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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)

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    Since Banach space must have uncountable basis,is it true that it is not easy to (explicitly) find an unbounded operator between two arbitrary Banach spaces?2012-12-17

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