Difference between revisions of "Bounded Sets and Bounded Functions in a Metric Space"
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= Bounded Functions= | = Bounded Functions= | ||
[[Image:Bounded and unbounded functions.svg|right|thumb|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.]] | [[Image:Bounded and unbounded functions.svg|right|thumb|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 | + | 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 |
:<math>|f(x)|\le M</math> | :<math>|f(x)|\le M</math> | ||
− | + | 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. | 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 | + | An important special case is a '''bounded sequence''', where ''X'' is taken to be the set '''N''' of natural numbers. Thus a sequence ''f'' = (''a''<sub>0</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>, ...) is bounded if there exists a real number ''M'' such that |
:<math>|a_n|\le M</math> | :<math>|a_n|\le M</math> | ||
− | for every natural number ''n''. The set of all bounded sequences forms the | + | for every natural number ''n''. The set of all bounded sequences forms the sequence space <math>l^\infty</math>. |
− | 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 | + | 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 == | == Related notions == | ||
− | Weaker than boundedness is | + | Weaker than boundedness is local boundedness. A family of bounded functions may be uniformly bounded. |
− | A | + | 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'' : ''X'' → ''Y'' if ''X'' and ''Y'' allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph. |
==Examples== | ==Examples== | ||
− | * The | + | * The sine function sin : '''R''' → '''R''' is bounded since <math>|\sin (x)| \le 1</math> for all <math>x \in \mathbf{R}</math>. |
− | * The function <math>f(x)=(x^2-1)^{-1}</math>, 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 <math>f(x)=(x^2-1)^{-1}</math>, 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 <math display="inline">f(x)= (x^2+1)^{-1}</math>, defined for all real ''x'', ''is'' bounded. | + | * The function <math display="inline">f(x)= (x^2+1)^{-1}</math>, defined for all real ''x'', ''is'' bounded. |
− | * The | + | * The inverse trigonometric function arctangent defined as: ''y'' = {{math|arctan(''x'')}} or ''x'' = {{math|[[Tangent (trigonometry)|tan]](''y'')}} is [[monotonic function|increasing]] for all real numbers ''x'' and bounded with −{{sfrac|{{pi}}|2}} < ''y'' < {{sfrac|{{pi}}|2}} radians |
− | * By the [[boundedness theorem]], every [[continuous function]] on a closed interval, such as ''f'' : [0, 1] → '''R''', is bounded. | + | * 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'' : '''C''' → '''C''' which are [[Entire function|entire]] are either unbounded or constant as a consequence of [[Liouville's theorem (complex analysis)|Liouville's theorem]]. | + | *All complex-valued functions ''f'' : '''C''' → '''C''' which are [[Entire function|entire]] are either unbounded or constant as a consequence of [[Liouville's theorem (complex analysis)|Liouville's theorem]]. In particular, the complex sin : '''C''' → '''C''' 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. [[Nowhere continuous function#Dirichlet function|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 function]]s on that interval.{{Citation needed|date= September 2021}} Moreover, continuous functions need not be bounded; for example, the functions <math>g:\mathbb{R}^2\to\mathbb{R}</math> and <math>h: (0, 1)^2\to\mathbb{R}</math> defined by <math>g(x, y) := x + y</math> and <math>h(x, y) := \frac{1}{x+y}</math> are both continuous, but neither is bounded. | + | * The function ''f'' which takes the value 0 for ''x'' [[rational number]] and 1 for ''x'' [[irrational number]] (cf. [[Nowhere continuous function#Dirichlet function|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 function]]s on that interval.{{Citation needed|date= September 2021}} Moreover, continuous functions need not be bounded; for example, the functions <math>g:\mathbb{R}^2\to\mathbb{R}</math> and <math>h: (0, 1)^2\to\mathbb{R}</math> defined by <math>g(x, y) := x + y</math> and <math>h(x, y) := \frac{1}{x+y}</math> are both continuous, but neither is bounded. (However, a continuous function must be bounded if its domain is both closed and bounded.) |
= Licensing = | = Licensing = |
Revision as of 21:55, 29 January 2022
Contents
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
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 : X → Y 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 : R → R 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 : C → C which are entire are either unbounded or constant as a consequence of Liouville's theorem. In particular, the complex sin : C → C 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.)
Licensing
Content obtained and/or adapted from:
- Bounded Sets in a Metric Space, mathonline.wikidot.com under a CC BY-SA license
- [https://en.wikipedia.org/wiki/Bounded_function Bounded function, Wikipedia) under a CC BY-SA license