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单词 ContinuousLinearMapping
释义

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 thatT(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

rx1T(x)2=T(rx1x)21,

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 (http://planetmath.org/Continuous) 0 [1].

References

  • 1 W. Rudin, Functional Analysis,McGraw-Hill Book Company, 1973.
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