Bounded operator

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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,

\|L(v)\|_{Y}\leq M\|v\|_{X}.\,

The smallest such M is called the operator norm \|L\|_{{op}} 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.
K:[a,b]\times [c,d]\to {{\mathbf  R}}
is a continuous function, then the operator L, defined on the space L^{1}[a,b] of Lebesgue integrable functions with values in the space L^{1}[c,d]
(Lf)(y)=\int _{{a}}^{{b}}\!K(x,y)f(x)\,dx,
is bounded.
\Delta :H^{2}({{\mathbf  R}}^{n})\to L^{2}({{\mathbf  R}}^{n})
(its domain is a Sobolev space and it takes values in a space of square integrable functions) is bounded.
L(x_{0},x_{1},x_{2},\dots )=(0,x_{0},x_{1},x_{2},\dots )
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

\|P\|=\int _{{-\pi }}^{{\pi }}\!|P(x)|\,dx.

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

v=e^{{inx}}

with n=1, 2, ...., we have \|v\|=2\pi , while \|L(v)\|=2\pi n\to \infty 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).

See also