This paper adresses the problem of testing for the equality of $k$ probability distributions on Hilbert spaces, with $k\geqslant 2$. We introduce a generalization of the maximum variance discrepancy called multiple maximum variance discrepancy (MMVD). Then, a consistent estimator of this measure is proposed as test statistic, and its asymptotic distribution under the null hypothesis is derived. A simulation study comparing the proposed test with existing ones is provided

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