= Solution
<Sylvester's law of inertia> states that a real <quadratic form> can be brought by an invertible real change of coordinates to
$$
Q=\sum_{j=1}^p u_j^2-\sum_{j=p+1}^{p+q}u_j^2,
$$
with $r$ remaining zero coordinates, and that the integers $(p,q,r)$ are uniquely determined. To prove existence, represent it by a <symmetric matrix>. The allowed orthogonal <diagonalization> yields real <eigenvalues>, and scaling each nonzero eigen-coordinate by the square root of its absolute <eigenvalue> changes its coefficient to $+1$ or $-1$.
For uniqueness, $p$ is the largest <dimension> of a subspace on which the form is positive definite. The positive coordinate subspace attains $p$. Any subspace of <dimension> greater than $p$ intersects the $(q+r)$-dimensional span of the negative and zero coordinates nontrivially, by the <dimension> formula. A nonzero vector in that intersection has nonpositive quadratic value, contradicting positive definiteness. Similarly, $q$ is the maximal <dimension> of a negative-definite subspace. These descriptions are independent of coordinates, and $r=\dim V-p-q$ is then fixed. This proves existence and uniqueness, including degenerate forms.
For the final application, nonsingularity gives $p+q=2m$. In diagonal coordinates write $v=(v_+,v_-)$, with $Q(v)=|v_+|^2-|v_-|^2$. If $v\in U$ projects to zero in the positive coordinates, $Q(v)=0$ forces $v_-=0$, so projection $U\to\mathbb R^p$ is injective. The same argument applies to the negative coordinates. Hence $m\leq p$ and $m\leq q$, and their sum is $2m$. Therefore
$$
\boxed{p=q=m.}
$$
The inertia signature is $(m,m)$, or signature difference $p-q=0$ if that convention is used. This is the <isotropic dimension bound from real inertia>.
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