the same for all $m \ge n$. In particular, $d_{\operatorname{Gr}(k, \infty)}$ is well-defined and yields a distance on $\operatorname{Gr}(k,\infty)$. We summarize these observations below.

\[cor:infty\] The Grassmann distance between two $k$-planes in $\operatorname{Gr}(k,n)$ is the geodesic distance in $\operatorname{Gr}(k,\infty)$ and is therefore independent of $n$. Also, the expression for a distance minimizing geodesic in $\operatorname{Gr}(k,n)$ extends to $\operatorname{Gr}(k,\infty)$.

Lemma \[lem:infty\] also holds for other distances on $\operatorname{Gr}(k,n)$ in Table \[tab:distances\], allowing us to define them on $\operatorname{Gr}(k, \infty)$.

\[lem:inclusion\] For all $m > n$, the inclusion $\iota_{nm}:\operatorname{Gr}(k,n)\to \operatorname{Gr}(k,m)$ is isometric when $\operatorname{Gr}(k,n)$ and $\operatorname{Gr}(k,m)$ are both equipped with the same distance in Table \[tab:distances\], i.e., $$d^*_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{B})=d^*_{\operatorname{Gr}(k,m)}(\iota_{nm}(\mathbf{A}),\iota_{nm}\bigl(\mathbf{B})\bigr),$$ $* = \alpha, \beta, \kappa, \phi, \mu, \rho, \pi, \sigma$. Consequently $d^{\ast}_{\operatorname{Gr}(k,\infty)}$ is well-defined.

$d^{\ast}_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{B})$ and $d^{\ast}_{\operatorname{Gr}(k,n+1)}\bigl(\iota_n(\mathbf{A}),\iota_n\bigl(\mathbf{B})\bigr)$ depend only on the principal angles between $\mathbf{A}$ and $\mathbf{B}$, so the distance remains unchanged under $\iota_n$. Repeated applications to $\iota_n \circ \iota_{n+1} \circ \dots \circ \iota_{m-1} = \iota_{nm}$ yield the required isometry.

Distances between subspaces of different dimensions {#sec:main}
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We now address our main problem. The proposed notion of distance will be that of a point $x \in X$ to a set $S \subset X$ in a metric space $(X,d)$. Recall that this is defined 