# Linear isomorphisms with dense graph

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? – rfauffar Sep 6 '11 at 17:42
@Robert: dense image would be trivial. – Willie Wong Sep 6 '11 at 17:48
@Willie: Of course, just wanted to be sure. – rfauffar Sep 6 '11 at 17:52
If I am not mistaken, the answer would follow from applying the answer you got for this question, by considering a bijection between two different Hamel bases. – Willie Wong Sep 6 '11 at 18:21
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. – Srivatsan Sep 27 '11 at 15:53