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

From Maths
Jump to: navigation, search
(Created page with "{{Stub page|grade=A|msg=Created to document, textbook stub}} __TOC__ ==Statement== {{/Statement}} ==Proof of claims== {{begin Inline Theorem}} /1 implies 2|{{...")
 
(Demoted stub grade, added note to link with metrically bounded set)
 
Line 1: Line 1:
−
{{Stub page|grade=A|msg=Created to document, textbook stub}}
+
{{Stub page|grade=D|msg=Created to document, textbook stub}}
 +
{{Requires work|grade=A*|msg=Cleanup required. New [[Metrically bounded set]] page could link to this in another form. Make sure the two are compatible [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 23:12, 18 March 2017 (UTC) }}
 
__TOC__
 
__TOC__
 
==[[/Statement|Statement]]==
 
==[[/Statement|Statement]]==

Latest revision as of 23:12, 18 March 2017

Stub grade: D
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Created to document, textbook stub
Grade: A*
This page requires some work to be carried out
Some aspect of this page is incomplete and work is required to finish it
The message provided is:
Cleanup required. New Metrically bounded set page could link to this in another form. Make sure the two are compatible Alec (talk) 23:12, 18 March 2017 (UTC)

Statement

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

  1. ∃C<∞ ∀a,b∈A[d(a,b)<C] - A is bounded (the definition)
  2. ∀x∈X∃C<∞∀a∈A[d(a,x)<C][1]

Proof of claims

[Expand]

1⟹2) (∃C<∞ ∀a,b∈A[d(a,b)<C])⟹(∀x∈X∃C<∞∀a∈A[d(a,x)<C]), that boundedness implies condition 2

[Expand]

2⟹1) (∀x∈X∃C<∞∀a∈A[d(a,x)<C])⟹(∃C<∞ ∀a,b∈A[d(a,b)<C]), that condition 2 implies boundedness

Notes

  1. Jump up ↑ Just in case the reader isn't sure what this means, if A and B are logically equivalent then:

References

  1. Jump up ↑ Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha
Retrieved from "https://www.maths.kisogo.com/index.php?title=Equivalent_conditions_to_a_set_being_bounded&oldid=4353"