&
\delta^{\rho}(\mathbf{A}, \mathbf{B}) &= \Bigl(2\sum\nolimits_{i=1}^k\sin^2(\theta_i/2)\Bigr)^{1/2},\\
\delta^{\pi}(\mathbf{A}, \mathbf{B}) &= \sin \theta_k, &
\delta^{\sigma}(\mathbf{A}, \mathbf{B}) &= 2\sin (\theta_k/2),\end{aligned}$$ or more generally with $\min(k,l)$ in place of the index $k$ when we do not require $k \le l$.

This follows from observing that our proof of Theorem \[thm1\] only involves principal angles between $\mathbf{A}$ and $\mathbf{B}$ and the diffeomorphism between $\operatorname{Gr}(k,n)$ and $\operatorname{Gr}(n-k,n)$ remains an isometry under these distances. In particular, both and still hold with any of these distances in place of the Grassmann distance.

We will see in Section \[sec:existing\] that the projection distance $\delta^\pi$ in Theorem \[thm:othermetrics\] is equivalent to the containment gap, a measure of distance between subspaces of different dimensions originally proposed in operator theory [@Kato].

Grassmannian of subspaces of all dimensions {#sec:dinfty}
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We view the equality of $d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A},\Omega_{-}(\mathbf{B})\bigr)$ and $d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B},\Omega_{+}(\mathbf{A})\bigr)$ as the strongest evidence that their common value $\delta(\mathbf{A}, \mathbf{B})$ provides the most natural notion of distance between subspaces of different dimensions. As we pointed out earlier, $\delta$ is a distance in the sense of a distance from a point to a set, but not a distance in the sense of a metric on the set of all subspaces of all dimensions. For instance, $\delta$ does 