Bounded $C^1$ 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 $n\in\mathbb{N}$, $C^{1}_{b}(\mathbb{R}^{n})$ is the set of bounded $C^{1}$ functions of Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded, taken with $q=n$.

1. (Cylindrical functions)¶ A function $\varphi:X\to\mathbb{R}$ is a bounded $C^{1}$ cylindrical function if there are $n\in\mathbb{N}$ and $\psi\in C^{1}_{b}(\mathbb{R}^{n})$ with $\varphi=\psi\circ p_{n}$. The set of all bounded $C^{1}$ cylindrical functions on $X$ is denoted $\mathcal{F}C^{1}_{b}(X)$.

2. (Representations)¶ Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$. A pair $(n,\psi)$ with $n\in\mathbb{N}$, $\psi\in C^{1}_{b}(\mathbb{R}^{n})$ and $\varphi=\psi\circ p_{n}$ is a representation of $\varphi$.