \mathrm{\infty }}{q}_s(u_j-v_j)=0$. The metric completion carries the unique compatible Banach-space structure supplied by Completion of a normed space and The metric completion of a normed space carries a unique compatible Banach-space structure; its norm is $\mathrm{\parallel }[u_j]{\mathrm{\parallel }}_{H^s}={\lim }_j{q}_s(u_j)$. The constant-sequence map $u\mapsto [(u,u,\dots )]$ is the canonical dense linear isometry from Schwartz space. At this definition stage $H^s$ is an abstract completion; no identification with a subset of ${\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^n)$ is implicit.

The only choice assumption is Countable Choice, used by the cited metric completion theorem in its countable-sequence construction and completeness argument. No full Axiom of Choice or dependent choice is assumed.