Sunday, 19 August 2007

big list - What is your favorite "strange" function?

Just a simple construction to illustrate Nate Eldredge's answer about functions with dense graphs. Pick any mathbbR-vector space E with a norm. On E, choose a non-continuous linear form L:EtomathbbR; now this can only be done if dim(E)=infty, of course.



Then, pick y such that L(y)=1, and let T:EtoE be defined by Tx=xL(x)y. Then obviously T maps E onto the kernel of L; it is not difficult to prove that ker(L) must be dense in E for any non-continuous L (the two conditions are even equivalent), and thus the graph of T must be dense in EtimesE.

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