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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 edit
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
Examples edit
- The function defined by
- The function defined by
- The function defined by
- 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 edit
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
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 edit
- The family of functions
- The family of functions
- The family of functions
Topological vector spaces edit
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 edit
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
Locally bounded functions edit
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 edit
- Bornological space – Space where bounded operators are continuous
- Bounded operator – Linear transformation between topological vector spaces
- Bounded set (topological vector space) – Generalization of boundedness