EricDarve
commented
3 years ago (1) You are correct; span is missing. (2) I added more info in the proof. $S$ has dimension $k$ and the span of ${q_k, \ldots, q_n}$ has dimension $n-k+1$. So their intersection is a subspace of dimension at least 1. y^* is a point in that intersection. Hence the equation.
On page 153-154, a partial proof of the minimax theorem is provided, but S is taken to be a k-dimensional subspace. (1) In the theorem statement, shouldn't it say "the set S that attains the maximum is the span of the set of eigenvectors..."? (2) "y in S" should be the linear combination of the k eigenvectors, but the y* given on page 154 is a linear combination of what appears to be n-k+1 eigenvectors (qk through qn). Should this make more explicit that we are choosing k of these vectors?