\to \operatorname{Gr}(k,m)$, where $\iota_{nm} \coloneqq \iota_n \circ \iota_{n+1} \circ \dots \circ \iota_{m-1}$. To be more concrete, if $A \in \mathbb{R}^{n \times k}$ has orthonormal columns, then $$\label{eq:iota}
\iota_{nm} : \operatorname{Gr}(k,n) \to \operatorname{Gr}(k,m), \qquad
\operatorname{span} (A) \mapsto \operatorname{span} \left( \begin{bmatrix}A \\ 0 \end{bmatrix}  \right),$$ where the zero block matrix is $(m-n) \times k$ so that $\left[ \begin{smallmatrix}A \\ 0 \end{smallmatrix} \right] \in \mathbb{R}^{m \times k}$.

For a fixed $k$, the family of Grassmannians $\{ \operatorname{Gr}(k,n): n\in \mathbb{N}, \; n \ge k \}$ together with the inclusion maps $\iota_{nm}:\operatorname{Gr}(k,n)\to \operatorname{Gr}(k,m)$ for $m > n$ form a direct system. The *infinite Grassmannian* of $k$-planes is defined to be the direct limit of this system in the category of topological spaces and denoted by $$\operatorname{Gr}(k,\infty)\coloneqq \varinjlim\operatorname{Gr}(k,n).$$ Those unfamiliar with the notion of direct limits may simply take $$\operatorname{Gr}(k, \infty) = \bigcup\nolimits_{n=k}^\infty \operatorname{Gr}(k,n),$$ where we regard $\operatorname{Gr}(k,n) \subset \operatorname{Gr}(k,n+1)$ by identifying $\operatorname{Gr}(k,n)$ with $\iota_n\bigl(\operatorname{Gr}(k,n)\bigr)$. With this identification, we no longer need to distinguish between $\mathbf{A} \in \operatorname{Gr}(k,n)$ and its image $\iota_{n}(\mathbf{A})\in \operatorname{Gr}(k,n+1)$ and may regard $\mathbf{A} \in \operatorname{Gr}(k,m)$ for all $m > n$.

We now define a distance $d_{\operatorname{Gr}(k, \infty)}$ on $\operatorname{Gr}(k,\infty)$ that is consistent with the Grassmann distance on $\operatorname{Gr}(k,n)$ for all $n$ sufficiently 