For two unit vectors $\vec{u}, \vec{v}$ in an inner product space $V$, we can gauge how "similar" they are by calculating the inner-product of the two. An inner product of $1$ means they are the exact same; an inner product between $0$ and $1$ means they are not the same vector, but they in some sense point in "similar directions"; an inner product of $0$ means they are orthogonal (as far from similar as possible); an inner product between $0$ and $-1$ implies that they point in "similar" directions, but opposite.
I was wondering if some sort of a similar metric could be defined on two subspaces of $V$. (For my purposes, we can assume $V = \mathbb{R}^n$ or $V = \mathbb{C}^n$.) Specifically, we want a function $f(U, W)$ with $U, W \subseteq V$ with at least the following properties:
- The output of $f$ is either a scalar, vector, or matrix.
- For any $U$, $\| f(U, U) \| = 1$ (for some appropriately defined norm $\| \cdot \|$).
- If all vectors in $U$ are orthogonal to all vectors in $V$, then $\|f(U, V)\| = 0$.
There may also be other desirable properties that I haven't thought of, but intuitively I want $\|f\|$ to be a measure of how "similar" to vector spaces are. One potential thought I had was to calculate $U$ and $V$ by taking two orthonormal bases $\{\vec{u}_1, \cdots, \vec{u}_m\}$ which spans $U$ and $\{\vec{v}_1, \cdots, \vec{v}_n\}$ which spans $V$ and have $f$ map to the matrix $F$, where $$F_{ij} = \langle \vec{u}_i, \vec{v}_j \rangle$$ but the trouble is that this will depend on the bases chosen.
EDIT: Alex R. suggested $f(U, W) = \sup_{u \in U, w \in W} \frac{\langle u, w \rangle}{\|u\| \|w\|}$. This does satisfy all the three criteria, but the problem is that as long as $U \cap V$ is non-trivial, $f$ evaluates to $1$. This leads to the following added criteria:
- We want $\|f(U, W)\| = 1$ if and only if $U = W$.
- (Not formal) If possible, we would want $\|f(U, W)\|$ to be positively correlated with $\text{dim}(U, W)$.