Isotropic dimension bound from real inertia ($\dim U\le \min (p,q)$)

For a nondegenerate real quadratic form of inertia $(p,q)$, projection of a totally isotropic subspace onto either its positive or negative coordinate subspace is injective: a vector with one projection zero would have strictly one-signed quadratic value unless it vanished. Thus its dimension is at most both $p$ and $q$. This bound is attained by pairing one positive and one negative coordinate for each of $\min (p,q)$ independent null vectors.