The negative Sobolev space $H^{-1}(\mathrm{\Omega })$

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) and let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, $n\ge 1$, with $H_0^1(\mathrm{\Omega };\mathbb{K})$ the zero-boundary Sobolev space $W_0^{1,2}$ over $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$ (Zero-boundary Sobolev space as a norm closure, Integer-order Sobolev spaces and their norms, The notation $H^k$ and the reserved zero-boundary symbol). Define $H^{-1}(\mathrm{\Omega }):=\{F:H_0^1(\mathrm{\Omega })\to \mathbb{K}:F\text{ is bounded and conjugate-linear}\},$ with $\mathrm{\parallel }F{\mathrm{\parallel }}_{H^{-1}}:={\sup }_{\mathrm{\parallel }v{\mathrm{\parallel }}_{H_0^1}\le 1}\mathrm{\mid }F(v)\mathrm{\mid }\mathrm{.}$ Pairing convention. The pairing $⟨F,v⟩:=F(v)$ is linear in $F$ and conjugate-linear in $v$; it is the dual pairing of $H^{-1}(\mathrm{\Omega })$ with $H_0^1(\mathrm{\Omega })$, not the $L^2$ inner product. The map $F\mapsto \overline{F(\cdot )}$ is an isometric conjugate-linear bijection of $H^{-1}(\mathrm{\Omega })$ onto the Banach dual $(H_0^1(\mathrm{\Omega }){)}^{\ast }$ of The dual space X^* of a normed space and its dual norm (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators); the design's notation $(H_0^1{)}^{\ast }$ is read through this identification, which is the one compatible with the page's sesquilinear convention (linear in the first argument, conjugate-linear in the second) fixed in Bounded, coercive and symmetric sesquilinear forms. The space $H^{-1}(\mathrm{\Omega })$ is a normed space, complete because $(H_0^1(\mathrm{\Omega }){)}^{\ast }$ is complete and $\overline{F(\cdot )}$ is an isometry in both directions (If (Y) is Banach then (\mathcal B(X,Y)) is Banach, Real and imaginary parts, complex conjugation, and modulus). For $f\in L^2(\mathrm{\Omega })$ the map $v\mapsto (f,v{)}_{L^2}$ is the corresponding element of $H^{-1}(\mathrm{\Omega })$ under the conventions of The space $L^p(\mu )$ as the quotient by null functions and Complex Lp classes and Euclidean test-function conventions, where complex integrability is in the sense of the latter; identifying a general element of $H^{-1}$ with an $L^2$ function is an embedding statement, never a definition. Elements of $H^{-1}$ are defined here by their action on Sobolev classes; this does not exclude their identification with distributions through smooth test functions.