Manuscript XV-B@2 · XV-B.tex · proposition · 4d3f12e8

Proposition: Descent onto the reconstructed image

Reconstruction and the exact relative-lifting criterion › The linear interface · lines 380–407 · proof included or linked

XV-B@2#prop:descent, XV-B@2#eq:descent

A linear map $F : P_{int} \to V$ has a unique factorization

F = M Rec, M : K_{int} \to V,

if and only if

F ∣_{H} = 0.

If $r F = c$ , the induced map satisfies $r M = m^{\circ}$ .

Proof. A factorization annihilates $ker Rec$ .

Conversely, suppose $F ∣_{H} = 0$ and define

M (Rec A) = F (A) .

If $Rec A = Rec B$ , then $A - B \in H$ , so $F (A) = F (B)$ . Thus $M$ is well defined and linear. Surjectivity of $Rec$ gives uniqueness. Finally,

r M (Rec A) = r F (A) = c (A) = m^{\circ} (Rec A) .

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