Linear Operator


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Related to Linear Operator: Linear differential operator

linear operator

[′lin·ē·ər ′äp·ə‚rād·ər]
(mathematics)
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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 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 xE, 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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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