containing $\mathbf{A}$, measured within $\operatorname{Gr}(l,n)$. In other words, we could have instead defined it as the Grassmann distance from $\mathbf{B}$ to the closed subset $\Omega_{+}(\mathbf{A})$, $$\label{eq:plus}
\delta(\mathbf{A}, \mathbf{B}) \coloneqq d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B}, \Omega_{+}\bigl(\mathbf{A})\bigr) 
= \min\bigl\{ d_{\operatorname{Gr}(l,n)}(\mathbf{B},\mathbf{X}) :  \mathbf{X}\in \Omega_{+}(\mathbf{A})\bigr\}.$$ It will have the same desirable features as the one in except that the distance is now measured in $d_{\operatorname{Gr}(l,n)}$ and within $\operatorname{Gr}(l,n)$.

It turns out that the two values in and are equal, allowing us to define $\delta(\mathbf{A}, \mathbf{B})$ as their common value. We will establish this equality and the properties of $\delta(\mathbf{A}, \mathbf{B})$ in the remainder of this section. The results are summarized in Theorem \[thm1\]. Our proof is constructive: In addition to showing the equality of and , it shows how one may explicitly find the closest points on Schubert varieties $\mathbf{X} \in \Omega_{-}(\mathbf{B})$ and $\mathbf{Y} \in \Omega_{+}(\mathbf{A})$ to any given point in the respective Grassmannians.

\[thm1\] Let $\mathbf{A}$ be a subspace of dimension $k$ and $\mathbf{B}$ be a subspace of dimension $l$ in $\mathbb{R}^n$. Suppose $k \le l \le n$. Then $$\label{eq:equal}
 d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A}, \Omega_{-}\bigl(\mathbf{B})\bigr) =  d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B}, \Omega_{+}\bigl(\mathbf{A})\bigr).$$ Their common value defines a distance $\delta(\mathbf{A}, \mathbf{B}) $ between the two subspaces with the following properties:

1.  \[indp\] $\delta(\mathbf{A}, \mathbf{B}) $ 