Manuscript III-B@1 · III-B-v1.tex · theorem · d2c98f62

Theorem: Hausdorff reduced Cauchy problem

The supported triangular Cauchy quotient › Closedness without admitting Newton tails · lines 1276–1303 · proof included or linked

III-B@1#thm:reduced

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Define

X_{r_{*}}^{red} = X_{r_{*}} / g (G_{r_{*}}),

S_{r_{*}}^{red} = S_{r_{*}} / {(K ξ, X_{ξ} Φ) : □ ξ = 0, (ξ, \partial_{s} ξ) ∣_{0} \in G_{r_{*}}} .

Both spaces are Hausdorff locally convex spaces, and evolution induces a topological linear isomorphism

X_{r_{*}}^{red} ≃ S_{r_{*}}^{red} .

Proof. Theorem thm:Cauchy maps the data gauge range exactly onto the solution gauge range, by covariance and uniqueness. The former is closed by Lemma lem:closed, so both quotients are Hausdorff. The continuous inverse unreduced maps induce continuous inverse quotient maps.

Displayed formulas in this passage

unlabeled · line 1279
$X_{r_{*}}^{red} = X_{r_{*}} / g (G_{r_{*}}),$
unlabeled · line 1283
$S_{r_{*}}^{red} = S_{r_{*}} / {(K ξ, X_{ξ} Φ) : □ ξ = 0, (ξ, \partial_{s} ξ) ∣_{0} \in G_{r_{*}}} .$
unlabeled · line 1291
$X_{r_{*}}^{red} ≃ S_{r_{*}}^{red} .$

Theorem: Hausdorff reduced Cauchy problem

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