It is well known that the differential operator is an unbounded operator on the space of all continuously differentiable function on $[0,1]$. However,I found difficulties in finding an unbounded operator from $C[0,1]$ to $C[0,1]$, where $C[0,1]$ is the space of continuous function under sup-norm. Can someone explicitly give me an example of such operator?

EDIT: Can someone provide an unbounded operator from $X$ to $Y$ where $X$ and $Y$ are Banach space?($X,Y$ are to be determined)

  • $\begingroup$ You mean, everywhere defined and unbounded? $\endgroup$ – abatkai Dec 17 '12 at 12:23
  • $\begingroup$ @abatkai: linear operator on the whole $C[0,1]$ but not bounded. $\endgroup$ – Ben Dec 17 '12 at 12:24
  • $\begingroup$ It's enough to find an unbounded linear functional, but I'm not sure we can give an explicit expression. $\endgroup$ – Davide Giraudo Dec 17 '12 at 12:26
  • $\begingroup$ You can definitely give one by using a Hamel basis. This is not constructive, but quite good. $\endgroup$ – abatkai Dec 17 '12 at 12:28
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    $\begingroup$ You have to use the axiom of choice to construct such an operator, so I would say this rules out having an "explicit" formula. $\endgroup$ – Nate Eldredge Dec 17 '12 at 12:34

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