Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space

In the setting of Real Hilbert Spaces: Standing Notation and Background and Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus, let $E$ be a real inner product space, with its inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$ and distance $d$ as fixed there, and let $\mathrm{Sym}(E)$ be the set of bounded symmetric bilinear forms on $E$, with its norm $\lVert\cdot\rVert$, as fixed there. The set $E$ is open in $(E,d)$, since every open ball of $(E,d)$ is a subset of $E$; accordingly the class $C^{2}(E)$ is defined, and a $\varphi\in C^{2}(E)$ has at each $x\in E$ a gradient $D\varphi(x)\in E$ and a Hessian $D^{2}\varphi(x)\in\mathrm{Sym}(E)$. Local maxima and local minima relative to a subset of $E$ are as fixed there, and $|s|$ is the absolute value of a real number $s$.

Let $A\subseteq E$, let $w:A\to\mathbb{R}$, let $\bar{x}\in A$, let $p\in E$ and let $X\in\mathrm{Sym}(E)$. For $\varphi\in C^{2}(E)$ let $w-\varphi$ denote the function from $A$ to $\mathbb{R}$ whose value at $x\in A$ is $w(x)-\varphi(x)$.

1. (From above)(i) The quadruple $\bigl(\bar{x},w(\bar{x}),p,X\bigr)$ is approximable by test data from above for $w$ on $A$ if for every positive $\varepsilon\in\mathbb{R}$ there exist $y\in A$ and $\varphi\in C^{2}(E)$ such that $w-\varphi$ has a local maximum at $y$ relative to $A$ and

$$|y-\bar{x}|<\varepsilon,\qquad |w(y)-w(\bar{x})|<\varepsilon,\qquad |D\varphi(y)-p|<\varepsilon,\qquad \lVert D^{2}\varphi(y)-X\rVert<\varepsilon .$$

2. (From below)(ii) The quadruple $\bigl(\bar{x},w(\bar{x}),p,X\bigr)$ is approximable by test data from below for $w$ on $A$ if for every positive $\varepsilon\in\mathbb{R}$ there exist $y\in A$ and $\varphi\in C^{2}(E)$ such that $w-\varphi$ has a local minimum at $y$ relative to $A$ and the four displayed inequalities of clause 1 hold.

This is the counterpart on a real inner product space of Quadruple Approximable by Test-Function Data, and differs from it in two respects: the ambient space is $E$ rather than a Euclidean space, and the set $A$ carrying $w$ is arbitrary rather than open, the test functions being of class $C^{2}$ on all of $E$.