Continuous Cylindrical Functions of Polynomial Growth on a Hilbert Space

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the coordinate maps $p_{n}:X\to\mathbb{R}^{n}$ and the Euclidean norm $\lVert\cdot\rVert$ on $\mathbb{R}^{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates; continuity of a map $\mathbb{R}^{n}\to\mathbb{R}$ is that of Continuous Map Between Metric Spaces for the Euclidean distance and the absolute-value distance; for real $u\ge0$ and $q\in\mathbb{N}$, $u^{q}$ is the natural power and $u^{0}=1$.

1. (Continuous cylindrical functions of polynomial growth)¶ A function $F:X\to\mathbb{R}$ is a continuous cylindrical function of polynomial growth if there are $n\in\mathbb{N}$, a continuous $\psi:\mathbb{R}^{n}\to\mathbb{R}$, a real $B\ge0$ and $q\in\mathbb{N}\cup\{0\}$ with $F=\psi\circ p_{n}$ and

$$|\psi(y)|\le B\bigl(1+\lVert y\rVert^{q}\bigr)\qquad\text{for every }y\in\mathbb{R}^{n}.$$

The set of these functions is $\mathcal{F}C_{\mathrm{pol}}(X)$.

2. (Representations)¶ For $F\in\mathcal{F}C_{\mathrm{pol}}(X)$, a quadruple $(n,\psi,B,q)$ as in clause 1 is a representation of $F$, and $q$ is its growth exponent.