conjunction with other common distances in Table \[tab:distances\].

We will see in Section \[sec:existing\] that the two existing notions of distance between subspaces of different dimensions are special cases of .

Evidently, the word ‘distance’ in ($\ast$) is used in the sense of a distance of a point to a set. For example, if a subspace is contained in another, then the distance between them is zero, even if they are distinct subspaces. Thus the distance in ($\ast$) is not a metric[^1]. In Section \[sec:metric\], we define a metric on the set of subspaces of all dimensions using an analogue of our main result: Given any two subspaces in $\mathbb{R}^n$, $\mathbf{A}$ of dimension $k$ and $\mathbf{B}$ of dimension $l$ with $k< l$, the distance from $\mathbf{A}$ to the furthest $k$-dimensional subspace contained in $\mathbf{B}$ *equals* the distance from $\mathbf{B}$ to the furthest $l$-dimensional subspace that contains $\mathbf{A}$. Their common value gives a metric between $\mathbf{A}$ and $\mathbf{B}$. The most interesting metrics for subspaces of different dimensions can be found in Table \[tab:metrics\].

In Section \[sec:volume\], we obtain a volumetric analogue of our main result: Given two arbitrary subspaces in $\mathbb{R}^n$, $\mathbf{A}$ of dimension $k$ and $\mathbf{B}$ of dimension $l$ with 