Twice Differentiability of a Sum in Separated Variables

We work in the setting of Euclidean Space and Real Matrices: Standing Notation, whose notation is fixed for every dimension and is used here with natural numbers $m$ and $n$ satisfying $1\le m$ and $1\le n$; we put $N=m+n$. The real numbers, the absolute value $|\cdot|$, the initial segments $[p]$, and Euclidean space with its sum and difference of points, scalar multiples, dot product, Euclidean norm $\lVert\,\cdot\,\rVert$, distance $d_{E}$, notion of openness and origins $0_{\mathbb{R}^{p}}$, the real matrices and the matrix-vector product $Ph$, the sets $\mathcal{S}(p)$ of symmetric real matrices and the concatenation map $\iota:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{N}$, a bijection, are all as fixed there. For $X\in\mathcal{S}(m)$ and $Y\in\mathcal{S}(n)$ the block diagonal matrix $X\oplus Y\in\mathcal{S}(N)$, and for $Z\in\mathcal{S}(N)$ the blocks $Z^{11}\in\mathcal{S}(m)$, $Z^{12}$ and $Z^{22}\in\mathcal{S}(n)$, are those of Block Diagonal Symmetric Matrices and the Blocks of a Symmetric Matrix (i) and Block Diagonal Symmetric Matrices and the Blocks of a Symmetric Matrix (v). Twice differentiability at a point with a given first-order coefficient and Hessian is as defined there.

Let $U_{1}\subseteq\mathbb{R}^{m}$ and $U_{2}\subseteq\mathbb{R}^{n}$ be open, put

$$U=\{\,\iota(\xi,\eta)\ :\ \xi\in U_{1},\ \eta\in U_{2}\,\}\subseteq\mathbb{R}^{N},$$

let $v_{1}:U_{1}\to\mathbb{R}$ and $v_{2}:U_{2}\to\mathbb{R}$, and let $w:U\to\mathbb{R}$ be the function determined by

$$w\bigl(\iota(\xi,\eta)\bigr)=v_{1}(\xi)+v_{2}(\eta)\qquad(\xi\in U_{1},\ \eta\in U_{2}),$$

which is well defined because $\iota$ is injective, so that each point of $U$ arises from exactly one such pair. Let $\xi_{0}\in U_{1}$ and $\eta_{0}\in U_{2}$, and put $x_{0}=\iota(\xi_{0},\eta_{0})$. Then the following hold.

1. (The product is open) (i) $U$ is open in $\mathbb{R}^{N}$.

2. (Sums of twice differentiable functions) (ii) Suppose $v_{1}$ is twice differentiable at $\xi_{0}$ with first-order coefficient $p_{1}\in\mathbb{R}^{m}$ and Hessian $X_{1}\in\mathcal{S}(m)$, and $v_{2}$ is twice differentiable at $\eta_{0}$ with first-order coefficient $p_{2}\in\mathbb{R}^{n}$ and Hessian $X_{2}\in\mathcal{S}(n)$. Then $w$ is twice differentiable at $x_{0}$ with first-order coefficient $\iota(p_{1},p_{2})$ and Hessian $X_{1}\oplus X_{2}$.

3. (Splitting of a twice differentiable sum) (iii) Conversely, suppose $w$ is twice differentiable at $x_{0}$ with first-order coefficient $p\in\mathbb{R}^{N}$ and Hessian $Z\in\mathcal{S}(N)$, and let $(p_{1},p_{2})\in\mathbb{R}^{m}\times\mathbb{R}^{n}$ be the unique pair with $\iota(p_{1},p_{2})=p$. Then $v_{1}$ is twice differentiable at $\xi_{0}$ with first-order coefficient $p_{1}$ and Hessian $Z^{11}$, the function $v_{2}$ is twice differentiable at $\eta_{0}$ with first-order coefficient $p_{2}$ and Hessian $Z^{22}$, every entry of $Z^{12}$ equals $0$, and

$$Z=Z^{11}\oplus Z^{22}.$$