Manuscript XIX-E@1 · XIX-E.tex · lemma · 702f146e

Lemma: Asymptotic transfer requires a map

Asymptotics, states, and quantum certificate boundaries › A source coefficient needs its actual comparison map · lines 941–961 · proof included or linked

XIX-E@1#lem:asymptotic

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Let $E, F$ be topological vector spaces and $B : E \to F$ continuous and linear. Suppose

t_{δ} = δ^{- r} C + R_{δ}, δ^{r} R_{δ} ⟶ 0.

Then

δ^{r} B t_{δ} ⟶ B C .

Nonvanishing of $C$ does not imply nonvanishing of $B C$ .

Proof. Apply $B$ to $δ^{r} t_{δ} = C + δ^{r} R_{δ}$ . Continuity proves the limit. A nonzero $C$ may lie in $ker B$ , even when $B \neq = 0$ .

Displayed formulas in this passage

unlabeled · line 945
$t_{δ} = δ^{- r} C + R_{δ}, δ^{r} R_{δ} ⟶ 0.$
unlabeled · line 950
$δ^{r} B t_{δ} ⟶ B C .$

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section: A source coefficient needs its actual comparison map