Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions

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 Real Matrices, Symmetric Matrices and the Semidefinite Ordering: 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 $\langle\cdot,\cdot\rangle$, norm $|\cdot|$, distance $d$ and zero vector $0_{H}$ as fixed there, and let $\mathrm{Sym}(H)$ be the set of bounded symmetric bilinear forms on $H$, with its norm, its order $\preceq$, its sums and scalar multiples, its identity form $I=I_{H}$ and its zero form $0_{\mathrm{Sym}}$, as fixed there; for $b\in\mathrm{Sym}(H)$ let $T_{b}$ be the operator represented by $b$. Let $\mathbb{R}^{m}$ with its dot product, Euclidean norm and Euclidean distance $d_{E}$, the set $\mathcal{S}(m)$ of symmetric real matrices containing $aI_{m}$ for every real $a$, its order $\preceq$, its norm and metric $d_{\mathcal{S}(m)}$, and the matrix-vector product $M\zeta$ and sums and scalar multiples of matrices, be as fixed there. As in those settings, the symbol $\lVert\cdot\rVert$ serves for the Euclidean norm on $\mathbb{R}^{m}$, for the norm of a symmetric real matrix and for the norm on $\mathrm{Sym}(H)$, and the symbol $\preceq$ for the order on $\mathcal{S}(m)$ and for the order on $\mathrm{Sym}(H)$; which is meant is in each case determined by the arguments. We write $\lVert\zeta\rVert^{2}=\lVert\zeta\rVert\lVert\zeta\rVert$ and $|x|^{2}=|x|\,|x|$, and $\tfrac{1}{2}$ for the multiplicative inverse of $2=1+1$, which exists by claim 8 of Elementary Order Arithmetic in an Ordered Field.

The set $H$ is open in $(H,d)$, since every open ball of $(H,d)$ is a subset of $H$; the class $C^{2}(H)$, and the gradient $Du(x)\in H$ and Hessian $D^{2}u(x)\in\mathrm{Sym}(H)$ of a member $u$ of it, are as fixed there.

Let $e\in H^{m}$ be an $m$-tuple in $H$ that is orthonormal, with components $e_{1},\dots,e_{m}$. Sums of vectors are finite sums in $H$ and sums of real numbers are finite sums in $\mathbb{R}$. Writing $\zeta=(\zeta_{1},\dots,\zeta_{m})$ for a point of $\mathbb{R}^{m}$, define maps $\Lambda:H\to\mathbb{R}^{m}$ and $\Lambda^{\sharp}:\mathbb{R}^{m}\to H$ by

and let $P:H\to H$ be the composite $P=\Lambda^{\sharp}\Lambda$, which is the map $Px=\sum_{i=1}^{m}\langle x,e_{i}\rangle e_{i}$ of Projection onto the Span of an Orthonormal Tuple, and Coordinates on a Finite-Dimensional Subspace. Then the following hold.

1. (Coordinates)¶ The maps $\Lambda$, $\Lambda^{\sharp}$ and $P$ are linear; $\Lambda\Lambda^{\sharp}\zeta=\zeta$ for every $\zeta\in\mathbb{R}^{m}$ and $\Lambda Px=\Lambda x$ for every $x\in H$; and

for all $x\in H$ and $\zeta\in\mathbb{R}^{m}$. Moreover, for $x,x'\in H$ one has $\Lambda x=\Lambda x'$ if and only if $Px=Px'$; and $\Lambda^{\sharp}$ maps the origin of $\mathbb{R}^{m}$ to $0_{H}$.

2. (Forms carried by the coordinate map)¶ For $M\in\mathcal{S}(m)$ let $M^{\Lambda}$ be the map assigning the real number $\Lambda z\cdot(M\,\Lambda w)$ to each pair $z,w\in H$. Then $M^{\Lambda}\in\mathrm{Sym}(H)$ and, for all $z,w\in H$, all $M,M'\in\mathcal{S}(m)$ and all $a\in\mathbb{R}$,

and $M\preceq M'$ implies $M^{\Lambda}\preceq(M')^{\Lambda}$.

3. (The projection form and the tail form)¶ The map $\Pi$ assigning the real number $\Lambda z\cdot\Lambda w$ to each pair $z,w\in H$ belongs to $\mathrm{Sym}(H)$, and so does $N=I-\Pi$. They satisfy, for all $z,w\in H$,

together with $T_{\Pi}z=Pz$; moreover $0_{\mathrm{Sym}}\preceq\Pi\preceq I$, $0_{\mathrm{Sym}}\preceq N\preceq I$, $\lVert\Pi\rVert\le1$ and $\lVert N\rVert\le1$. The form $N$ is called the tail form of $e$ and $\Pi$ its projection form.

4. (Coordinate quadratics)¶ Let $\zeta_{1},q\in\mathbb{R}^{m}$, let $M\in\mathcal{S}(m)$ and let $c,\eta\in\mathbb{R}$. Let $T_{0}:\mathbb{R}^{m}\to\mathbb{R}$ and $T:H\to\mathbb{R}$ be given by

Put $b=M^{\Lambda}+2\eta\,\Pi\in\mathrm{Sym}(H)$, with $\Pi$ as in clause 3. Then $T\in C^{2}(H)$ and, for every $x\in H$,

In particular $DT(x)=\Lambda^{\sharp}q$ for every $x\in H$ with $\Lambda x=\zeta_{1}$, and more generally

Finally $T(x)=T(x')$ whenever $\Lambda x=\Lambda x'$, and

5. (Quadratic majorant and minorant of a function of class $C^{2}$)¶ Let $\Omega\subseteq\mathbb{R}^{m}$ be open, let $\chi:\Omega\to\mathbb{R}$ be of class $C^{2}$ on $\Omega$, with gradient $D\chi(\zeta)\in\mathbb{R}^{m}$ and Hessian $D^{2}\chi(\zeta)\in\mathcal{S}(m)$ at $\zeta\in\Omega$, let $\zeta_{1}\in\Omega$ and let $\eta\in\mathbb{R}$. Let $T_{0}$ be as in clause 4 with

Then $T_{0}(\zeta_{1})=\chi(\zeta_{1})$. If $0<\eta$, there is a positive $\rho\in\mathbb{R}$ such that every $\zeta\in\mathbb{R}^{m}$ with $\lVert\zeta-\zeta_{1}\rVert<\rho$ belongs to $\Omega$ and satisfies $\chi(\zeta)\le T_{0}(\zeta)$. If $\eta<0$, there is a positive $\rho\in\mathbb{R}$ such that every $\zeta\in\mathbb{R}^{m}$ with $\lVert\zeta-\zeta_{1}\rVert<\rho$ belongs to $\Omega$ and satisfies $T_{0}(\zeta)\le\chi(\zeta)$.

6. (Stability of the squared Euclidean distance)¶ For all $\zeta,\zeta',\omega,\omega'\in\mathbb{R}^{m}$,

and