Bounded $C^1$ Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions

In the settings of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, with the coordinates $x_{k}$ and the coordinate maps $p_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, let $C^{1}_{b}(\mathbb{R}^{n})$ be the set of Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded with $q=n$, and let the set $\mathcal{F}C^{1}_{b}(X)$ of bounded $C^{1}$ cylindrical functions, their representations and the partial derivatives $\partial_{k}\varphi$ of functions differentiable on $X$ be those of Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis and Partial Derivatives along an Orthonormal Basis of a Function Differentiable on a Hilbert Space respectively, with differentiability on $X$ and the gradient $D\varphi(x)$ of Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space, applied with $E=U=X$. For $\mu\in\mathcal{P}(X)$, $L^{2}(\mu)=L^{2}(X,\mathcal{B}(X),\mu)$ is the real Hilbert space of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §hilbert, with inner product $\langle\cdot,\cdot\rangle_{L^{2}(\mu)}$, and $\mathcal{P}_{2}(X)$ is the set of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space.

1. (Linear structure)¶ Every constant function $X\to\mathbb{R}$ belongs to $\mathcal{F}C^{1}_{b}(X)$, and $\mathcal{F}C^{1}_{b}(X)$ is closed under pointwise sums and real multiples.

2. (Gradient)¶ Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$ have a representation $(n,\psi)$. Then $\varphi$ is differentiable on $X$, with gradient

$$D\varphi(x)=\sum_{i=1}^{n}\partial_{i}\psi\bigl(p_{n}(x)\bigr)\,e_{i}\qquad(x\in X).$$

3. (Partial derivatives)¶ Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$, differentiable on $X$ by claim 2, and let $k\in\mathbb{N}$. If $(n,\psi)$ is a representation of $\varphi$, then $\partial_{k}\varphi=(\partial_{k}\psi)\circ p_{n}$ for $k\le n$ and $\partial_{k}\varphi(x)=0$ for $k>n$ and every $x\in X$.

4. (Bounded and Borel)¶ Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$ and $k\in\mathbb{N}$. Then $\varphi$ and $\partial_{k}\varphi$ are Borel and bounded.

5. (Bounded gradient)¶ Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$. Then the map $D\varphi:X\to X$ is Borel, and there is a real number $B\ge0$ with $|D\varphi(x)|\le B$ for every $x\in X$.

6. (Square-integrability)¶ Let $\mu\in\mathcal{P}(X)$, $\varphi\in\mathcal{F}C^{1}_{b}(X)$ and $k\in\mathbb{N}$. Then $\varphi$ and $\partial_{k}\varphi$ are integrable with respect to $\mu$ and $2$-integrable with respect to $\mu$ in the sense of Power-Integrable Functions and the p-Seminorm §space, so that their classes belong to $L^{2}(\mu)$.

7. (Coordinate moments)¶ Let $\mu\in\mathcal{P}_{2}(X)$, $\varphi\in\mathcal{F}C^{1}_{b}(X)$ and $k\in\mathbb{N}$. Then the function $x\mapsto x_{k}\varphi(x)$ is integrable with respect to $\mu$.

8. (Density)¶ Let $\mu\in\mathcal{P}(X)$. The set of the classes in $L^{2}(\mu)$ of the functions in $\mathcal{F}C^{1}_{b}(X)$ is dense in $L^{2}(\mu)$. In particular, if $g\in L^{2}(\mu)$ satisfies $\langle g,\varphi\rangle_{L^{2}(\mu)}=0$ for every $\varphi\in\mathcal{F}C^{1}_{b}(X)$, then $g$ is the zero vector of $L^{2}(\mu)$.