# Distributional PDE solutions as topological linear duals of PDE solutions

Let

$$P \;\colon\; \Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast)$$

be a formally self-adjoint hyperbolic linear differential operator ($\tilde E^\ast$ denotes the densitized dual of a smooth vector bundle $E$) which has unique advanced and retarded Green functions $\mathrm{G}_{\pm}$. Then by prop. 2.1 in

• Igor Khavkine, "Covariant phase space, constraints, gauge and the Peierls formula", Int. J. Mod. Phys. A, 29, 1430009 (2014) (arXiv:1402.1282)

there is an exact sequence of vector spaces of smooth sections the form

$$0 \to \Gamma_{\Sigma,cp}(E) \overset{P}{\longrightarrow} \Gamma_{\Sigma,cp}(\tilde E^\ast) \overset{\mathrm{G}}{\longrightarrow} \Gamma_{\Sigma,scp}(E) \overset{P}{\longrightarrow} \Gamma_{\Sigma,scp}(\tilde E^\ast) \to 0 \,,$$

where $\mathrm{G} := \mathrm{G}_+ - \mathrm{G}_-$ is the causal Green function, where "cp" means compact support and "scp" means spacelike compact support.

This implies in particular that linear functionals on the space $ker_{scp}(P)$ of solutions are equivalently those linear functionals on all compactly supported sections which are "distributional weak solutions" in that they vanish on the image of $P$:

$$\left( ker_{scp}(P) \right)^\ast \underset{\simeq}{\overset{(-)\circ \mathrm{G}}{\longrightarrow}} \left( \Gamma_{\Sigma,cp}(\tilde E^\ast)/im(P) \right)^\ast$$

Question: With the usual topologies from distribition theory, is the analogous map of linear topological duals of topological vector spaces still an isomorphism?

• Igor Khavkine has kindly provided the proof here: ncatlab.org/nlab/show/… Nov 14, 2017 at 11:09
• Why not make this an answer? Nov 14, 2017 at 21:05