In mathematics, a **functional calculus** is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory. (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative. Sometimes it is used in relation to types of functional equations, or in logic for systems of predicate calculus.)

If is a function, say a numerical function of a real number, and is an operator, there is no particular reason why the expression should make sense. If it does, then we are no longer using on its original function domain. In the tradition of operational calculus, algebraic expressions in operators are handled irrespective of their meaning. This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of and an matrix. The idea of a functional calculus is to create a *principled* approach to this kind of overloading of the notation.

The most immediate case is to apply polynomial functions to a square matrix, extending what has just been discussed. In the finite-dimensional case, the polynomial functional calculus yields quite a bit of information about the operator. For example, consider the family of polynomials which annihilates an operator . This family is an ideal in the ring of polynomials. Furthermore, it is a nontrivial ideal: let be the finite dimension of the algebra of matrices, then is linearly dependent. So for some scalars , not all equal to 0. This implies that the polynomial lies in the ideal. Since the ring of polynomials is a principal ideal domain, this ideal is generated by some polynomial . Multiplying by a unit if necessary, we can choose to be monic. When this is done, the polynomial is precisely the minimal polynomial of . This polynomial gives deep information about . For instance, a scalar is an eigenvalue of if and only if is a root of . Also, sometimes can be used to calculate the exponential of efficiently.

The polynomial calculus is not as informative in the infinite-dimensional case. Consider the unilateral shift with the polynomials calculus; the ideal defined above is now trivial. Thus one is interested in functional calculi more general than polynomials. The subject is closely linked to spectral theory, since for a diagonal matrix or multiplication operator, it is rather clear what the definitions should be.

In mathematics, any vector space * has a corresponding ***dual vector space** consisting of all linear forms on *, together with the vector space structure of pointwise addition and scalar multiplication by constants.*

In mathematics, a **linear map** is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism.

**Linear algebra** is the branch of mathematics concerning linear equations such as:

In mathematics, an **operator** is generally a mapping or function that acts on elements of a space to produce elements of another space. There is no general definition of an *operator*, but the term is often used in place of *function* when the domain is a set of functions or other structured objects. Also, the domain of an operator is often difficult to be explicitly characterized, and may be extended to related objects. See Operator (physics) for other examples.

A **vector space** is a set of objects called *vectors*, which may be added together and multiplied ("scaled") by numbers, called *scalars*. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector *axioms*. To specify that the scalars are real or complex numbers, the terms **real vector space** and **complex vector space** are often used.

In algebra and algebraic geometry, the **spectrum** of a commutative ring *R*, denoted by , is the set of all prime ideals of *R*. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an **affine scheme**.

In mathematics, particularly linear algebra and functional analysis, a **spectral theorem** is a result about when a linear operator or matrix can be diagonalized. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.

In mathematics, a **self-adjoint operator** on a finite-dimensional complex vector space *V* with inner product is a linear map *A* that is its own adjoint: for all vectors v and w. If *V* is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of *A* is a Hermitian matrix, i.e., equal to its conjugate transpose *A*^{∗}. By the finite-dimensional spectral theorem, *V* has an orthonormal basis such that the matrix of *A* relative to this basis is a diagonal matrix with entries in the real numbers. In this article, we consider generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.

In mathematics, the **exterior product** or **wedge product** of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors and , denoted by , is called a bivector and lives in a space called the *exterior square*, a vector space that is distinct from the original space of vectors. The magnitude of can be interpreted as the area of the parallelogram with sides and , which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning that for all vectors and , but, unlike the cross product, the exterior product is associative.

In mathematics, a **differential operator** is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function.

In linear algebra, a **Jordan normal form**, also known as a **Jordan canonical form** or **JCF**, is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to some basis. Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal, and with identical diagonal entries to the left and below them.

In mathematics, especially in the field of algebra, a **polynomial ring** or **polynomial algebra** is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

In mathematics, a **linear differential equation** is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form

In mathematics, **operator theory** is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis.

In functional analysis, a branch of mathematics, the **Borel functional calculus** is a *functional calculus*, which has particularly broad scope. Thus for instance if *T* is an operator, applying the squaring function *s* → *s*^{2} to *T* yields the operator *T*^{2}. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator −Δ or the exponential

This page lists some **examples of vector spaces**. See vector space for the definitions of terms used on this page. See also: dimension, basis.

In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry.

In mathematics, **holomorphic functional calculus** is functional calculus with holomorphic functions. That is to say, given a holomorphic function *f* of a complex argument *z* and an operator *T*, the aim is to construct an operator, *f*(*T*), which naturally extends the function *f* from complex argument to operator argument. More precisely, the functional calculus defines a continuous algebra homomorphism from the holomorphic functions on a neighbourhood of the spectrum of *T* to the bounded operators.

In functional analysis, the concept of a **compact operator on Hilbert space** is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach.

**Vector logic** is an algebraic model of elementary logic based on matrix algebra. Vector logic assumes that the truth values map on vectors, and that the monadic and dyadic operations are executed by matrix operators. "Vector logic" has also been used to refer to the representation of classical propositional logic as a vector space, in which the unit vectors are propositional variables. Predicate logic can be represented as a vector space of the same type in which the axes represent the predicate letters and . In the vector space for propositional logic the origin represents the false, F, and the infinite periphery represents the true, T, whereas in the space for predicate logic the origin represents "nothing" and the periphery represents the flight from nothing, or "something".

- "Functional calculus",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]

- Media related to Functional calculus at Wikimedia Commons

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