LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Adapted norm balls are ordinary Euclidean neighbourhoods

Statement

Let $m$ be a nonnegative integer, let $p \in R^{m}$ , let $N$ be any norm on $R^{m}$ , and let $ρ > 0$ . Write $∣ \cdot ∣_{2}$ for the Euclidean norm and $d_{2} (y, z) = ∣ y - z ∣_{2}$ for its metric. The set $B_{N} (0, ρ) = {y \in R^{m} : N (y) < ρ}$ is an open Euclidean neighbourhood of zero. Its translate $p + B_{N} (0, ρ) = {p + y : y \in B_{N} (0, ρ)}$ is an open Euclidean neighbourhood of $p$ .

The following specifies the shrinking assertion for orbit trapping. Let $B$ be either of these two neighbourhoods, let $B^{'} \subseteq B$ , let $X$ be a set, let $Φ_{n} : X \to R^{m}$ be maps for every integer $n \geq 0$ , and let $W \subseteq X$ . If, for every $x \in X$ , the condition $Φ_{n} (x) \in B$ for every $n \geq 0$ implies $x \in W$ , then the condition $Φ_{n} (x) \in B^{'}$ for every $n \geq 0$ also implies $x \in W$ . In particular, $B$ may be replaced by any smaller open neighbourhood of its centre in such an implication.

Facts & Assumptions

[F1]

For $m \geq 1$ , norm equivalence gives constants $c, C > 0$ with $c ∣ y ∣_{2} \leq N (y) \leq C ∣ y ∣_{2}$ for every $y \in R^{m}$ For n ≥ 1 all norms on ℝⁿ are equivalent.

[F2]

The norm axioms give $N (0) = 0$ and the induced metric $d_{N} (y, z) = N (y - z)$ A norm on a real vector space, the induced metric, and the dictionary with the metric axioms. The reverse triangle inequality gives $∣ N (y) - N (z) ∣ \leq N (y - z)$ The reverse triangle inequality in a normed space.

[F3]

The Euclidean norm is $∣ y ∣_{2} = (\sum_{k < m} y_{k}^{2})^{1 / 2}$ , also when $m = 0$ , in which case its value is zero The p-norms ‖ x‖ₚ for rational p ≥ 1, and ‖ x‖_∞. It is a norm for every $m \geq 0$ ; for $m \geq 1$ its induced metric is the usual Euclidean metric Each ‖·‖ₚ is a norm on ℝⁿ, and the induced metrics are exactly d₁, d₂ and d_∞ of the published metric-spaces page.

[F4]

A metric ball is $B_{d} (a, r) = {z : d (a, z) < r}$ for $r > 0$ Open ball, closed ball and sphere in a metric space. A set is open when every point has a positive-radius metric ball inside it; an open neighbourhood of $a$ is an open set containing $a$ The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement.

Proof

Given: The dimension, norm, radius, point, sets and family of maps specified in the Statement.

Goal: Prove both neighbourhood assertions and the quantified shrinking implication.

1.1givenF1F2F3F4

Suppose $m \geq 1$ . By the reverse triangle inequality and the upper estimate in [F1], $∣ N (y) - N (z) ∣ \leq N (y - z) \leq C ∣ y - z ∣_{2}$ . For each $y \in B_{N} (0, ρ)$ set $ε = (ρ - N (y)) / C > 0$ . If $d_{2} (z, y) < ε$ , then $N (z) \leq N (y) + C ∣ z - y ∣_{2} < ρ$ . Thus $B_{d_{2}} (y, ε) \subseteq B_{N} (0, ρ)$ , proving openness by [F4]. Also $N (0) = 0 < ρ$ , and $∣ z ∣_{2} < ρ / C$ implies $N (z) < ρ$ , so this is an open neighbourhood of zero containing $B_{d_{2}} (0, ρ / C)$ .

2.1givenF2F3F4step 1.1

For $m \geq 1$ , translation satisfies $d_{2} (p + z, p + y) = ∣ (p + z) - (p + y) ∣_{2} = ∣ z - y ∣_{2} = d_{2} (z, y)$ . Hence the ball about $y$ constructed in step 1.1 translates to a Euclidean ball about $p + y$ inside $p + B_{N} (0, ρ)$ ; that set is open and contains $p$ . For $m = 0$ , $R^{0} = {0}$ , $p = 0$ , and $N (0) = 0 < ρ$ , so both sets are the whole singleton space, which is open: each positive-radius metric ball about its only point is the singleton. Both neighbourhood assertions therefore also hold in dimension zero.

3.1given∎

Fix $x \in X$ such that $Φ_{n} (x) \in B^{'}$ for every integer $n \geq 0$ . For each such $n$ , the inclusion $B^{'} \subseteq B$ gives $Φ_{n} (x) \in B$ . The assumed implication for $B$ therefore gives $x \in W$ . This holds for every $x \in X$ and every subset $B^{'} \subseteq B$ , including every smaller open neighbourhood of the centre. For an orbit-trapping application, take $Φ_{n} (x)$ to be the $n$ th forward iterate on a set $X$ where all these iterates are defined.

Depends on

Used by · 1

Transport local invariance and forward trapping to a fixed point

Dependency tree · 2 levels

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