Schwartz kernel theorem

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Problem 442 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection III.4.4 "Generalized Functions"

Prove that every continuous linear mapping A:D(Ω1)→D′(Ω2)A:\mathcal D(\Omega_1)\to \mathcal D'(\Omega_2) has the form

⟨Aφ1,φ2⟩=∫Ω1×Ω2K(x,y)φ1(x)φ2(y) dx dy, \langle A\varphi _1,\varphi _2\rangle =\int _{\Omega _1\times \Omega _2}K(x,y)\varphi _1(x)\varphi _2(y)\, dx\, dy,

where K∈D′(Ω1×Ω2)K\in \mathcal D'(\Omega_1\times \Omega_2).

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Problem 443 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection III.4.4 "Generalized Functions"

In the conditions of Problem 442 let Ω1=Ω2=R\Omega_1=\Omega_2=\mathbb {R}. Find explicitly the generalized function K∈D′(R2)K\in \mathcal D'(\mathbb {R}^{2}) if the mapping AA

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(a)

is the natural imbedding of D(R)\mathcal D(\mathbb {R}) into D′(R)\mathcal D'(\mathbb {R});

(b)

has the form φ↦φ(a)⋅δb\varphi \mapsto \varphi (a)\cdot \delta_b.

References

[KiGv88] Kirillov A.A., Gvishiani A.D. Теоремы и задачи функционального анализа. 2-е изд., Наука, 1988.