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The reverse triangle inequality in a normed space
Statement
Let be a normed space. For all , In particular, the norm map is -Lipschitz for the norm metric.
Facts & Assumptions
Given: A normed space and vectors .
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
Since , the triangle inequality in [L1] gives .
Since and by absolute homogeneity at the scalar , [L1] also gives .
Step 1.1 yields , and step 1.2 yields ; together these are exactly .
The displayed inequality says precisely that the map is -Lipschitz for the metric .
Depends on
Used by
- The L² unit sphere is not weakly sequentially closed in infinite dimensions Counterexample
- Cutoff extension across a puncture in complex dimension two Example
- Property (T) for finite groups via normalized counting measure Example
- A nonzero compact scalar identity forces finite dimension Lemma
- Mean value inequality for a differentiable Banach-valued curve Lemma
- Neumann series and small perturbations of bounded inverses Lemma
- Square-summable orthogonal families have norm-convergent finite sums Lemma
- The Cauchy-class operations of a normed-space completion are well defined Lemma
- Unit spheres in finite-dimensional normed spaces are sequentially compact Lemma
- Vector addition and scalar multiplication are continuous in a normed space Lemma
- Weak H¹ convergence plus Rellich preserves the L² unit normalisation Lemma
- Fourier expansion in a Hilbert space Theorem
- Parseval equivalences for an orthonormal family Theorem
- Projection onto a nonempty closed convex set Theorem
- SLn(R) has property (T) for n at least three Theorem
- The norm completion of an inner-product space is a Hilbert space Theorem
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)