In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, bounded function

Bounded function

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...

s are functions

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 for which there exists a lower bound and an upper bound

Upper bound

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...

, in other words, a constant which is larger than the absolute value

Absolute value

In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

 of any value of this function. If we consider a family of bounded functions, this constant can vary between functions. If it is possible to find one constant which bounds all functions, this family of functions is uniformly bounded.

The uniform boundedness principle

Uniform boundedness principle

In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field...

 in functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

 provides sufficient conditions for uniform boundedness of a family of operators.

Real line and complex plane

Let
be a family of functions indexed

Index set

In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index set...

 by , where is an arbitrary set and is the set of real

Real

Real may also refer to:* Reality, the state of things as they actually exist, rather than as they may appear or may be thought to be.-Finance:* Inflation adjusted amountsCurrencies:* Brazilian realFormer currencies:* Mexican real* Portuguese real...

 or complex

Complex

A complex is a whole that comprehends a number of intricate parts, especially one with interconnected or mutually related parts; for example, a complex of buildings.Complex may refer to:-Biology:...

 numbers. We call uniformly bounded if there exists a real number such that

Metric space

In general let be a metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

 with metric , then the set
is called uniformly bounded if there exists an element from and a real number such that

Examples

• Every uniformly convergent sequence of bounded functions is uniformly bounded.

• The family of functions defined for real
  Real number
  In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
  
    with traveling through the integer
  Integer
  The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
  
  s, is uniformly bounded by 1.

• The family of derivative
  Derivative
  In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
  
  s of the above family, is not uniformly bounded. Each is bounded by but there is no real number such that for all integers