A Fubini Theorem for Grothendieck Functional Integrals

Abstract: This paper systematically studies the subset of continuous linear functionals on the projective tensor product of Banach spaces whose norms are bounded by Grothendieck's constant $K_G$. We term such functionals Grothendieck functional integrals. The integral is defined as a linear functional on the projective tensor product space that satisfies the boundedness condition $|μ(x)| \leq K_G |x|_π$, where $K_G$ denotes Grothendieck's constant. We prove that such integrals admit a Hilbert space representation theorem and establish the corresponding abstract Fubini theorem to demonstrate that the order of integration may be interchanged. Furthermore, we extend this theory to the setting of multiple tensor products and provide integral representations in concrete function spaces. Our work offers a unified framework for bilinear and multilinear analysis, with a universal constant serving as the fundamental bound.