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.
- 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.
- 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.
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).