LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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The reverse triangle inequality in a normed space

Statement

Let $V$ be a normed space. For all $x, y \in V$ , $∣ ∥ x ∥ - ∥ y ∥ ∣ \leq ∥ x - y ∥ .$ In particular, the norm map is $1$ -Lipschitz for the norm metric.

Facts & Assumptions

Given: A normed space $V$ and vectors $x, y \in V$ .

[L1]

A norm satisfies the triangle inequality and absolute homogeneity (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms).

Proof

technique · direct

1.1

Since $x = (x - y) + y$ , the triangle inequality in [L1] gives $∥ x ∥ \leq ∥ x - y ∥ + ∥ y ∥$ .

1.2

Since $y = (y - x) + x$ and $∥ y - x ∥ = ∥ x - y ∥$ by absolute homogeneity at the scalar $- 1$ , [L1] also gives $∥ y ∥ \leq ∥ x - y ∥ + ∥ x ∥$ .

L1algebra

2.1

Step 1.1 yields $∥ x ∥ - ∥ y ∥ \leq ∥ x - y ∥$ , and step 1.2 yields $∥ y ∥ - ∥ x ∥ \leq ∥ x - y ∥$ ; together these are exactly $∣ ∥ x ∥ - ∥ y ∥ ∣ \leq ∥ x - y ∥$ .

step 1.1step 1.2algebra

3.1

The displayed inequality says precisely that the map $x \mapsto ∥ x ∥$ is $1$ -Lipschitz for the metric $d (u, v) = ∥ u - v ∥$ .

step 2.1∎

Depends on

A norm on a real vector space, the induced metric, and the dictionary with the metric axioms

Used by

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Theo Buhler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)
Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis (standard reference, not scraped)