Local boundedness

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 f defined on some topological space X is called locally bounded if for any x₀ in X there exists a neighborhood A of x₀ such that f(A) is a bounded set. That is, for some number M > 0 one has

|f(x)|\leq M

for all x in A.

In other words, 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 (see below).

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

d\left(f(x),a\right)\leq M

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.

Examples

The function f: R → R defined by

f(x)={\frac {1}{x^{2}+1}}

is bounded, because 0 ≤ f(x) ≤ 1 for all x. Therefore, it is also locally bounded.

The function f: R → R defined by

f(x)=2x+3

is not bounded, as it becomes arbitrarily large. However, it is locally bounded because for each a, |f(x)| ≤ M in the neighborhood (a − 1, a + 1), where M = 2|a| + 5.

The function f: R → R defined by

f(x)={\begin{cases}{\frac {1}{x}},&{\mbox{if }}x\neq 0,\\0,&{\mbox{if }}x=0\end{cases}}

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 f: U → R be continuous where U ⊆ R, and we will show that f is locally bounded at a for all a in U. Taking ε = 1 in the definition of continuity, there exists δ > 0 such that |f(x) − f(a)| < 1 for all x in U with |x − a| < δ. Now by the triangle inequality, |f(x)| = |f(x) − f(a) + f(a)| ≤ |f(x) − f(a)| + |f(a)| < 1 + |f(a)|, which means that f is locally bounded at a (taking M = 1 + |f(a)| and the neighborhood (a − δ, a + δ)). This argument generalises easily to when the domain of f is any topological space.

The converse of the above result is not true however, i.e. a discontinuous function may be locally bounded. For example consider the function f: R → R given by f(0) = 1 and f(x) = 0 for all x ≠ 0. Then f is discontinuous at 0 but f is locally bounded; it is locally constant apart from at zero, where we can take M = 1 and the neighborhood (−1, 1), for example.

Locally bounded family

A set (also called a family) U of real-valued or complex-valued functions defined on some topological space X is called locally bounded if for any x₀ in X there exists a neighborhood A of x₀ and a positive number M such that

|f(x)|\leq M

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.

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

The family of functions f_n: R → R

f_{n}(x)={\frac {x}{n}}

where n = 1, 2, ... is locally bounded. Indeed, if x₀ is a real number, one can choose the neighborhood A to be the interval (x₀ − 1, x₀ + 1). Then for all x in this interval and for all n ≥ 1 one has

|f_{n}(x)|\leq M

with M = |x₀| + 1. Moreover, the family is uniformly bounded, because neither the neighborhood A nor the constant M depend on the index n.

The family of functions f_n: R → R

f_{n}(x)={\frac {1}{x^{2}+n^{2}}}

is locally bounded, if n is greater than zero. For any x₀ one can choose the neighborhood A to be R itself. Then we have

|f_{n}(x)|\leq M

with M = 1. Note that the value of M does not depend on the choice of x₀ or its neighborhood A. This family is then not only locally bounded, it is also uniformly bounded.

The family of functions f_n: R → R

f_{n}(x)=x+n

is not locally bounded. Indeed, for any x₀ the values f_n(x₀) cannot be bounded as n tends toward infinity.

Topological vector spaces

Local boundedness may also refer to a property of topological vector spaces, 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 neighborhood U of 0 in X there exists a scalar s > 0 such that

B ⊂ tU for all t > s.

A topological vector space is said to be locally bounded if X admits a bounded 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 map id_X: X → X is locally bounded.

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