Fibre Suprema along a Coordinate Map and Their Semicontinuous Envelopes

We work in the settings of Real Hilbert Spaces: Standing Notation and Background, Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, the last together with Euclidean Space and Real Matrices: Standing Notation on which it rests, used in the dimension $m$ for a natural number $m$ with $1\le m$.

Let $H$ be a real Hilbert space, with its inner product $\langle\cdot,\cdot\rangle$ and norm $|\cdot|$ as fixed there, let $\mathbb{R}^{m}$ carry its Euclidean norm $\lVert\cdot\rVert$ and Euclidean distance $d_{E}$, and regard $\mathbb{R}^{m}$ as the metric space $(\mathbb{R}^{m},d_{E})$. We write $\lVert\zeta\rVert^{2}=\lVert\zeta\rVert\lVert\zeta\rVert$, and $\tfrac{\alpha}{2}$ for the quotient of $\alpha\in\mathbb{R}$ by $2=1+1$.

Let $A$ be a linear subspace of $H$, let $e\in H^{m}$ be an $m$-tuple in $H$ that is orthonormal and all of whose components lie in $A$, and let $\Lambda:H\to\mathbb{R}^{m}$ be the coordinate map determined by $e$.

Let $\hat{u},\hat{v}:A\to\mathbb{R}$ and let $\alpha\in\mathbb{R}$ be positive. Assume that the set of values of $\hat{u}$ is bounded above and that the set of values of $\hat{v}$ is bounded below. Let $\bar{x},\bar{y}\in A$ and assume that

$$\hat{u}(x)-\hat{v}(y)-\tfrac{\alpha}{2}\lVert\Lambda x-\Lambda y\rVert^{2} \ \le\ \hat{u}(\bar{x})-\hat{v}(\bar{y})-\tfrac{\alpha}{2}\lVert\Lambda\bar{x}-\Lambda\bar{y}\rVert^{2} \qquad\text{for all }x,y\in A .$$

Put $\bar{\zeta}=\Lambda\bar{x}$ and $\bar{\omega}=\Lambda\bar{y}$, and for $\zeta\in\mathbb{R}^{m}$ put $A_{\zeta}=\{\,x\in A\ :\ \Lambda x=\zeta\,\}$. Then the following hold.

1. (The fibre suprema)(i) For every $\zeta\in\mathbb{R}^{m}$ the set $A_{\zeta}$ is nonempty, the set $\{\hat{u}(x):x\in A_{\zeta}\}$ has a least upper bound and the set $\{\hat{v}(y):y\in A_{\zeta}\}$ has a greatest lower bound, unique by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below and Uniqueness of the Supremum and of the Infimum. The functions $U,\mathcal{V}:\mathbb{R}^{m}\to\mathbb{R}$ given by

$$U(\zeta)=\sup\{\hat{u}(x):x\in A_{\zeta}\}, \qquad \mathcal{V}(\zeta)=\inf\{\hat{v}(y):y\in A_{\zeta}\}$$

are therefore defined; the set of values of $U$ is bounded above and that of $\mathcal{V}$ is bounded below. Consequently $U$ is bounded above near each point of $\mathbb{R}^{m}$ and $\mathcal{V}$ is bounded below near each point of $\mathbb{R}^{m}$.

2. (Envelopes and the maximum)(ii) Let $U^{*}$ be the upper semicontinuous envelope of $U$ and let $\mathcal{V}_{*}$ be the lower semicontinuous envelope of $\mathcal{V}$, both taken in the metric space $(\mathbb{R}^{m},d_{E})$. Then $U^{*}$ is upper semicontinuous on $\mathbb{R}^{m}$, $\mathcal{V}_{*}$ is lower semicontinuous on $\mathbb{R}^{m}$, and

$$U^{*}(\zeta)-\mathcal{V}_{*}(\omega)-\tfrac{\alpha}{2}\lVert\zeta-\omega\rVert^{2} \ \le\ U^{*}(\bar{\zeta})-\mathcal{V}_{*}(\bar{\omega})-\tfrac{\alpha}{2}\lVert\bar{\zeta}-\bar{\omega}\rVert^{2} \qquad\text{for all }\zeta,\omega\in\mathbb{R}^{m}.$$

3. (Values at the base point)(iii) $U^{*}(\bar{\zeta})=U(\bar{\zeta})=\hat{u}(\bar{x})$ and $\mathcal{V}_{*}(\bar{\omega})=\mathcal{V}(\bar{\omega})=\hat{v}(\bar{y})$.

4. (Approximation over a fibre neighbourhood)(iv) Let $\zeta_{1}\in\mathbb{R}^{m}$ and let $r,\varepsilon\in\mathbb{R}$ be positive. Then there are $x\in A$ and $y\in A$ with

$$\lVert\Lambda x-\zeta_{1}\rVert\le r, \qquad U^{*}(\zeta_{1})-\varepsilon<\hat{u}(x), \qquad \lVert\Lambda y-\zeta_{1}\rVert\le r, \qquad \hat{v}(y)<\mathcal{V}_{*}(\zeta_{1})+\varepsilon .$$