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

Each result cited is universally quantified over the data in its own statement, and is applied here to the data named in the statement of the lemma.

Notation. Let $\Lambda$, $\Lambda^{\sharp}$, $P$, the forms $M^{\Lambda}$ for $M\in\mathcal{S}(m)$, the projection form $\Pi$ and the tail form $N$ be those determined by $e$ as in Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions, all of whose clauses are used throughout; in particular $\Lambda$ and $\Lambda^{\sharp}$ are linear, $P=\Lambda^{\sharp}\Lambda$, $\lVert\Lambda x\rVert=|Px|\le|x|$, $|\Lambda^{\sharp}\xi|=\lVert\xi\rVert$, $|x|^{2}=\Pi(x,x)+N(x,x)$ and $N(x,x)=|x-Px|^{2}$, so that $|x-Px|\le|x|$. Since $|e_{1}|=1$ we have $H\ne\{0_{H}\}$, so $\lVert I\rVert=1$ by claim 3 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity. Write

and let $g,k:H\to\mathbb{R}$ be given by

By clause 1 of Sequentially Strict Maxima and Minima on a Subset of a Metric Space §maximum, $\Phi(x,y)\le M_{0}$ for all $(x,y)\in A\times A$. Every component of $e$ lies in $A$, so $A$ is nonempty.

Claim 1 (splitting off the tail of the penalty). For all $x,y\in H$,

with equality when $x=\bar{x}$ and $y=\bar{y}$; moreover $g(x)\le\tfrac{\alpha}{4}|\bar{z}|^{2}$ and $k(y)\le\tfrac{\alpha}{4}|\bar{z}|^{2}$ for all $x,y\in H$.

Proof. Put $a=(x-\bar{x})-P(x-\bar{x})$ and $b=(y-\bar{y})-P(y-\bar{y})$. Since $P$ is linear,

because $(x-y)=(x-\bar{x})-(y-\bar{y})+(\bar{x}-\bar{y})$. By Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions §tail,

and, using the second expression $N(z,w)=\langle z-Pz,w\rangle$ there together with the first,

By Elementary Identities in a Real Inner Product Space §expansion applied twice, and by Elementary Identities in a Real Inner Product Space §bilinear,

and $|a-b|^{2}=|a|^{2}-2\langle a,b\rangle+|b|^{2}$ while $0\le|a+b|^{2}=|a|^{2}+2\langle a,b\rangle+|b|^{2}$, so $-2\langle a,b\rangle\le|a|^{2}+|b|^{2}$ and therefore $|a-b|^{2}\le2|a|^{2}+2|b|^{2}$. Hence

Multiplying by the positive number $\tfrac{\alpha}{2}$ (claim 5 of Elementary Arithmetic in an Ordered Field) and reversing the sign (claim 4 of Elementary Order Arithmetic in an Ordered Field) gives

by the identities displayed above. If $x=\bar{x}$ and $y=\bar{y}$ then $a=b=0_{H}$, so $g(\bar{x})=0$, $k(\bar{y})=0$ and $N(\bar{x}-\bar{y},\bar{x}-\bar{y})=|\bar{z}|^{2}$, and both sides equal $-\tfrac{\alpha}{2}|\bar{z}|^{2}$.

For the upper bounds, The Cauchy-Schwarz Inequality in a Real Inner Product Space and claims 1 and 3 of Properties of the Absolute Value in an Ordered Field give $-\langle\bar{z},a\rangle\le|\langle\bar{z},a\rangle|\le|\bar{z}|\,|a|$, so $g(x)\le\alpha|\bar{z}||a|-\alpha|a|^{2}$ by claim 5 of Elementary Arithmetic in an Ordered Field. Since $0\le\alpha\bigl(|a|-\tfrac{|\bar{z}|}{2}\bigr)^{2}=\alpha|a|^{2}-\alpha|\bar{z}||a|+\tfrac{\alpha}{4}|\bar{z}|^{2}$, we get $\alpha|\bar{z}||a|-\alpha|a|^{2}\le\tfrac{\alpha}{4}|\bar{z}|^{2}$, whence $g(x)\le\tfrac{\alpha}{4}|\bar{z}|^{2}$. The same computation with $b$ in place of $a$ bounds $k$. This proves Claim 1.

Claim 2 (the decoupled functional). Let $\hat{u},\hat{v}:A\to\mathbb{R}$ be given by $\hat{u}(x)=u(x)+g(x)$ and $\hat{v}(y)=v(y)-k(y)$. Then the set of values of $\hat{u}$ is bounded above and that of $\hat{v}$ is bounded below;

with equality when $x=\bar{x}$, $y=\bar{y}$; and

Proof. By Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions §coordinates and Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions §tail, $\lVert\Lambda x-\Lambda y\rVert=\lVert\Lambda(x-y)\rVert$ and $|x-y|^{2}=\lVert\Lambda(x-y)\rVert^{2}+N(x-y,x-y)$. Hence, by Claim 1,

which is the first display; equality at $(\bar{x},\bar{y})$ follows from the equality case of Claim 1. For the second display, the first one and $\Phi(x,y)\le M_{0}$ give

and by the equality case the right-hand side equals $\hat{u}(\bar{x})-\hat{v}(\bar{y})-\tfrac{\alpha}{2}\lVert\Lambda\bar{x}-\Lambda\bar{y}\rVert^{2}$. Finally, let $C_{u}$ be an upper bound for the values of $u$ and $C_{v}$ one for the values of $-v$; by Claim 1, $\hat{u}(x)\le C_{u}+\tfrac{\alpha}{4}|\bar{z}|^{2}$ and $-\hat{v}(y)=-v(y)+k(y)\le C_{v}+\tfrac{\alpha}{4}|\bar{z}|^{2}$, so the values of $\hat{v}$ are bounded below. This proves Claim 2.

Claim 3 (the two forms, and clauses 2 to 5). Claim 2 supplies the hypotheses of Fibre Suprema along a Coordinate Map and Their Semicontinuous Envelopes for the data $A$, $e$, $\hat{u}$, $\hat{v}$, $\alpha$, $\bar{x}$, $\bar{y}$. Let $U$, $\mathcal{V}$, $U^{*}$, $\mathcal{V}_{*}$, $\bar{\zeta}=\Lambda\bar{x}$ and $\bar{\omega}=\Lambda\bar{y}$ be as there. By Fibre Suprema along a Coordinate Map and Their Semicontinuous Envelopes §maximum, $U^{*}$ is upper semicontinuous on $\mathbb{R}^{m}$, $\mathcal{V}_{*}$ is lower semicontinuous on $\mathbb{R}^{m}$, and

and by Fibre Suprema along a Coordinate Map and Their Semicontinuous Envelopes §values, $U^{*}(\bar{\zeta})=\hat{u}(\bar{x})$ and $\mathcal{V}_{*}(\bar{\omega})=\hat{v}(\bar{y})$.

The set $\mathbb{R}^{m}$ is open in $\mathbb{R}^{m}$ by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, so Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference applies with $m$ in the role of $n$, with $\Omega=\mathbb{R}^{m}$, with $U^{*}$ and $\mathcal{V}_{*}$ in the roles of $u$ and $v$, with $\alpha$, and with $\bar{\zeta}$ and $\bar{\omega}$ in the roles of the two points; its local hypothesis holds with $\delta=1$ because the displayed inequality holds for all $\zeta,\omega$. Writing $p_{0}=\alpha(\bar{\zeta}-\bar{\omega})$, we obtain $X_{0},Y_{0}\in\mathcal{S}(m)$ such that (i) the quadruple $(\bar{\zeta},U^{*}(\bar{\zeta}),p_{0},X_{0})$ is approximable by test data from above for $U^{*}$ and $(\bar{\omega},\mathcal{V}_{*}(\bar{\omega}),p_{0},Y_{0})$ is approximable by test data from below for $\mathcal{V}_{*}$, the open set being $\mathbb{R}^{m}$ in both cases (clause 1 there); (ii) for all $\xi,\eta\in\mathbb{R}^{m}$,

(clause 3 there); (iii) $X_{0}\preceq Y_{0}$ (clause 4 there); and (iv) $\lVert X_{0}\rVert\le6\alpha$ and $\lVert Y_{0}\rVert\le6\alpha$ (clause 5 there).

Put $X=X_{0}^{\Lambda}$ and $Y=Y_{0}^{\Lambda}$, elements of $\mathrm{Sym}(H)$ by Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions §forms. Clause 5 is then the identity $M^{\Lambda}(z,w)=M^{\Lambda}(Pz,Pw)$ of that clause, clause 4 follows from (iv) and the bound $\lVert M^{\Lambda}\rVert\le\lVert M\rVert$ there, and clause 3 follows from (iii) and the order transfer there. For clause 2, apply (ii) with $\xi=\Lambda z$ and $\eta=\Lambda w$, so that $\xi\cdot(X_{0}\xi)=X(z,z)$ and $\eta\cdot(Y_{0}\eta)=Y(w,w)$. Since $0\le\lVert\Lambda(z-w)\rVert\le|z-w|$, two applications of claim 5 of Elementary Arithmetic in an Ordered Field give $\lVert\Lambda z-\Lambda w\rVert^{2}=\lVert\Lambda(z-w)\rVert^{2}\le|z-w|^{2}$, and multiplying by the nonnegative $3\alpha$ gives the upper bound of clause 2. Likewise $\lVert\Lambda z\rVert^{2}+\lVert\Lambda w\rVert^{2}\le|z|^{2}+|w|^{2}$, so multiplying by $3\alpha$ and reversing the sign gives $-3\alpha(|z|^{2}+|w|^{2})\le-3\alpha(\lVert\Lambda z\rVert^{2}+\lVert\Lambda w\rVert^{2})$, which with (ii) gives the lower bound.

We record for later use that $\bar{\zeta}-\bar{\omega}=\Lambda(\bar{x}-\bar{y})$, so that by the linearity of $\Lambda^{\sharp}$ and $P=\Lambda^{\sharp}\Lambda$,

This proves Claim 3.

Claim 4 (clause 1). Let $\varepsilon\in\mathbb{R}$ be positive. We produce $x_{1},y_{1}\in A$ and $\varphi,\psi\in C^{2}(H)$ such that the function on $A$ with value $u(x)-\varphi(x)$ at $x$ has a local maximum at $x_{1}$ relative to $A$, the function on $A$ with value $v(y)-\psi(y)$ at $y$ has a local minimum at $y_{1}$ relative to $A$, and

Since $\varepsilon$ is arbitrary, this is clause 1, by Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space §above and Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space §below.

Step 1 (the localisation radius). Let $\varepsilon_{1}$ be the lesser of $\varepsilon$ and $\tfrac{\varepsilon}{8\alpha}$, positive because it is one of them. The function $u$ and the function $-v$ have closed superlevel sets in $H$, $A$ is nonempty, and $\Phi$ attains a sequentially strict maximum on $A\times A$ at $(\bar{x},\bar{y})$, so Localisation at a Sequentially Strict Maximum of a Quadratically Penalised Difference §localisation provides a positive $\eta_{0}\in\mathbb{R}$ such that every $(x,y)\in A\times A$ with $M_{0}-\eta_{0}<\Phi(x,y)$ satisfies

Step 2 (accuracy of the Euclidean data). Put $D_{0}=\lVert\bar{\zeta}-\bar{\omega}\rVert$. Let $\varepsilon'$ be a positive real number with

which exists: the first three are satisfied by the least of $1$, $\tfrac{\varepsilon}{16}$ and $\tfrac{\eta_{0}}{16}$, and the fourth by any positive number at most $\tfrac{\eta_{0}}{16\,(2|\alpha|(D_{0}+2)+1)}$, so the least of these four positive numbers serves. Let $\eta_{1}$ be a positive real with $2\eta_{1}\le\tfrac{\varepsilon}{16}$.

Step 3 (the Euclidean test data). By (i) of Claim 3 and Quadruple Approximable by Test-Function Data §above, applied with the accuracy $\varepsilon'$, there are $\zeta_{1}\in\mathbb{R}^{m}$ and a function $\chi:\mathbb{R}^{m}\to\mathbb{R}$ of class $C^{2}$ on $\mathbb{R}^{m}$ such that the function with value $U^{*}(\zeta)-\chi(\zeta)$ at $\zeta$ has a local maximum at $\zeta_{1}$ relative to $\mathbb{R}^{m}$ and

Likewise, by Quadruple Approximable by Test-Function Data §below, there are $\omega_{1}\in\mathbb{R}^{m}$ and $\theta:\mathbb{R}^{m}\to\mathbb{R}$ of class $C^{2}$ on $\mathbb{R}^{m}$ such that the function with value $\mathcal{V}_{*}(\omega)-\theta(\omega)$ at $\omega$ has a local minimum at $\omega_{1}$ relative to $\mathbb{R}^{m}$ and the four analogous inequalities hold with $\omega_{1},\theta,\mathcal{V}_{*},\bar{\omega},Y_{0}$ in place of $\zeta_{1},\chi,U^{*},\bar{\zeta},X_{0}$.

Step 4 (replacing the test functions by coordinate quadratics). Apply Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions §majorant with $\Omega=\mathbb{R}^{m}$, with $\chi$, $\zeta_{1}$ and the positive $\eta_{1}$: the associated quadratic

satisfies $T_{0}(\zeta_{1})=\chi(\zeta_{1})$, and there is a positive $\rho_{1}$ with $\chi(\zeta)\le T_{0}(\zeta)$ whenever $\lVert\zeta-\zeta_{1}\rVert<\rho_{1}$. Let $\rho_{0}$ be positive and such that $U^{*}(\zeta)-\chi(\zeta)\le U^{*}(\zeta_{1})-\chi(\zeta_{1})$ for every $\zeta$ with $d_{E}(\zeta,\zeta_{1})<\rho_{0}$, as the local maximum property provides. Then for every $\zeta$ with $\lVert\zeta-\zeta_{1}\rVert$ less than both $\rho_{0}$ and $\rho_{1}$,

Let $T=T_{0}\circ\Lambda$. By Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions §quadratic, $T\in C^{2}(H)$, $T(x)$ depends only on $\Lambda x$, the Hessian $D^{2}T(x)$ equals the constant form

and $\bigl|DT(x)-\Lambda^{\sharp}D\chi(\zeta_{1})\bigr|\le\bigl(\lVert D^{2}\chi(\zeta_{1})\rVert+2\eta_{1}\bigr)\lVert\Lambda x-\zeta_{1}\rVert$ for every $x\in H$. By Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions §forms and Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions §tail, together with claims 1 and 3 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity,

Applying Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions §majorant instead with $\theta$, $\omega_{1}$ and the negative number $-\eta_{1}$ gives a quadratic $S_{0}$ with $S_{0}(\omega_{1})=\theta(\omega_{1})$ and a positive $\rho_{1}'$ with $S_{0}(\omega)\le\theta(\omega)$ for $\lVert\omega-\omega_{1}\rVert<\rho_{1}'$; with $\rho_{0}'$ a radius for the local minimum of $\mathcal{V}_{*}-\theta$ at $\omega_{1}$, we get

whenever $\lVert\omega-\omega_{1}\rVert$ is less than both $\rho_{0}'$ and $\rho_{1}'$. Setting $S=S_{0}\circ\Lambda$, we have $S\in C^{2}(H)$, $S(y)$ depends only on $\Lambda y$, $D^{2}S(y)$ is the constant form $b_{v}=(D^{2}\theta(\omega_{1}))^{\Lambda}-2\eta_{1}\Pi$ with $\lVert b_{v}-Y\rVert\le\varepsilon'+2\eta_{1}$, and $\bigl|DS(y)-\Lambda^{\sharp}D\theta(\omega_{1})\bigr|\le(\lVert D^{2}\theta(\omega_{1})\rVert+2\eta_{1})\lVert\Lambda y-\omega_{1}\rVert$.

Step 5 (the fibre radius). By (iv) of Claim 3 and Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §norm, $\lVert D^{2}\chi(\zeta_{1})\rVert\le\lVert X_{0}\rVert+\varepsilon'\le6\alpha+\varepsilon'$, and likewise for $\theta$; put $K=6\alpha+\varepsilon'+2\eta_{1}$, so that the two gradient estimates of Step 4 hold with the factor $K$. Let $r$ be a positive real number with

and such that

Such an $r$ exists: the first seven conditions hold for every positive $r$ at most the least of the seven positive numbers

the sixth of which is defined because $K$ is positive; and for the last two it suffices, by the final estimate of Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions §quadratic, that $r\le1$ and $(\lVert q\rVert+\tfrac12\lVert M\rVert+|\eta|)\,r\le\tfrac{\eta_{0}}{16}$ for each of the two quadratics, since then $\lVert\zeta-\zeta_{1}\rVert\le r\le1$ gives $\lVert\zeta-\zeta_{1}\rVert^{2}\le\lVert\zeta-\zeta_{1}\rVert\le r$. Put

Step 6 (the auxiliary functions). Let $\Theta_{u},\Theta_{v}:A\to\mathbb{R}$ be given by $\Theta_{u}(x)=\hat{u}(x)-T(x)$ and $\Theta_{v}(y)=-\hat{v}(y)+S(y)$, and put

(a) Upper bounds. Let $x\in A_{u}$. Then $\hat{u}(x)\le U(\Lambda x)\le U^{*}(\Lambda x)$, by the definition of $U$ in Fibre Suprema along a Coordinate Map and Their Semicontinuous Envelopes §defined and claim 1 of Properties of the Upper Semicontinuous Envelope; and $T(x)=T_{0}(\Lambda x)$ with $\lVert\Lambda x-\zeta_{1}\rVert\le r$, which is less than both $\rho_{0}$ and $\rho_{1}$. So Step 4 gives $\Theta_{u}(x)\le U^{*}(\Lambda x)-T_{0}(\Lambda x)\le M_{u}$. Symmetrically $\Theta_{v}(y)\le M_{v}$ for every $y\in A_{v}$, using $-\hat{v}(y)\le-\mathcal{V}(\Lambda y)\le-\mathcal{V}_{*}(\Lambda y)$ (claim 2 of Properties of the Lower Semicontinuous Envelope, by Duality).

(b) Near-maximisers on a prescribed fibre neighbourhood. Let $\tau$ and $s$ be positive with $s\le r$. Let $s'$ be a positive number, at most $s$, with $\bigl|T_{0}(\zeta)-T_{0}(\zeta_{1})\bigr|\le\tfrac{\tau}{2}$ whenever $\lVert\zeta-\zeta_{1}\rVert\le s'$; such an $s'$ exists by the argument used for $r$ in Step 5. Applying Fibre Suprema along a Coordinate Map and Their Semicontinuous Envelopes §approximation with $\zeta_{1}$, the radius $s'$ and the accuracy $\tfrac{\tau}{2}$ gives $x\in A$ with $\lVert\Lambda x-\zeta_{1}\rVert\le s'\le s\le r$, hence $x\in A_{u}$, and $U^{*}(\zeta_{1})-\tfrac{\tau}{2}<\hat{u}(x)$. Since $T(x)=T_{0}(\Lambda x)\le T_{0}(\zeta_{1})+\tfrac{\tau}{2}$,

Applying the same clause of Fibre Suprema along a Coordinate Map and Their Semicontinuous Envelopes again, this time with $\omega_{1}$ in the role of $\zeta_{1}$, and using the second half of its conclusion (the one concerning $\hat{v}$ and $\mathcal{V}_{*}$) together with $S_{0}$ in place of $T_{0}$, gives in the same way a $y\in A_{v}$ with $\lVert\Lambda y-\omega_{1}\rVert\le s$ and $M_{v}-\tau<\Theta_{v}(y)$. In particular $A_{u}$ and $A_{v}$ are nonempty.

(c) Closed superlevel sets. The functions $g$, $k$, $T$ and $S$ belong to $C^{2}(H)$: for $T$ and $S$ this is Step 4, and for $g$ note that

by the bilinearity and symmetry of $N$ and claim 1 of The Bounded Symmetric Operator Represented by a Bounded Symmetric Bilinear Form on a Real Hilbert Space, which exhibits $g$ as a sum of an affine function and a scalar multiple of $x\mapsto\tfrac12N(x,x)$; so claims 1 and 2 of Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ and claims 2 and 3 of Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space give $g\in C^{2}(H)$ with

and likewise $k\in C^{2}(H)$ with $Dk(y)=\alpha\bar{z}-2\alpha T_{N}(y-\bar{y})$ and $D^{2}k(y)=-2\alpha N$. By claim 4 of The Bounded Symmetric Operator Represented by a Bounded Symmetric Bilinear Form on a Real Hilbert Space and Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions §tail, $T_{N}z=z-Pz$, so $|T_{N}z|\le|z|$.

Members of $C^{2}(H)$ are differentiable on $H$ and hence continuous on $H$, by The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c2 and Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space §continuous. Thus $T-g$ is continuous on $H$, and $\Theta_{u}(x)=u(x)-\bigl(T(x)-g(x)\bigr)$, so $\Theta_{u}$ has closed superlevel sets in $H$ as a function on $A$, by claim 3 of Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits.

The restriction of $\Theta_{u}$ to $A_{u}$ also has closed superlevel sets in $H$. Indeed, let $(x_{j})_{j\in\mathbb{N}}$ be a sequence in $A_{u}$ converging to $x\in H$ and let $t\in\mathbb{R}$ satisfy $t\le\Theta_{u}(x_{j})$ for every $j$. By claim 1 of Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits applied to $\Theta_{u}$ on $A$ we get $x\in A$ and $t\le\Theta_{u}(x)$. Moreover, for each $j$,

by Euclidean Distance is a Metric on $\mathbb{R}^n$ and Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions §coordinates; since $|x-x_{j}|$ is smaller than any prescribed positive number for suitable $j$, Comparison of Real Numbers with Arbitrary Positive Slack gives $\lVert\Lambda x-\zeta_{1}\rVert\le r$, so $x\in A_{u}$. By claim 1 of Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits again, in the converse direction, the restriction has closed superlevel sets in $H$. The same argument applies to $\Theta_{v}$ on $A_{v}$.

Step 7 (the perturbed maxima). Let $\lambda$ be the lesser of $\tfrac{r}{8}$ and $1$, and let $\mu$ be a positive real with

which exists as the least of three positive numbers of the required form. Then $4\lambda\le\tfrac{r}{2}$.

By (b) with $\tau=\mu\lambda^{2}$ and $s=\tfrac{r}{4}$ there is $x_{0}\in A_{u}$ with $\lVert\Lambda x_{0}-\zeta_{1}\rVert\le\tfrac{r}{4}$ and $M_{u}-\mu\lambda^{2}<\Theta_{u}(x_{0})$. By (a), $M_{u}$ is an upper bound for the values of $\Theta_{u}$ on $A_{u}$, so the least upper bound of those values is at most $\Theta_{u}(x_{0})+\mu\lambda^{2}$. Hence A Perturbed Maximum Principle of Borwein-Preiss Type in a Real Hilbert Space applies to $A_{u}$, $\Theta_{u}$, $\mu$, $\lambda$ and $x_{0}$, and provides two points, written here $c_{1}\in H$ and $x_{1}\in A_{u}$ — they are the points called $\bar{y}$ and $\bar{x}$ in that theorem, and are unrelated to the point $\bar{y}$ of the present lemma — such that

the function on $A_{u}$ with value $\Theta_{u}(x)-\mu|x-c_{1}|^{2}$ at $x$ attains a sequentially strict maximum on $A_{u}$ at $x_{1}$ (claim 2 there), and the least upper bound of the values of $\Theta_{u}$ on $A_{u}$ is at most $\Theta_{u}(x_{1})+2\mu\lambda^{2}$ (claim 3 there). Combining the last bound with $M_{u}-\mu\lambda^{2}<\Theta_{u}(x_{0})$ gives

Also

The same construction on the $v$ side provides $c_{2}\in H$ and $y_{1}\in A_{v}$ with $|y_{1}-c_{2}|\le8\lambda$, with $\Theta_{v}-\mu|\cdot-c_{2}|^{2}$ attaining a sequentially strict maximum on $A_{v}$ at $y_{1}$, with $M_{v}-3\mu\lambda^{2}<\Theta_{v}(y_{1})$ and with $\lVert\Lambda y_{1}-\omega_{1}\rVert<r$.

Step 8 (the test functions and the local extrema). Define $\varphi,\psi:H\to\mathbb{R}$ by

so that, for $x,y\in A$,

Put $\rho_{2}=r-\lVert\Lambda x_{1}-\zeta_{1}\rVert$, positive by Step 7. If $x\in A$ and $|x-x_{1}|<\rho_{2}$ then