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Is it true that for each infinite dimensional Banach space $X$ there exists a linear bijection $f: X \rightarrow X$ with a dense graph?

A graph of $f$ it is the set $\Gamma(f):=\{(x, f(x)): x \in X \} \subset X \times X$.

($X\times X$ is a Banach space with natural addition and multiplication by scalars and norm defined by $\|(x,y)\|=\|x\|+\|y\|$ for $x,y \in X$.)

It seems that it is true when $X$ is separable.

Thanks.

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    Do you mean with dense image?2011-09-06
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    @Robert: dense image would be trivial.2011-09-06
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    @Willie: Of course, just wanted to be sure.2011-09-06
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    If I am not mistaken, the answer would follow from applying the answer you got for [this question](http://math.stackexchange.com/q/60057/1543), by considering a bijection between two different Hamel bases.2011-09-06
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    Is the term "graphic" standard? Actually, "graph" sounds a little better to me. Also, perhaps you can add the definition of the graph of $f$ into the question text itself.2011-09-27
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    @WillieWong, why don't you post this as answer?2013-11-27
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    @Norbert: go ahead if you feel like it. It being two years on I will have to re-read both questions, that answer, and try to solve the hint I left...2013-11-28

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