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Minkowski functional of an absorbing set
Definition
For an absorbing subset of a normed space , its Minkowski functional or gauge is
The defining set is nonempty by absorption, so is finite and nonnegative. It is not asserted to be a norm or a seminorm: those conclusions need additional hypotheses on .
Depends on
Used by
- A gauge of a nonbalanced set need not be a seminorm Counterexample
- Gauges of norm balls and finite-dimensional ellipsoids Example
- An open convex neighbourhood is recovered from its gauge Lemma
- The gauge of a convex absorbing set is sublinear Lemma
- The gauge of an absolutely convex absorbing set is a seminorm Lemma
- Separate a point from an open convex set Theorem
Dependency tree · two levels
3 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
- Gerald Teschl, Topics in Real and Functional Analysis, §5.1 (standard reference, not scraped)