The Z 1-eigenvalue of tensors (hypermatrices) was widely used to discuss the properties of higher order Markov chains and transition probability tensors. In this paper, we extend the concept of Z 1-eigenvalue from finite-dimensional tensors to infinite-dimensional tensors, and discuss the upper bound of such eigenvalues for infinite-dimensional generalized Hilbert tensors. Furthermore, an upper bound of Z 1-eigenvalue for finite-dimensional generalized Hilbert tensor is obtained also.