The author generalizes the Paley-Wiener theorem (known from the distributional Fourier transform theory) to an infinite-dimensional white noise space \(({\mathcal E}',\mu)\), where \({\mathcal E}'\) is the dual of nuclear space \({\mathcal E}\) and \(\mu\) is the Gaussian measure on \({\mathcal E}'\). Let \(E\) be a real separated Hilbert space with norm \(\|\cdot\|_0\) and complexification \(E_c\). For each \(p\geq 0\) the space \({\mathcal E}_p= \{f\in E:\|A^pf\|_00\) such that for all \(n\in\mathbb{N}\) and for all \(\zeta_1\in{\mathcal E}_c\), \(\zeta_1\perp V^n_{1,c}\) we have \[ 1/\pi^{n- 1}\int|F(\zeta+ \zeta_1)|^2 e^{-|\zeta|^2_0} d\zeta= C e^{2R|\text{Re }\zeta_1|_p- \text{Re}\langle\zeta_1, \zeta_1\rangle} \] over \(V^n_{1,c}\). A part of the present paper is devoted to a description of the compact subsets of \({\mathcal E}'\) and to some points of the probability. The paper is based on \textit{H.-H. Kuo's} book ``White noise distribution theory'', Boca Raton/FL. (1996; Zbl 0853.60001). The author beliefs that this paper ``opens the gate for further research and a deeper understanding of the infinite-dimensional space in the world of white noise analysis''.