Local boundedness

Summary

In mathematics, a function is locally bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around that point and by the same number.

Locally bounded function

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A real-valued or complex-valued function   defined on some topological space   is called a locally bounded functional if for any   there exists a neighborhood   of   such that   is a bounded set. That is, for some number   one has  

In other words, for each   one can find a constant, depending on   which is larger than all the values of the function in the neighborhood of   Compare this with a bounded function, for which the constant does not depend on   Obviously, if a function is bounded then it is locally bounded. The converse is not true in general (see below).

This definition can be extended to the case when   takes values in some metric space   Then the inequality above needs to be replaced with   where   is some point in the metric space. The choice of   does not affect the definition; choosing a different   will at most increase the constant   for which this inequality is true.

Examples

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  • The function   defined by   is bounded, because   for all   Therefore, it is also locally bounded.
  • The function   defined by   is not bounded, as it becomes arbitrarily large. However, it is locally bounded because for each     in the neighborhood   where  
  • The function   defined by   is neither bounded nor locally bounded. In any neighborhood of 0 this function takes values of arbitrarily large magnitude.
  • Any continuous function is locally bounded. Here is a proof for functions of a real variable. Let   be continuous where   and we will show that   is locally bounded at   for all   Taking ε = 1 in the definition of continuity, there exists   such that   for all   with  . Now by the triangle inequality,   which means that   is locally bounded at   (taking   and the neighborhood  ). This argument generalizes easily to when the domain of   is any topological space.
  • The converse of the above result is not true however; that is, a discontinuous function may be locally bounded. For example consider the function   given by   and   for all   Then   is discontinuous at 0 but   is locally bounded; it is locally constant apart from at zero, where we can take   and the neighborhood   for example.

Locally bounded family

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A set (also called a family) U of real-valued or complex-valued functions defined on some topological space   is called locally bounded if for any   there exists a neighborhood   of   and a positive number   such that   for all   and   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.

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.

Examples

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  • The family of functions     where   is locally bounded. Indeed, if   is a real number, one can choose the neighborhood   to be the interval   Then for all   in this interval and for all   one has   with   Moreover, the family is uniformly bounded, because neither the neighborhood   nor the constant   depend on the index  
  • The family of functions     is locally bounded, if   is greater than zero. For any   one can choose the neighborhood   to be   itself. Then we have   with   Note that the value of   does not depend on the choice of x0 or its neighborhood   This family is then not only locally bounded, it is also uniformly bounded.
  • The family of functions     is not locally bounded. Indeed, for any   the values   cannot be bounded as   tends toward infinity.

Topological vector spaces

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Local boundedness may also refer to a property of topological vector spaces, or of functions from a topological space into a topological vector space (TVS).

Locally bounded topological vector spaces

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A subset   of a topological vector space (TVS)   is called bounded if for each neighborhood   of the origin in   there exists a real number   such that   A locally bounded TVS is a TVS that possesses a bounded neighborhood of the origin. By Kolmogorov's normability criterion, this is true of a locally convex space if and only if the topology of the TVS is induced by some seminorm. In particular, every locally bounded TVS is pseudometrizable.

Locally bounded functions

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Let   a function between topological vector spaces is said to be a locally bounded function if every point of   has a neighborhood whose image under   is bounded.

The following theorem relates local boundedness of functions with the local boundedness of topological vector spaces:

Theorem. A topological vector space   is locally bounded if and only if the identity map   is locally bounded.

See also

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  • PlanetMath entry for Locally Bounded
  • nLab entry for Locally Bounded Category