Manuscript IX-Free-2@1 · IX-Free-2.tex · lemma · abb2397a

Lemma: Vanishing of the old overlap of an anchored sum

Gate G: simultaneous new-domain and old-domain gluing › An anchored-domain lemma · lines 1276–1296 · proof included or linked

IX-Free-2@1#lem:anchored

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Let $A$ be a complex vector space and let $B, P \subset A$ satisfy $B \cap P = {0}$ . Then

(A \otimes B + B \otimes A) \cap (P \otimes P) = {0} .

Proof. On $P \oplus B$ , define $π (p + b) = p$ . Extend this linear map algebraically to $A$ , using a vector-space complement, with values in $P$ . Then

π ∣_{P} = id, π ∣_{B} = 0.

The map $π \otimes π$ is the identity on $P \otimes P$ and vanishes on $A \otimes B + B \otimes A$ . An element in their intersection must therefore be both unchanged and annihilated by this map, hence zero.

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$(A \otimes B + B \otimes A) \cap (P \otimes P) = {0} .$
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$π ∣_{P} = id, π ∣_{B} = 0.$

Lemma: Vanishing of the old overlap of an anchored sum

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