This is not possible as you formulate it. Consider the following two bases in $\mathbb R^3:

\begin{gather*} (e_1,e_1+\epsilon e_2) \\ (e_1, e_1+\epsilon e_3). \end{gather*}

These are close together by your definition. However, there is no orthonormal basis of the first subspace that is close to any orthonormal basis of the second subspace (in the topology of the Stiefel manifold, or in any reasonable sense of closeness).

This is because, regardless of $\epsilon$, the first basis spans $(e_1,e_2)$ and the second basis spans $(e_1, e_3)$. These are different subspaces, corresponding to different points on the Grassmanian, and so the fibers over them in the Stiefel manifold are separated by a nonzero distance.

This is not possible as you formulate it. Consider the following two bases in $\mathbb R^3$:

\begin{gather*} (e_1,\ e_1+\epsilon\, e_2) \\ (e_1,\ e_1+\epsilon\, e_3). \end{gather*}

These are close together by your definition. However, there is no orthonormal basis of the first subspace that is close to any orthonormal basis of the second subspace (in the topology of the Stiefel manifold, or in any reasonable sense of closeness).

This is because, regardless of $\epsilon$, the first basis spans $(e_1,e_2)$ and the second basis spans $(e_1, e_3)$. These are different subspaces, corresponding to different points on the Grassmanian, and so the fibers over them in the Stiefel manifold are separated by a nonzero distance.

This is not possible as you formulate it. Consider the following two bases in $\mathbb R^3:

$$(e_1,e_1+\epsilon e_2)$$

$$(e_1, e_1+\epsilon e_3)$$

These are close together by your definition. However, there is no orthonormal basis of the first subspace that is close to any orthonormal basis of the second subspace (in the topology of the Stiefel manifold, or in any reasonable sense of closeness).

This is because, regardless of $\epsilon$, the first basis spans $(e_1,e_2)$ and the second basis spands $e_1, e_3$. These are different subspaces, corresponding to different points on the Grassmanian, and so the fibers over them in the Stiefel manifold are separated by a nonzero distance.

This is not possible as you formulate it. Consider the following two bases in $\mathbb R^3:

\begin{gather*} (e_1,e_1+\epsilon e_2) \\ (e_1, e_1+\epsilon e_3). \end{gather*}

These are close together by your definition. However, there is no orthonormal basis of the first subspace that is close to any orthonormal basis of the second subspace (in the topology of the Stiefel manifold, or in any reasonable sense of closeness).

This is because, regardless of $\epsilon$, the first basis spans $(e_1,e_2)$ and the second basis spans $(e_1, e_3)$. These are different subspaces, corresponding to different points on the Grassmanian, and so the fibers over them in the Stiefel manifold are separated by a nonzero distance.