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linear operator[′lin·ē·ər ′äp·ə‚rād·ər]
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 E1 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 E1 are normed spaces and ǀǀF(x)ǀǀ / ǀǀxǀǀ 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.