Local boundedness explained
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
A real-valued or complex-valued function
defined on some
topological space
is called a
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
defined by
is bounded, because
for all
Therefore, it is also locally bounded.
defined by
is bounded, as it becomes arbitrarily large. However, it locally bounded because for each
in the neighborhood
where
defined by
is neither bounded 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,
|f(x)|=|f(x)-f(a)+f(a)|\leq|f(x)-f(a)|+|f(a)|<1+|f(a)|,
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
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
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
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 x
0 or its neighborhood
This family is then not only locally bounded, it is also uniformly bounded.
is locally bounded. Indeed, for any
the values
cannot be bounded as
tends toward infinity.
Topological vector spaces
See also: Bounded set (topological vector space) and Kolmogorov's normability criterion. 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
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
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
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.
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