Manuscript III@2 · III.tex · theorem · 82dad3e5

Theorem: Well-posed reduced Cauchy problem

The C1 triangular Cauchy quotient › A closed-range membership test · lines 1979–2010 · proof included or linked

III@2#thm:reduced-Cauchy

Let

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

and

S_{C}^{red} = S_{C} / {(K ξ, X_{ξ} Φ) : □_{η} ξ = 0, ξ has compact Cauchy data} .

These are Hausdorff locally convex quotient spaces. Evolution induces a continuous linear bijection with continuous inverse:

X_{C}^{red} ≃ S_{C}^{red} .

Proof. The unreduced Cauchy isomorphism maps $g (α, β)$ precisely to $(K ξ, X_{ξ} Φ)$ , by covariance and uniqueness. It intertwines the equivalence relations.

The data range is closed by Lemma lem:closed-range. Its image under the isomorphism is closed as well. Both quotients are Hausdorff. The continuous inverse maps induce continuous inverse quotient maps.

Displayed formulas in this passage

unlabeled · line 1982
$X_{C}^{red} = X_{C} / g (G_{C}),$
unlabeled · line 1986
$S_{C}^{red} = S_{C} / {(K ξ, X_{ξ} Φ) : □_{η} ξ = 0, ξ has compact Cauchy data} .$
unlabeled · line 1995
$X_{C}^{red} ≃ S_{C}^{red} .$

Theorem: Well-posed reduced Cauchy problem

Displayed formulas in this passage

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