Integrability on the Abstract Wiener Space of Double Sequences and Fernique Theorem

Abstract

The integrability of a function defined on the abstract Wiener space of double Fourier coefficients is explored. The abstract Wiener space is also a Hilbert space. We define an orthonormal system of the Hilbert space to establish a measure and integration on the abstract Wiener space. We examine the integrability of a function $e^{\alpha {‖·‖}^2}$ defined on the abstract Wiener space for Fernique theorem. With respect to the abstract Wiener measure, the integral of the function turns out to be convergent for $\alpha <1/2$ . The result provides a wider choice of the constant $\alpha$ than that of Fernique.