Some distances on subspaces

Abstract

Let $H$ be an infinite separable complex Hilbert space. Let $V$ be a closed subspace of $H$ with $\dim V=n$ and let $MO_n(V)$ be the set of $n$--tuples $(e_1,\cdots,e_n)$ such that $e_1,\cdots,e_n$ are mutually orthonormal in $V$. In this short note, we first show that for an $n$--tuple of linearly independent unit vectors $(\alpha_1,\cdots,\alpha_n)$ in $H$, there is an $n$--tuple of mutually orthonormal vectors $(\gamma_1,\cdots,\gamma_n)=
(\alpha_1,\cdots,\alpha_n)A^{-1/2}$ such that
$$
\sum\limits^n_{j=1}\|\alpha_j-\gamma_j\|^2=\inf\big\{\sum\limits^n_{j=1}\|\alpha_j-e_j\|^2\vert\,(e_1,\cdots,e_n)\in MO_n(H)\big\},
$$
where $A$ is the Gram matrix of $(\alpha_1,\cdots,\alpha_n)$; then we define a new distance $d_1(M,N)$ for two subspaces $M$ and $N$ with $\dim M=\dim N<+\infty$ and give a formula to compute $d_2(M,N)$.