Manuscript VII-C@1 · VII-C.tex · section · 8b2ade29

Normal-topology stages

The named topology and regular approximations › Normal-topology stages · lines 883–919

For a compact $K \subset M \times M$ , write

E^{'}_{Γ_{B}} (K) = {u \in E^{'} (M \times M) : supp u \subset K, WF (u) \subset Γ_{B}},

componentwise in the finite-dimensional tensor indices. It carries the normal topology.

Explicitly, this topology is generated by strong distribution seminorms

p_{B} (u) = ψ \in B sup ∣ ⟨ u, ψ ⟩ ∣,

where $B$ is a bounded set of test functions, and by

p_{χ, V, N} (u) = ζ \in V sup (1 + ∣ ζ ∣)^{N} ∣ χ u (ζ) ∣,

where $V$ is a closed cone and

(supp χ \times V) \cap Γ_{B} = \emptyset .

The normal topology and the distinction between continuity and hypocontinuity are important here DB,BDH.

The translation-invariant angular cone $Λ_{B}$ in (eq:GammaB) is deliberately an upper-bound cone. The theorem does not presume that it is the smallest possible cone.

Displayed formulas in this passage

unlabeled · line 886
$E^{'}_{Γ_{B}} (K) = {u \in E^{'} (M \times M) : supp u \subset K, WF (u) \subset Γ_{B}},$
unlabeled · line 897
$p_{B} (u) = ψ \in B sup ∣ ⟨ u, ψ ⟩ ∣,$
unlabeled · line 902
$p_{χ, V, N} (u) = ζ \in V sup (1 + ∣ ζ ∣)^{N} ∣ χ u (ζ) ∣,$
unlabeled · line 909
$(supp χ \times V) \cap Γ_{B} = \emptyset .$