Good evening to everyone. I am new to Manifold Theory, so I am trying the last weeks to study some chapters from the book of John M. Lee 's Introduction to Smooth Manifolds.
I was trying to understand this evening the concept of the Grassmannian manifold, which he introduced on pages 23–24 in the book. More specifically, there he was trying to endow the space of $k$-dimensional subspaces of a $n$-dimensional space $V$ (obviously $k \leq n$) with a smooth structure using a preceding Lemma he had mentioned earlier in the book.
I understood most of the part, but I was stuck with the proof of the Hausdorff property of the induced topology (which is the last condition remaining for the application of his Lemma).So, he mentions the following paragraph:
Hausdorff condition (v) is easily verified by noting that for any two $k$-dimensional subspaces $P, P' \leq V$, it is possible to find a subspace $Q$ of dimension $n-k$ whose intersections with both $P$ and $P'$ are trivial.
I do not understand this statement. If we suppose that $\dim V = 5$ and $P = \operatorname{span}\{e_1, e_2, e_3\}$ and $P' = \operatorname{span} \{e_3, e_4, e_5\}$, how do we find a subspace $Q$ of dimension $2$ that has trivial intersections with both $P, P'$?
Thank you for your help.
As already stated in the comments, $Q = \operatorname{span}\{e_1 + e_4, e_2 + e_5\}$ does the trick, since no vector in $P$ has a non-trivial fourth of fifth while no vector in $P'$ has a non-trivial first or second coordinate.