example of a linear map in real space for which $0$ is the only eigenvalue but the map is not nilpotent? What would be an example of a linear map in real space for which $0$ is the only eigenvalue but the map is not nilpotent? Only hint is enough.
If

$$A=\left(\begin{matrix}
0&amp;1&amp;0\\
-1&amp;0&amp;0\\
0&amp;0&amp;0
\end{matrix}\right)$$
then $\chi_A=-x(x^2+1)$ (which we calculate easily using block matrices) so $0$ is the only eigenvalue in $\mathbb R$.
The matrix $A$ isn't nilpotent since $A^3=-A$.
This cannot happen in finite dimension. 

Hint: Such a map would have a Jordan normal form with zeros on the diagonal.