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

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(Created page with "<noinclude> ==Statement== </noinclude>Let {{M|(X,d)}} be a metric space and let {{M|A\in\mathcal{P}(X)}} be an arbitrary subset of {{M|X}}. Then the following are all...")
 
m (Added symbolic form of bounded)
 
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</noinclude>Let {{M|(X,d)}} be a [[metric space]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}}. Then the following are all logical equivalent to each other<ref group="Note">Just in case the reader isn't sure what this means, if {{M|A}} and {{M|B}} are logically equivalent then:
 
</noinclude>Let {{M|(X,d)}} be a [[metric space]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}}. Then the following are all logical equivalent to each other<ref group="Note">Just in case the reader isn't sure what this means, if {{M|A}} and {{M|B}} are logically equivalent then:
 
* {{M|A\iff B}}. In words "{{M|A}} {{iff}} {{M|B}}"</ref>:
 
* {{M|A\iff B}}. In words "{{M|A}} {{iff}} {{M|B}}"</ref>:
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# {{M|A}} is ''bounded''
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# {{M|1=\exists C<\infty\ \forall a,b\in A[d(a,b)<C]}} - {{M|A}} is [[bounded]] (the definition)
 
# {{M|1=\forall x\in X\exists C<\infty\forall a\in A[d(a,x)<C]}}{{rFAVIDMH}}
 
# {{M|1=\forall x\in X\exists C<\infty\forall a\in A[d(a,x)<C]}}{{rFAVIDMH}}
 
<noinclude>
 
<noinclude>

Latest revision as of 23:41, 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]\exists C<\infty\ \forall a,b\in A[d(a,b)<C][/ilmath] - [ilmath]A[/ilmath] is bounded (the definition)
  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
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