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In [[mathematics]], a [[function (mathematics)|function]] is '''locally bounded''' if it is [[bounded function|bounded]] around every point. A [[Family (disambiguation)#Mathematics|family]] of functions is '''locally bounded''' if for any point in their [[domain (mathematics)|domain]] all the functions are bounded around that point and by the same number.  


==Locally bounded function==
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A [[real-valued function|real-valued]] or [[complex-valued function|complex-valued]] function ''f'' defined on some [[topological space]] ''X'' is called '''locally bounded'''
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if for any ''x''<sub>0</sub> in ''X'' there exists a [[neighborhood (mathematics)|neighborhood]] ''A'' of ''x''<sub>0</sub> such that
''f''&nbsp;(''A'') is a [[bounded set]], that is, for some number ''M''>0 one has
:<math>|f(x)|\le M</math>
for all ''x'' in ''A''.


That is to say, for each ''x'' one can find a constant, depending on ''x'', which is larger than all the values of the function in the neighborhood of ''x''. Compare this with a bounded function, for which the constant does not depend on ''x''. Obviously, if a function is bounded then it is locally bounded. The converse is not true in general.
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This definition can be extended to the case when ''f'' takes values in some [[metric space]]. Then the inequality above needs to be replaced with
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:<math>d\left(f(x), a\right)\le M</math>
for all ''x'' in ''A'', where ''d'' is the distance function in the metric space, and ''a'' is some point in the metric space. The choice of ''a'' does not affect the definition. Choosing a different ''a'' will at most increase the constant ''M'' for which this inequality is true.


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* The function ''f'': '''R''' → '''R''' defined by
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:<math>f(x)=\frac{1}{x^2+1}\,</math>
is bounded, because 0≤ ''f''&nbsp;(''x'') ≤ 1 for all ''x''. Therefore it is also locally bounded.


*  The function ''f'': '''R''' → '''R''' defined by
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:<math>f(x)=2x+3\,</math>
 
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* The function ''f'':'''R''' → '''R''' defined by
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:<math>f(x)=\frac{1}{x}\,</math>
for ''x'' ≠ 0 and taking the value 0 for ''x''=0 is ''not'' locally bounded. In any neighborhood of 0 this function takes values of arbitrarily large magnitude.


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A set (also called a [[Index set|family]]) ''U'' of real-valued or complex-valued functions defined on some topological space ''X'' is called '''locally bounded''' if for any ''x''<sub>0</sub> in ''X'' there exists a [[neighborhood (topology)|neighborhood]] ''A'' of ''x''<sub>0</sub> and a positive number ''M'' such that
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:<math>|f(x)|\le M</math>
 
for all ''x'' in ''A'' and ''f'' in ''U''. In other words, all the functions in the family must be locally bounded, and around each point they need to be bounded by the same constant.
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This definition can also be extended to the case when the functions in the family ''U''  take values in some metric space, by again replacing the absolute value with the distance function.  
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* The family of functions ''f''<sub>n</sub>:'''R'''→'''R'''
</ul>
:<math>f_n(x)=\frac{x}{n}</math>
where ''n'' = 1, 2, ... is uniformly bounded. Indeed, if ''x''<sub>0</sub> is a real number, one can choose the neighborhood ''A'' to be the interval (''x''<sub>0</sub>-1''x''<sub>0</sub>+1). Then for all ''x'' in this interval and for all ''n''≥1 one has
:<math>|f_n(x)|\le M</math>
with ''M''=|''x''<sub>0</sub>|+1.
 
* The family of functions ''f''<sub>n</sub>:'''R'''→'''R'''
:<math>f_n(x)=\frac{1}{x^2+n^2}</math>
is locally bounded. For any ''x''<sub>0</sub> one can choose the neighborhood ''A'' to be '''R''' itself. Then we have
:<math>|f_n(x)|\le M</math>
with ''M''=1. Note that the value of ''M'' does not depend on the choice of x<sub>0</sub> or its neighborhood ''A''. This family is then not only locally bounded, it is also [[uniformly bounded]].
 
* The family of functions ''f''<sub>n</sub>:'''R'''→'''R'''
:<math>f_n(x)=x+n</math>
is ''not'' locally bounded. Indeed, for any ''x''<sub>0</sub> the values ''f''<sub>n</sub>(''x''<sub>0</sub>) cannot be bounded as ''n'' tends toward infinity.  
 
==Topological vector spaces==
Local boundedness may also refer to a property of [[topological vector space]]s, or of functions from a [[topological space]] into a topological vector space.
 
===Locally bounded topological vector spaces===
Let ''X'' be a topological vector space. Then a subset ''B'' ⊂ ''X'' is '''bounded''' if for each open neighborhood ''U'' of 0 in ''X'' there exists a number ''m'' > 0 such that
:''B'' &sub; ''xU'' for all ''x'' > ''m''.
A topological vector space is said to be '''locally bounded''' if ''X'' admits a bounded open neighborhood of 0.
 
===Locally bounded functions===
Let ''X'' be a topological space, ''Y'' a topological vector space, and ''f'' : ''X'' → ''Y'' a function. Then ''f'' is '''locally bounded''' if each point of ''X'' has a neighborhood whose image under ''f'' is bounded.
 
The following theorem relates local boundedness of functions with the local boundedness of topological vector spaces:
:'''Theorem.''' A topological vector space ''X'' is locally bounded if and only if the identity mapping '''1''' : ''X'' &rarr; ''X'' is locally bounded.
 
==External links==
* {{planetmath reference|title=Locally bounded|id=5752}}
 
{{DEFAULTSORT:Local Boundedness}}
[[Category:Mathematical analysis]]

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