By using the notations of question 3b, deduce that $$\sup_{\mathcal{V} \subset \mathbb{R}^{n},\, \operatorname{dim} \mathcal{V} = j} \inf_{x \in \mathcal{V},\, \|x\|=1} \langle x, Mx \rangle = m_{j}.$$ Is this supremum attained?
By using the notations of question 3b, deduce that
$$\sup_{\mathcal{V} \subset \mathbb{R}^{n},\, \operatorname{dim} \mathcal{V} = j} \inf_{x \in \mathcal{V},\, \|x\|=1} \langle x, Mx \rangle = m_{j}.$$
Is this supremum attained?