Proof of Wang’s conjecture on subspaces of an inner product space

Abstract:

B.Y. Wang conjectured that if $R_{t}$ and $S_{t}$ $(t=1,\ldots ,k)$ are subspaces of an $n$-dimensional complex inner product space $V$, and their dimensions are $i_{t}$ and $n-i_{t}+1$, respectively, where $1\le i_{1}<i_{2}<\cdots <i_{k}\le n$, then there exists a $k$-dimensional subspace $W$ having two orthonormal bases $\{x_{1},\ldots ,x_{k}\}$ and $\{y_{1},\ldots ,y_{k}\}$ with $x_{t}\in R_{t}$ and $y_{t}\in S_{t}$ for all $t$.

We prove this conjecture and its real counterpart. The proof is in essence an application of a real version of the Bézout Theorem for the product of several projective spaces.