= Solution
The restriction of a <positive definiteness>[positive definite] form to each $V_m$ is positive definite. By the <spectral theorem for real symmetric matrices>, choose an <orthogonal matrix> $P$ such that $P^TA_mP=\operatorname{diag}(d_1,\ldots,d_m)$.
Each $d_i=\varphi(f_i,f_i)>0$, since a basis vector is nonzero. Taking <determinants> in part (i) gives
$$
\det(P^TA_mP)=(\det P)^2\det A_m=\det A_m,
$$
because an <orthogonal matrix> has determinant $\pm1$. Consequently
$$
\boxed{\det A_m=\prod_{i=1}^m d_i>0\quad(1\leq m\leq n).}
$$
These determinants are the leading <principal minors> in <Sylvester's criterion>.
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