# Linear Operator

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## linear operator

[′lin·ē·ər ′äp·ə‚rād·ər]## Linear Operator

a generalization of the concept of linear transformation to vector spaces. *F* is called a linear operator on a vector space *E* if it is a function on *E* with values in some vector space *E*_{1} and has the linearity property

*F*(*αx* + *βy*) = *αF*(*x*) + *βF*(*y*)

where *x* and *y* are elements of *E*, and *α* and *β* are numbers. If the *E* and *E*_{1} are normed spaces and ǀǀ*F*(*x*)ǀǀ / ǀǀ*x*ǀǀ is uniformly bounded for all *x* ∈ *E*, then the linear operator *F* is said to be bounded and

is called its norm.

The most important linear operators on function spaces are the differential linear operators

and the integral linear operators

The Laplace operator is an example of a linear operator on a space of functions of many variables. The theory of linear operators finds numerous applications in various problems of mathematical physics and applied mathematics.