Segment Derivatives, the Second-Order Taylor Expansion, and the Second-Order Condition at a Local Extremum

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 $|\cdot|$, distance $d$ and zero vector $0_{E}$ as fixed there, let $\mathrm{Sym}(E)$ be the set of bounded symmetric bilinear forms on $E$, with its order $\preceq$ and zero form $0_{\mathrm{Sym}}$. Let $U\subseteq E$ be open in $(E,d)$ and let $u:U\to\mathbb{R}$. That $u$ is differentiable on $U$, its gradient $Du(x)$, that a form is a second derivative of $u$ at a point, and the Hessian $D^{2}u(x)$, are as defined there; and the class $C^{2}(U)$ is as defined there. Local maxima and local minima of $u$ relative to $U$ are understood with the ambient metric space $(E,d)$.

Then the following hold.

1. (Derivative along a segment)(i) Let $x\in U$, let $r\in\mathbb{R}$ be positive and such that the open ball $B_{d}(x,r)$ is contained in $U$ and $u$ is differentiable at every point of $B_{d}(x,r)$, and let $z\in E$ satisfy $2\,|z|<r$. Then $x+tz\in U$ for every $t$ in the open interval $(-2,2)$, so that the function $g:(-2,2)\to\mathbb{R}$ with $g(t)=u(x+tz)$ is defined; and $g$ is differentiable at every $t\in(-2,2)$, with

$$g'(t)=\langle Du(x+tz),z\rangle .$$

2. (Second-order Taylor expansion)(ii) Suppose $u$ has a second derivative $D^{2}u(x)$ at $x\in U$. Then for every positive $\varepsilon\in\mathbb{R}$ there is a positive $\delta\in\mathbb{R}$ such that every $z\in E$ with $|z|<\delta$ satisfies $x+z\in U$ and

$$\Bigl|u(x+z)-u(x)-\langle Du(x),z\rangle-\tfrac{1}{2}\,D^{2}u(x)(z,z)\Bigr|\le\varepsilon\,|z|^{2} .$$

In particular this holds for every $u\in C^{2}(U)$ and every $x\in U$.

3. (Second-order condition at a local extremum)(iii) Suppose $u$ has a second derivative $D^{2}u(x)$ at $x\in U$. If $u$ has a local maximum at $x$ relative to $U$, then

$$Du(x)=0_{E}\qquad\text{and}\qquad D^{2}u(x)\preceq 0_{\mathrm{Sym}} .$$

If $u$ has a local minimum at $x$ relative to $U$, then

$$Du(x)=0_{E}\qquad\text{and}\qquad 0_{\mathrm{Sym}}\preceq D^{2}u(x) .$$