Convergence of a sequence of subspaces

Question

Let $E_n\subset \mathbb R^n$ be a sequence of subspaces. What does it mean $E_n$ convergence to a subspace $E\subset \mathbb R^n$? I saw this when reading about hyperbolic sets.

Where can I read about this subject?

Thanks.

Answer 1

If all the subspaces have a common dimension $r$, then a natural object to look at is the Grassmannian $Gr(r,\mathbb{R}^n)$.

In particular one may define a metric $d(E,F) = \|P_E - P_F\|$, where $\|\cdot\|$ is the operator norm and $P_E \colon \mathbb{R}^n \to E$ is the orthogonal projection on $E$. The convergence of $E_n \to E$ can now be taken to mean that $\lim_{n \to \infty} d(E_n, E) = 0$. One more concrete way of computing this metric is via the formula: $$d(E,F) = \sup \{|\langle u,v \rangle| : u \in E, v \in F^\bot, \|u\| = \|v\| = 1\}$$