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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* = (*a*_{0}, *a*_{1}, *a*_{2}, ...) 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*.

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.

- 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 −π/2 <*y*< π/2 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.

In mathematics, a **continuous function** is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be *discontinuous*. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.

In mathematics, a **metric space** is a set together with a metric on the set. The metric is a function that defines a concept of *distance* between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:

In mathematics, **real analysis** is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

In mathematics, a function *f* is **uniformly continuous** if, roughly speaking, it is possible to guarantee that *f*(*x*) and *f*(*y*) be as close to each other as we please by requiring only that *x* and *y* be sufficiently close to each other; unlike ordinary continuity, where the maximum distance between *f*(*x*) and *f*(*y*) may depend on *x* and *y* themselves.

In mathematical analysis, **Lipschitz continuity**, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the *Lipschitz constant* of the function. For instance, every function that has bounded first derivatives is Lipschitz continuous.

In mathematics, an infinite series of numbers is said to **converge absolutely** if the sum of the absolute values of the summands is finite. More precisely, a real or complex series is said to **converge absolutely** if for some real number . Similarly, an improper integral of a function, , is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if

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 its basic form, it asserts that for a family of continuous linear operators whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.

In calculus, **absolute continuity** is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: *absolute continuity of functions* and *absolute continuity of measures.* These two notions are generalized in different directions. The usual derivative of a function is related to the *Radon–Nikodym derivative*, or *density*, of a measure.

In calculus, the **extreme value theorem** states that if a real-valued function is continuous on the closed interval , then must attain a maximum and a minimum, each at least once. That is, there exist numbers and in such that:

In functional analysis, a **bounded linear operator** is a linear transformation *L* : *X* → *Y* between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector spaces, then L is bounded if and only if there exists some *M* ≥ 0 such that for all x in X,

The **Arzelà–Ascoli theorem** is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.

In Fourier analysis, a **multiplier operator** is a type of linear operator, or transformation of functions. These operators act on a function by altering its Fourier transform. Specifically they multiply the Fourier transform of a function by a specified function known as the **multiplier** or **symbol**. Occasionally, the term *multiplier operator* itself is shortened simply to *multiplier*. In simple terms, the multiplier reshapes the frequencies involved in any function. This class of operators turns out to be broad: general theory shows that a translation-invariant operator on a group which obeys some regularity conditions can be expressed as a multiplier operator, and conversely. Many familiar operators, such as translations and differentiation, are multiplier operators, although there are many more complicated examples such as the Hilbert transform.

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a **function of a real variable** is a function whose domain is the real numbers ℝ, or a subset of ℝ that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the **real functions**, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.

In mathematics, more specifically functional analysis and operator theory, the notion of **unbounded operator** provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.

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.

In mathematics, **weak convergence** in a Hilbert space is convergence of a sequence of points in the weak topology.

In mathematics, **subharmonic** and **superharmonic** functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.

In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions. If the spaces involved are also topological spaces, then it makes sense to ask whether all linear maps are continuous. It turns out that for maps defined on infinite-dimensional topological vector spaces, the answer is generally no: there exist **discontinuous linear maps**. If the domain of definition is complete, it is trickier; such maps can be proven to exist, but the proof relies on the axiom of choice and does not provide an explicit example.

In mathematics, a real or complex-valued function *f* on *d*-dimensional Euclidean space satisfies a **Hölder condition**, or is **Hölder continuous**, when there are nonnegative real constants *C*, α>0, such that

This is a glossary for the terminology in a mathematical field of functional analysis.

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Images, videos and audio are available under their respective licenses.