# Bounded operator

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*,

The smallest such *M* is called the operator norm of *L*.

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.

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*.

## 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 defined on the space of Lebesgue integrable functions with values in the space
- 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
*l*^{2}space of all sequences (*x*_{0},*x*_{1},*x*_{2}...) 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

Define the operator *L*:*X*→*X* which acts by taking the derivative, so it maps a polynomial *P* to its derivative *P*′. Then, for

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).