Constants, Sums, Scalar Multiples and Differences of Differentiable Functions 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 $|\cdot|$, distance $d$ and zero vector $0_{E}$ 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 $(E,d)$. That a function $U\to\mathbb{R}$ is differentiable at a point of $U$ or differentiable on $U$, its gradient, that a form is a second derivative at a point, the Hessian, and the classes $C^{1}(U)$ and $C^{2}(U)$, are as defined there.

For $u,v:U\to\mathbb{R}$ and $\lambda\in\mathbb{R}$ let $u+v$, $\lambda u$ and $u-v$ denote the functions from $U$ to $\mathbb{R}$ whose values at $x\in U$ are $u(x)+v(x)$, $\lambda\,u(x)$ and $u(x)-v(x)$; nothing is asserted about them beyond this reading of the symbols. Then the following hold.

1. (Constants)(i) For every $c\in\mathbb{R}$ the function $U\to\mathbb{R}$ with constant value $c$ belongs to $C^{2}(U)$, and its gradient and Hessian at every $x\in U$ are $0_{E}$ and $0_{\mathrm{Sym}}$.

2. (Sums)(ii) Let $u,v:U\to\mathbb{R}$ and let $x\in U$. If $u$ and $v$ are differentiable at $x$, then $u+v$ is differentiable at $x$ and

$$D(u+v)(x)=Du(x)+Dv(x).$$

If in addition $u$ and $v$ have second derivatives at $x$, then $u+v$ has the second derivative $D^{2}u(x)+D^{2}v(x)$ at $x$. If $u,v\in C^{1}(U)$ then $u+v\in C^{1}(U)$, and if $u,v\in C^{2}(U)$ then $u+v\in C^{2}(U)$.

3. (Scalar multiples)(iii) Let $u:U\to\mathbb{R}$, let $\lambda\in\mathbb{R}$ and let $x\in U$. If $u$ is differentiable at $x$, then $\lambda u$ is differentiable at $x$ and

$$D(\lambda u)(x)=\lambda\,Du(x).$$

If in addition $u$ has a second derivative at $x$, then $\lambda u$ has the second derivative $\lambda\,D^{2}u(x)$ at $x$. If $u\in C^{1}(U)$ then $\lambda u\in C^{1}(U)$, and if $u\in C^{2}(U)$ then $\lambda u\in C^{2}(U)$.

4. (Differences)(iv) Let $u,v:U\to\mathbb{R}$ and let $x\in U$. If $u$ and $v$ are differentiable at $x$, then $u-v$ is differentiable at $x$ and

$$D(u-v)(x)=Du(x)-Dv(x).$$

If in addition $u$ and $v$ have second derivatives at $x$, then $u-v$ has the second derivative $D^{2}u(x)-D^{2}v(x)$ at $x$. If $u,v\in C^{1}(U)$ then $u-v\in C^{1}(U)$, and if $u,v\in C^{2}(U)$ then $u-v\in C^{2}(U)$.