Let $\Lambda\_{n}$ be the set of all Lagrangian subspaces of $C^{n}$, and $P\in\Lambda\_{n}$. Put $U\_{P}= ( Q\in\Lambda\_{n} : Q\cap (iP)=0 )$. There is an assertion that the set $U\_{P}$ is homeomorphic to the real vector space of all symmetric endomorphisms of $P$. And then in the proof of it there is a fact that the subspaces $Q$ that intersect $iP$ only at $0$ are the graphs of the linear maps $\phi : P\to iP$. This is what I don't understand, any explanation or reference where I can find it would be helpful.



Lagrangian submanifolds 
It is very elementary; the graph representation refers to the cartesian product $P\times iP$ coming from the real vector space direct sum decomposition $\mathbb{C}^n = P\oplus iP\sim P\times iP$.


Here the direct sum decomposition is possible because $\dim \_ \mathbb{R}(P)=n $ and $P\cap iP=0$, from the definition of Lagrangian subspace (thus any $z$ writes uniquely as $z=x+iy$ with $x$ and $y$ in $P$). 


The second projection $\pi\_2: P\times iP \to iP$, restricted to a subspace $Q$, has kernel $Q\cap iP$, so it is injective exactly for $Q\in U\_P$, in which case $Q=\operatorname{graph}\\ \pi\_1({\pi\_2}{|Q})^{-1}$. 


For other details check e.g. the introductory pages of Hofer-Zehnder's book, *Symplectic Invariants and Halmiltonian Dynamics*. 


