x↦eix,
is an eigenfunction of the differential operator
(H, Φ) with Φ is given a topological vector space structure for which the inclusion map
H, the adjoint to Φ and *</sup>"}},"i":0}}]}'/>Φ is then compatible with the inner product on u∈Φ⊂H and v (physics convention), and conjugate-linear (complex anti-linear) in the other variable.
The triple Φ is isomorphic to *</sup>"}},"i":0}}]}'/>Φ (via Riesz representation) if it happens that i*
Φ which carries a finer topology, that is one for which the natural inclusion
H for the Hilbert norm. We consider the inclusion of dual spaces *</sup>"}},"i":0}}]}'/>H in *</sup>"}},"i":0}}]}'/>Φ. The latter, dual to ϕ↦⟨v,ϕ⟩
for Φ dense).
Now by applying the Riesz representation theorem we can identify *</sup>"}},"i":0}}]}'/>H with Φ is a nuclear space; this comment is an abstract expression of the idea that Φ and its dual 𝒮′(ℝ), respectively, rigging the Hilbert space of square-integrable functions. As such, the rigged Hilbert space is given by
H=L2(ℝn), Φ=Hs(ℝn), Φ∗=H−s(ℝn),
where 0"}}'>s>0.