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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 α · 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  α &lt; 1 / 2 . The result provides a wider choice of the constant  α than that of Fernique.

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