The Second Derivative and the Hessian on an Open Subset of a Real Inner Product Space

In the setting of Real Hilbert Spaces: Standing Notation and Background, 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 the sums and scalar multiples of forms and the zero form $0_{\mathrm{Sym}}$ fixed there. Let $U\subseteq E$ be open in $(E,d)$, let $u:U\to\mathbb{R}$ and let $x\in U$. That $u$ is differentiable at a point of $U$, and its gradient $Du$, are as defined there. For $w\in E$ one has $d(x,x+w)=|w|$, by Real Inner Product Spaces (iii) and Elementary Identities in a Real Inner Product Space (iv), so that $x+w$ lies in the open ball $B_{d}(x,\rho)$ exactly when $|w|<\rho$.

1. (Second derivative at a point)(i) Let $b\in\mathrm{Sym}(E)$ and let $\rho\in\mathbb{R}$ be positive and such that $B_{d}(x,\rho)\subseteq U$ and $u$ is differentiable at every point of $B_{d}(x,\rho)$. The form $b$ is a second derivative of $u$ at $x$ if for every positive $\varepsilon\in\mathbb{R}$ there is a positive $\delta\in\mathbb{R}$ with $\delta\le\rho$ such that all $w,y\in E$ with $|w|<\delta$ satisfy

$$\bigl|\langle Du(x+w)-Du(x),y\rangle-b(w,y)\bigr|\le\varepsilon\,|w|\,|y| ;$$

the gradients occurring here are defined because $|w|<\delta\le\rho$ places $x+w$ in $B_{d}(x,\rho)$. This condition does not depend on the choice of $\rho$: if $\rho'$ is a second positive real number with $B_{d}(x,\rho')\subseteq U$ and $u$ differentiable at every point of $B_{d}(x,\rho')$, then a radius $\delta$ witnessing the condition for $\rho$, replaced by the lesser of $\delta$ and $\rho'$ (claim 9 of Elementary Order Arithmetic in an Ordered Field), which is positive because it is one of them, witnesses it for $\rho'$, and symmetrically. We say that $u$ has a second derivative at $x$ if some $b\in\mathrm{Sym}(E)$ is a second derivative of $u$ at $x$ for some such $\rho$.

2. (The Hessian)(ii) At most one $b\in\mathrm{Sym}(E)$ is a second derivative of $u$ at $x$. Indeed, suppose $b$ and $b'$ both are and let $\varepsilon\in\mathbb{R}$ be positive. Let $\delta$ and $\delta'$ be radii provided by clause 1 for $b$ and for $b'$ with $\tfrac{\varepsilon}{2}$, positive by claim 8 of Elementary Order Arithmetic in an Ordered Field, in place of $\varepsilon$, and let $\delta''$ be the lesser of $\delta$ and $\delta'$ (claim 9 of Elementary Order Arithmetic in an Ordered Field), which is positive because it is one of them. The form $b-b'$ belongs to $\mathrm{Sym}(E)$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (i) and satisfies $(b-b')(w,y)=b(w,y)-b'(w,y)$ by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iv), and all $w,y\in E$ with $|w|<\delta''$ satisfy

$$\bigl|(b-b')(w,y)\bigr|\le\bigl|\langle Du(x+w)-Du(x),y\rangle-b'(w,y)\bigr|+\bigl|\langle Du(x+w)-Du(x),y\rangle-b(w,y)\bigr|\le\varepsilon\,|w|\,|y|$$

by claims 2 and 5 of Properties of the Absolute Value in an Ordered Field. Since $\varepsilon$ was an arbitrary positive real number, Vanishing of Uniformly Small Linear, Quadratic and Bilinear Terms in a Real Inner Product Space (iii) gives $b-b'=0_{\mathrm{Sym}}$, so that $b(w,y)=b'(w,y)$ for all $w,y\in E$, that is $b=b'$. When $u$ has a second derivative at $x$ we write $D^{2}u(x)$ for this unique form and call it the Hessian of $u$ at $x$.

3. (The Hessian map)(iii) If $u$ has a second derivative at every point of $U$, its Hessian map is the map $D^{2}u:U\to\mathrm{Sym}(E)$ sending $x\in U$ to $D^{2}u(x)$.