Manuscript IX-C@1 · IX-C.tex · lemma · 03ce8e6b

Lemma: Gluing linear coefficient prescriptions

Actual-functional descent of the second stage › Bulk agreement and overlap consistency · lines 975–990 · proof included or linked

IX-C@1#lem:gluing

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Let $D_{1}, D_{2}$ be subspaces of a vector space and $a_{i} : D_{i} \to Y$ linear. A linear map on $D_{1} + D_{2}$ extending both $a_{i}$ exists if and only if

a_{1} = a_{2} on D_{1} \cap D_{2} .

When it exists, it is unique.

Proof. Necessity is immediate. For sufficiency, define $a (d_{1} + d_{2}) = a_{1} (d_{1}) + a_{2} (d_{2})$ . Any two decompositions differ by an element of the intersection, on which the maps agree.

Displayed formulas in this passage

unlabeled · line 980
$a_{1} = a_{2} on D_{1} \cap D_{2} .$

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section: Bulk agreement and overlap consistency