Difference between revisions of "Equivalent conditions to a set being bounded/Statement"

Revision as of 23:37, 29 October 2016

Statement

Let [ilmath](X,d)[/ilmath] be a metric space and let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath]. Then the following are all logical equivalent to each other^{[Note 1]}:

1. [ilmath]A[/ilmath] is bounded
2. [ilmath]\forall x\in X\exists C<\infty\forall a\in A[d(a,x)<C][/ilmath]^[1]

Notes

1. ↑ Just in case the reader isn't sure what this means, if [ilmath]A[/ilmath] and [ilmath]B[/ilmath] are logically equivalent then:
   • [ilmath]A\iff B[/ilmath]. In words "[ilmath]A[/ilmath] if and only if [ilmath]B[/ilmath]"

References

1. ↑ Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha