Proof of Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space

We use repeatedly that $d(x,x+z)=|z|$ for $x,z\in E$, by Real Inner Product Spaces (iii) and Elementary Identities in a Real Inner Product Space (iv), and record two remarks.

Remark (i). If $f,g:U\to E$ are continuous on $U$ as maps into $(E,d)$ and $\lambda\in\mathbb{R}$, then so are the maps $x\mapsto f(x)+g(x)$ and $x\mapsto\lambda\,f(x)$. Indeed, for $x,y\in U$ the vector space identity $(f(x)+g(x))-(f(y)+g(y))=(f(x)-f(y))+(g(x)-g(y))$ and The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (i) give

$$d\bigl(f(x)+g(x),f(y)+g(y)\bigr)\le d\bigl(f(x),f(y)\bigr)+d\bigl(g(x),g(y)\bigr),$$

so that, given a positive $\varepsilon$, radii for $f$ and for $g$ at the positive number $\varepsilon\cdot 2^{-1}$ (claim 8 of Elementary Order Arithmetic in an Ordered Field) and the lesser of the two (claim 9 there) witness continuity of the sum at $y$ relative to $U$, by Continuous Map Between Metric Spaces. For the scalar multiple, $d(\lambda f(x),\lambda f(y))=|\lambda|\,d(f(x),f(y))$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (iii); if $\lambda=0$ the map is constant, and otherwise a radius for $f$ at $\varepsilon\,|\lambda|^{-1}$ serves.

Remark (ii). The same holds for maps $U\to\mathrm{Sym}(E)$ continuous into $(\mathrm{Sym}(E),d_{\mathrm{Sym}})$: by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (i) the set $\mathrm{Sym}(E)$ is a vector space, so $(b_{1}+b_{1}')-(b_{2}+b_{2}')=(b_{1}-b_{2})+(b_{1}'-b_{2}')$ and $\lambda b_{1}-\lambda b_{2}=\lambda(b_{1}-b_{2})$, and Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (iii) gives

$$d_{\mathrm{Sym}}(b_{1}+b_{1}',b_{2}+b_{2}')\le d_{\mathrm{Sym}}(b_{1},b_{2})+d_{\mathrm{Sym}}(b_{1}',b_{2}'),\qquad d_{\mathrm{Sym}}(\lambda b_{1},\lambda b_{2})=|\lambda|\,d_{\mathrm{Sym}}(b_{1},b_{2}).$$

The argument of remark (i) then applies verbatim.

Claim 1. Let $c\in\mathbb{R}$ and let $k:U\to\mathbb{R}$ have constant value $c$, and let $x\in U$. Since $U$ is open, Open Subset of a Metric Space provides a positive $\rho$ with $B_{d}(x,\rho)\subseteq U$, so that $x+z\in U$ whenever $|z|<\rho$. For such $z$, $k(x+z)-k(x)-\langle 0_{E},z\rangle=0$ by Elementary Identities in a Real Inner Product Space (ii), and $0\le\varepsilon|z|$ for every positive $\varepsilon$ by claim 5 of Elementary Arithmetic in an Ordered Field; taking $\delta=\rho$ for every $\varepsilon$ shows that $k$ is differentiable at $x$ with gradient $0_{E}$. The gradient map is constant, hence continuous on $U$, so $k\in C^{1}(U)$ by The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space (i). Moreover $k$ is differentiable at every point of $B_{d}(x,\rho)$ and, for all $w,y\in E$ with $|w|<\rho$,

$$\langle Dk(x+w)-Dk(x),y\rangle-0_{\mathrm{Sym}}(w,y)=\langle 0_{E},y\rangle-0=0 ,$$

so taking $\delta=\rho$ for every $\varepsilon$ shows that $0_{\mathrm{Sym}}$ is a second derivative of $k$ at $x$. The Hessian map is constant, hence continuous, and $k\in C^{2}(U)$ by The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space (ii).

Claim 2. Suppose $u$ and $v$ are differentiable at $x$, with gradients $p$ and $q$. Let $\varepsilon\in\mathbb{R}$ be positive, let $\delta_{u}$ and $\delta_{v}$ be radii supplied by Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space (i) for the positive number $\varepsilon\cdot 2^{-1}$, and let $\delta$ be the lesser of the two, which is positive. Every $z\in E$ with $|z|<\delta$ satisfies $x+z\in U$ and, using $\langle p+q,z\rangle=\langle p,z\rangle+\langle q,z\rangle$ (condition (b) of Real Inner Product Spaces (i)), claim 5 of Properties of the Absolute Value in an Ordered Field and claim 8 of Elementary Order Arithmetic in an Ordered Field,

$$\bigl|(u+v)(x+z)-(u+v)(x)-\langle p+q,z\rangle\bigr|\le\bigl|u(x+z)-u(x)-\langle p,z\rangle\bigr|+\bigl|v(x+z)-v(x)-\langle q,z\rangle\bigr|\le\varepsilon\,|z| .$$

Hence $u+v$ is differentiable at $x$ with gradient $p+q$.

Suppose in addition that $u$ and $v$ have second derivatives $b_{u}=D^{2}u(x)$ and $b_{v}=D^{2}v(x)$ at $x$, with radii $\rho_{u}$ and $\rho_{v}$ as in The Second Derivative and the Hessian on an Open Subset of a Real Inner Product Space (i), and let $\rho$ be the lesser of the two, which is positive. Both $u$ and $v$ are differentiable at every point of $B_{d}(x,\rho)\subseteq U$, hence so is $u+v$ by the previous paragraph, with $D(u+v)(y)=Du(y)+Dv(y)$ for every such $y$. The form $b_{u}+b_{v}$ lies in $\mathrm{Sym}(E)$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (i). Given a positive $\varepsilon$, let $\delta$ be the lesser of $\rho$ and of the two radii supplied by that clause for $\varepsilon\cdot 2^{-1}$; for all $w,y\in E$ with $|w|<\delta$, the identity

$$\langle D(u+v)(x+w)-D(u+v)(x),y\rangle-(b_{u}+b_{v})(w,y)=\bigl(\langle Du(x+w)-Du(x),y\rangle-b_{u}(w,y)\bigr)+\bigl(\langle Dv(x+w)-Dv(x),y\rangle-b_{v}(w,y)\bigr),$$

which uses condition (b) of Real Inner Product Spaces (i) and Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iv), together with claim 5 of Properties of the Absolute Value in an Ordered Field, bounds the left-hand side in absolute value by $\varepsilon\,|w|\,|y|$. Hence $b_{u}+b_{v}$ is a second derivative of $u+v$ at $x$.

Finally, if $u,v\in C^{1}(U)$ then $u+v$ is differentiable on $U$ with gradient map $Du+Dv$, continuous by remark (i), so $u+v\in C^{1}(U)$; and if $u,v\in C^{2}(U)$ then in addition $u+v$ has at every $x\in U$ the second derivative $D^{2}u(x)+D^{2}v(x)$, and its Hessian map is continuous by remark (ii), so $u+v\in C^{2}(U)$.

Claim 3. If $\lambda=0$ then $\lambda u$ is the function with constant value $0$, and claim 1 applies; moreover $\lambda\,Du(x)=0_{E}$ and $\lambda\,D^{2}u(x)=0_{\mathrm{Sym}}$ by claim 3 of Elementary Identities in a Vector Space, applied in $E$ and in the vector space $\mathrm{Sym}(E)$ of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (i), so the asserted formulas hold.

Suppose $\lambda\ne 0$, so that $0<|\lambda|$ by claim 1 of Properties of the Absolute Value in an Ordered Field, and let $u$ be differentiable at $x$ with gradient $p$. Given a positive $\varepsilon$, let $\delta$ be the radius supplied by Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space (i) for the positive number $\varepsilon\,|\lambda|^{-1}$. For $z$ with $|z|<\delta$, condition (c) of Real Inner Product Spaces (i) gives $\langle\lambda p,z\rangle=\lambda\langle p,z\rangle$, so claim 4 of Properties of the Absolute Value in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field give

$$\bigl|(\lambda u)(x+z)-(\lambda u)(x)-\langle\lambda p,z\rangle\bigr|=|\lambda|\,\bigl|u(x+z)-u(x)-\langle p,z\rangle\bigr|\le|\lambda|\,\varepsilon\,|\lambda|^{-1}\,|z|=\varepsilon\,|z| .$$

Hence $\lambda u$ is differentiable at $x$ with gradient $\lambda p$. The second-derivative assertion follows in the same way from the identity

$$\langle D(\lambda u)(x+w)-D(\lambda u)(x),y\rangle-(\lambda\,b_{u})(w,y)=\lambda\,\bigl(\langle Du(x+w)-Du(x),y\rangle-b_{u}(w,y)\bigr),$$

where $b_{u}=D^{2}u(x)$ and $\lambda b_{u}\in\mathrm{Sym}(E)$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (i), using condition (c) of Real Inner Product Spaces (i) and Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iv). The class assertions follow from remarks (i) and (ii) as in claim 2.

Claim 4. For every $y\in U$ one has $u(y)-v(y)=u(y)+(-1)v(y)$ by claim 5 of Elementary Identities in a Vector Space applied in the vector space $\mathbb{R}$, so $u-v$ is the function $u+(-1)v$. Claims 2 and 3 therefore give every assertion, the gradient being $Du(x)+(-1)Dv(x)=Du(x)-Dv(x)$ and the second derivative $D^{2}u(x)+(-1)D^{2}v(x)=D^{2}u(x)-D^{2}v(x)$, by claim 5 of Elementary Identities in a Vector Space in $E$ and by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iv) in $\mathrm{Sym}(E)$.