I'm working on an error estimate for a numerical method, and in the process I've stumbled across the following abstract inequality which I think is true, but am having a hard time proving.
Suppose $\{\phi_i\}_{i=1}^N$ and $\{\psi_i\}_{i=1}^N$ are orthonormal bases of $\mathbb{R}^N$ with $(\phi_i,\psi_i) \ge 0$, and call the "error" between these basis vectors $e_i=\phi_i-\psi_i$. Is there a constant $C$ independent of N such that $$ \sum_{i \neq j} (e_i,e_j) \le C \sum_i ||e_i||^2? $$ I've written a matlab script to test it for thousands of random sets of orthonormal vectors and found no counterexamples so far for C=1.