Lions' Lemma on a Hilbert Space: Test Data and Form Bounds at a Sequentially Strict Maximum of a Quadratically Penalised Difference

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 $1\le m$. We write $3=2+1$ and $6=3+3$, $|x|^{2}=|x|\,|x|$, and $\tfrac{\alpha}{2}$ for the quotient of $\alpha\in\mathbb{R}$ by $2=1+1$.

Let $H$ be a real Hilbert space, with its inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$ and distance $d$ as fixed there, let $\mathrm{Sym}(H)$ be the set of bounded symmetric bilinear forms on $H$, with its norm $\lVert\cdot\rVert$, its order $\preceq$ and its sums and scalar multiples as fixed there, and let $H\times H$ be the product of $H$ with itself, a real inner product space by Properties of the Product of Two Real Inner Product Spaces (i), whose distance is written $d_{\times}$.

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 $P:H\to H$ and $N\in\mathrm{Sym}(H)$ be the projection and the tail form determined by $e$.

Let $u,v:A\to\mathbb{R}$ and let $-v:A\to\mathbb{R}$ be the function whose value at $y\in A$ is the additive inverse of $v(y)$. Assume that the set of values of $u$ is bounded above, that the set of values of $-v$ is bounded above, and that $u$ and $-v$ have closed superlevel sets in $H$. Let $\alpha\in\mathbb{R}$ be positive and let $\Phi:A\times A\to\mathbb{R}$ be given by

$$\Phi(x,y)=u(x)-v(y)-\tfrac{\alpha}{2}\,|x-y|^{2}.$$

Suppose $\Phi$ attains a sequentially strict maximum on $A\times A$ at a point $(\bar{x},\bar{y})$, the ambient metric space being $(H\times H,d_{\times})$. Put $p=\alpha(\bar{x}-\bar{y})\in H$. That a quadruple is approximable by test data from above or from below for a function on $A$ is as defined there.

Then there exist $X,Y\in\mathrm{Sym}(H)$ for which the following hold.

1. (Test data)(i) The quadruple $\bigl(\bar{x},\,u(\bar{x}),\,p,\,X+2\alpha N\bigr)$ is approximable by test data from above for $u$ on $A$, and the quadruple $\bigl(\bar{y},\,v(\bar{y}),\,p,\,Y-2\alpha N\bigr)$ is approximable by test data from below for $v$ on $A$.

2. (Quadratic form bound)(ii) For all $z,w\in H$,

$$-3\alpha\bigl(|z|^{2}+|w|^{2}\bigr)\ \le\ X(z,z)-Y(w,w)\ \le\ 3\alpha\,|z-w|^{2}.$$

3. (Ordering)(iii) $X\preceq Y$.

4. (Norm bound)(iv) $\lVert X\rVert\le6\alpha$ and $\lVert Y\rVert\le6\alpha$.

5. (Dependence on the tuple alone)(v) $X(z,w)=X(Pz,Pw)$ and $Y(z,w)=Y(Pz,Pw)$ for all $z,w\in H$.

The forms $X$ and $Y$ depend on $e$, and the second-order data in clause 1 carry, besides them, the tail term $2\alpha N$, whose quadratic form is $z\mapsto2\alpha|z-Pz|^{2}$ by Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions (iii). No bound on $X(z,z)-Y(w,w)$ of the kind in clause 2 holds for the shifted forms $X+2\alpha N$ and $Y-2\alpha N$ themselves; a dependent result must therefore either be insensitive to the tail term or control it by a further hypothesis on $e$.