In functional analysis (a branch of mathematics), a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in X. In other words, there exists some M > 0 such that for all v in X, Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ...

The smallest such M is called the operator norm of L. In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...

Let us note that a bounded linear operator is not necessarily a bounded function; the latter would require that the norm of L(v) is bounded for all v. Rather, a bounded linear operator is a locally bounded function. If we have a map where B is a poset, then f is a bounded function if the image of f, as a subset of B has an upper and lower bound. ...

It is quite easy to prove that a linear operator L is bounded if and only if it is a continuous function from X to Y. In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

Examples

• Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrix.

• Many integral transforms are bounded linear operators. For instance, if

is a continuous function, then the operator L, defined on the space L¹[a,b] of Lebesgue integrable functions with values in the space L¹[c,d]

is bounded.

• The Laplacian operator

(its domain is a Sobolev space and it takes values in a space of square integrable functions) is bounded.

• The shift operator on the l² space of all sequences (x₀, x₁, x₂...) of real numbers with

is bounded. Its norm is easily seen to be 1.

Not every linear operator between normed spaces is bounded. Let X be the space of all trigonometric polynomials defined on [−π, π], with the norm For the square matrix section, see square matrix. ... In mathematics, an integral transform is any transform T of the following form: The input of this transform is a function f, and the output is another function Tf. ... The integral can be interpreted as the area under a curve. ... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ... In mathematics, a Sobolev space is a normed space of functions obtained by imposing on a function f and its derivatives up to some order k the condition of finite Lp norm, for given p ≥ 1. ... In mathematics, the term integrable function refers to a function whose integral may be calculated. ... In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... This is a page about mathematics. ... In the mathematical subfield of numerical analysis, a trigonometric polynomial is a finite linear linear combination of sin(nx) and cos(nx) with n a natural number. ...

Define the operator L:X→X which acts by taking the derivative, so it maps a polynomial P to its derivative P′. Then, for In mathematics, the derivative is one of the two central concepts of calculus. ...

v = e^inx

with n=1, 2, ...., we have while as n→∞, so this operator is not bounded.

Further properties

A common procedure for defining a bounded linear operator between two given Banach spaces is as follows. First, define a linear operator on a dense subset of the domain, such that it is locally bounded. Then, extend the operator by continuity to a continuous linear operator on the whole domain (see continuous linear extension). In mathematics, the term dense has at least three different meanings. ... In functional analysis, it is often convenient to define something on a normed vector space by defining it on a dense set and extending it to the whole space. ...

See also

• operator norm
• operator algebra
• operator theory
• closed operator