I'm considering vector spaces over $\mathbb{R}$ or $\mathbb{C}$ : $(E_n)_n$ is a sequence of vector space (all included in $\mathbb{R}^p$ for example, with $p$ fixed). What meaning(s) do we usually give to :
- $E_n$ converges to $E$ a vector space of $\mathbb{R}^p$.
Does the dimension of $E_n$ then converge to $\text{dim} E$?
I am not really an expert, but you might use the following notion of convergence: a sequence $(F_j)_{j \in \mathbb{N}}$ of non-empty closed subsets of $\mathbb{R}^m$ is said to converge to another non-empty closed subset of $F$ of $\mathbb{R}^m$ if $$ \lim_{j \to \infty} \sup_{x \in F_j \cap B(0,r)} \operatorname{dist}(x,F)=0$$ AND $$ \lim_{j \to \infty} \sup_{x \in F \cap B(0,r)} \operatorname{dist}(x,F_j)=0$$ for all $r>0$. Of course the suprema are understood to vanish when $F_j \cap B(0,r)$ and $F \cap B(0,r)$ are empty. This is used often in the context of metric spaces. You can find further info e.g. in Fractured Fractals and Broken Dreams by Semmes.