sense of a metric (can be zero for a pair of distinct subspaces) and thus not listed. The spectral distance $d^{\sigma}_{\operatorname{Gr}(k,n)}$ is also called chordal $2$-norm distance [@BN]. For each distance in Table \[tab:distances\] defined for equidimensional $\mathbf{A}$ and $\mathbf{B}$, Theorem \[thm:othermetrics\] provides a corresponding version for when $\dim \mathbf{A} \ne \dim \mathbf{B}$.

The fact that all these distances in Table \[tab:distances\] depend on the principal angles is not a coincidence — the result [@Wong Theorem 3] implies the following.

\[thm:wong\] Any notion of distance between $k$-dimensional subspaces in $\mathbb{R}^n$ that depends only on the relative positions of the subspaces, i.e., invariant under any rotation in $\operatorname{O}(n)$, must be a function of their principal angles. To be more specific, if a distance $d : \operatorname{Gr}(k,n) \times \operatorname{Gr}(k,n) \to [0, \infty)$ satisfies $$d(Q \cdot \mathbf{A}, Q \cdot \mathbf{B}) =d(\mathbf{A}, \mathbf{B}),$$ for all $\mathbf{A}, \mathbf{B} \in \operatorname{Gr}(k,n)$ and all $Q \in \operatorname{O}(n)$, where the action is as defined in , then $d$ must be a function of $\theta_i(\mathbf{A},\mathbf{B})$, $i=1,\dots,k$.

We will next introduce the *infinite Grassmannian* $\operatorname{Gr}(k,\infty)$ to show that these distances between subspaces are independent of the dimension of their ambient space.

The Infinite Grassmannian {#sec:infty}
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One way of defining 