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

Revision as of 23:37, 29 October 2016

Statement

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

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

Notes

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

References

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