linear transformation is continuous if its domain is finite dimensional

Theorem 1.

A linear transformation is continuousMathworldPlanetmathPlanetmath if the domain is finite dimensional.


Suppose L:XY is the transformation, dimX=n, and X, Y are the norms on X, Y, respectively. By this result ( and this result (, it suffices to prove that L:XL(X) is continuous when L(X) is equipped with the topology given by Y restricted onto L(X). Also, since continuity and boundedness are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath, it suffices to prove that L is bounded. Let e1,,en be a basis for X such that L is invertiblePlanetmathPlanetmathPlanetmath on span{e1,,ek} and kerL=span{ek+1,,en} for k=1,,n. (The zero map is always continuous.) Let fi=L(ei) for i=1,,k, so that span{f1,,fk}=L(X). Let us define new norms on X and L(X),

xX = i=1nαi2,
yX = i=1kβi2,

for x=i=1nαieiX and y=i=1kβifiY. Since norms on finite dimensional vector spacesMathworldPlanetmath are equivalent, it follows that


for some constants C,D>0. For x=i=1nαieiX,

L(x)Y Di=1kαifiY
= Di=1kαi2
= DxX
= CDxX.

Thus L:XL(X) is bounded. ∎

Title linear transformation is continuous if its domain is finite dimensional
Canonical name LinearTransformationIsContinuousIfItsDomainIsFiniteDimensional
Date of creation 2013-03-22 15:17:59
Last modified on 2013-03-22 15:17:59
Owner matte (1858)
Last modified by matte (1858)
Numerical id 7
Author matte (1858)
Entry type Theorem
Classification msc 15A04