Linear isomorphisms with dense graph

Asked Sep 06 '11 at 17:24

Active Nov 28 '13 at 10:12

Viewed 184 times

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.

edited Sep 28 '11 at 09:30

asked Sep 06 '11 at 17:24

Richard

Do you mean with dense image? – rfauffar Sep 06 '11 at 17:42
@Robert: dense image would be trivial. – Willie Wong Sep 06 '11 at 17:48
1

@Willie: Of course, just wanted to be sure. – rfauffar Sep 06 '11 at 17:52
1

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 06 '11 at 18:21
1

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
@WillieWong, why don't you post this as answer? – Norbert Nov 27 '13 at 23:23
@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... – Willie Wong Nov 28 '13 at 09:27

0 Answers0