Abstract: This paper addresses the problem of two-sample paired testing for multivariate functional data. Initially, four novel global test statistics are proposed based on the concepts of the L² norm and the F-test, utilizing integration and supremum techniques, and their asymptotic null distributions are derived. The asymptotic null distributions of these test statistics are then approximated using the non-parametric bootstrap method and the Welch-Satterthwaite chi-square approximation method. Additionally, the √n-consistency of the proposed testing methods is established. Finally, an empirical analysis of the finite sample properties of the proposed test methods is conducted through numerical simulations and real data from air pollution.