Bounded $C^2$ Cylindrical Functions on a Hilbert Space with an Orthonormal Basis

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 coordinate maps $p_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates. For $n\in\mathbb{N}$, $C^{2}_{b}(\mathbb{R}^{n})$ is the set of Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded, taken with $q=n$.

(Bounded $C^{2}$ cylindrical functions)¶ A function $\varphi:X\to\mathbb{R}$ is a bounded $C^{2}$ cylindrical function if there are $n\in\mathbb{N}$ and $\psi\in C^{2}_{b}(\mathbb{R}^{n})$ with $\varphi=\psi\circ p_{n}$. The set of all bounded $C^{2}$ cylindrical functions on $X$ is denoted $\mathcal{F}C^{2}_{b}(X)$.