continuous linear mapping

If (V1,1) and (V2,2) are normed vector spacesPlanetmathPlanetmath, a linear mapping T:V1V2 is continuousMathworldPlanetmathPlanetmath if it is continuous in the metric induced by the norms.

If there is a nonnegative constant c such that T(x)2cx1 for each xV1, we say that T is . This should not be confused with the usual terminology referring to a bounded function as one that has boundedPlanetmathPlanetmathPlanetmathPlanetmath range. In fact, bounded linear mappings usually have unboundedPlanetmathPlanetmath ranges.

The expression bounded linear mapping is often used in functional analysisMathworldPlanetmathPlanetmath to refer to continuous linear mappings as well. This is because the two definitions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

If T is bounded, then T(x)-T(y)2=T(x-y)2cx-y1, so T is a Lipschitz function. Now suppose T is continuous. Then there exists r>0 such that T(x)21 when x1r. For any xV1, we then have


hence T(x)2rx1; so T is bounded.

It can be shown that a linear mapping between two topological vector spacesMathworldPlanetmath is continuous if and only if it is continuous at ( 0 [1].


  • 1 W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.
Title continuous linear mapping
Canonical name ContinuousLinearMapping
Date of creation 2013-03-22 13:15:41
Last modified on 2013-03-22 13:15:41
Owner Koro (127)
Last modified by Koro (127)
Numerical id 7
Author Koro (127)
Entry type Definition
Classification msc 46B99
Synonym bounded linear mapping
Related topic HomomorphismsOfCAlgebrasAreContinuous
Related topic CAlgebra
Related topic BoundedLinearFunctionalsOnLpmu
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