Bounded Sets and Bounded Functions in a Metric Space

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Bounded Sets in a Metric Space

Definition: Let be a metric space. A subset is said to be Bounded if there exists a positive real number such that for some . The set is said to be Unbounded if it is not bounded.

By the definition above, we see that is bounded if there exists some open ball with a finite radius that contains .

For example, consider the metric space where is the discrete metric defined for all by:

Let . Then by the definition of the discrete metric, for all we have that . Therefore, if we consider any point and take then:

Therefore, is bounded. This shows that every subset of is bounded with respect to the discrete metric. In fact, the wholeset is also bounded and for any .

For another example, consider the metric space where is the Euclidean metric. Consider the following set:

The set above is the first octant of , and is actually unbounded. To prove this, suppose that instead is bounded. Then there exists a maximal distance between some pair of points , say:

Then . For consider the point . Then:

Since for each we see that:

But then implies which is a contradiction. Therefore our assumption that was bounded is false.

Bounded Functions

A schematic illustration of a bounded function (red) and an unbounded one (blue). Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does not.

In mathematics, a function f defined on some set X with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number M such that

for all x in X. A function that is not bounded is said to be unbounded.

If f is real-valued and f(x) ≤ A for all x in X, then the function is said to be bounded (from) above by A. If f(x) ≥ B for all x in X, then the function is said to be bounded (from) below by B. A real-valued function is bounded if and only if it is bounded from above and below.

An important special case is a bounded sequence, where X is taken to be the set N of natural numbers. Thus a sequence f = (a0, a1, a2, ...) is bounded if there exists a real number M such that

for every natural number n. The set of all bounded sequences forms the sequence space .

The definition of boundedness can be generalized to functions f : X → Y taking values in a more general space Y by requiring that the image f(X) is a bounded set in Y.

Related notions

Weaker than boundedness is local boundedness. A family of bounded functions may be uniformly bounded.

A bounded operator T : X → Y is not a bounded function in the sense of this page's definition (unless T = 0), but has the weaker property of preserving boundedness: Bounded sets M ⊆ X are mapped to bounded sets T(M) ⊆ Y. This definition can be extended to any function f : XY if X and Y allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph.

Examples

  • The sine function sin : RR is bounded since for all .
  • The function , defined for all real x except for −1 and 1, is unbounded. As x approaches −1 or 1, the values of this function get larger and larger in magnitude. This function can be made bounded if one considers its domain to be, for example, [2, ∞) or (−∞, −2].
  • The function , defined for all real x, is bounded.
  • The inverse trigonometric function arctangent defined as: y = arctan(x) or x = tan(y) is increasing for all real numbers x and bounded with −Template:Sfrac < y < Template:Sfrac radians
  • By the boundedness theorem, every continuous function on a closed interval, such as f : [0, 1] → R, is bounded. More generally, any continuous function from a compact space into a metric space is bounded.
  • All complex-valued functions f : CC which are entire are either unbounded or constant as a consequence of Liouville's theorem. In particular, the complex sin : CC must be unbounded since it is entire.
  • The function f which takes the value 0 for x rational number and 1 for x irrational number (cf. Dirichlet function) is bounded. Thus, a function does not need to be "nice" in order to be bounded. The set of all bounded functions defined on [0, 1] is much larger than the set of continuous functions on that interval.Template:Citation needed Moreover, continuous functions need not be bounded; for example, the functions and defined by and are both continuous, but neither is bounded. (However, a continuous function must be bounded if its domain is both closed and bounded.)

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