Suppose I have two countably infinite sets of orthonormal basis functions $\tilde{F}=\{f_1,f_2,\ldots\}$ and $\tilde{H}=\{h_1,h_2,\ldots\}$ that both span the same function space of functions defined on a Riemannian manifold $(M,g)$. In other words, given some $r:M\rightarrow \mathbb{R}$, we can write $$ r(x) = \sum_i a_i f_i(x) = \sum_i b_i h_i(x) $$ in terms of the basis functions.
Next consider the truncated basis sets $\tilde{F}_T=\{f_1,f_2,\ldots, f_n\}$ and $\tilde{H}_T=\{h_1,h_2,\ldots h_n\}$. I am interested in quantifying how similar the function subspaces spanned by $\tilde{H}_T$ and $\tilde{F}_T$ are.
So my first question is: what sort of "metrics" exist to quantify how similar these subspaces are?
One idea could be something like this: $$ D[\tilde{H}_T,\tilde{F}_T]= \max_{r\in S} \int_M \left[ \sum_{j=1}^n a_jf_j(x) - b_jh_j(x) \right]^2dx \tag{1} $$ In words, for a function $r$, take its representation in each basis, and see how different they are over the manifold - then take the measure of the difference for the "worst" function in some set of smooth functions $S$. This seems difficult to compute to practice though.
However, in reality, I have a discretized environment. Any function $P:M\rightarrow\mathbb{R}$ is thus represented by a vector: $P=(P(x_1), \ldots, P(x_m))$ of its values at a set of points $X=\{x_1,\ldots,x_m\}$. The sets of basis functions then become sets of basis vectors: ${F}=\{f_1,f_2,\ldots, f_n\}$ and ${H}=\{h_1,h_2,\ldots h_n\}$, which are orthonormal with respect to the (discretized) inner product on the function space $\langle f_i, f_j\rangle=f_i^TCf_j=\delta_{ij}$, where $f_k\in\mathbb{R}^m$, $C\in\mathbb{R}^{m\times m}$, and $\delta_{ij}$ is the Kronecker Delta.
So again I have a very similar question: how can I quantify the distance between the spanned subspaces of $F$ and $H$ in computationally efficient and principled manner?
Consider the basis matrices: $\Phi_F=[f_1,\ldots,f_m]$ and $\Phi_H=[h_1,\ldots,h_m]$, where $h_k,f_\ell$ are column vectors. Clearly I should not simply do something like $D_\mathcal{F}[F,H]=||\Phi_F-\Phi_H||_F$, since switching the order of the vectors might cause issues. I guess we could do: $$ D_P[F,H]=\min_{E\in\mathcal{P_M}} ||\Phi_F-E\Phi_H||_F $$ where $E$ is a matrix that permutes columns. This will only go to zero if $F$ is exactly a permutation of $H$ though.
Separately I could try to do something like equation (1). This could look something like $$ D[F,H]= \max_{v\in \mathcal{V}} || u(v)||_2^2 $$ where $\mathcal{V}$ is some vector space (e.g. $\{v:||v||_2=1\}$) and $$ u(v)= \sum_{i=1}^m (v^TCf_i)f_i - (v^TCh_i)h_i $$ But I'm not sure this is the best way to go about this.