Dales and Polyakov introduced a multi-norm (∥⋅∥(2,2)n:n∈N) based on a Banach space X and showed that it is equal with the Hilbert-multi-norm (∥⋅∥Hn:n∈N) based on an infinite-dimensional Hilbert space H. We enrich the theory and present three extensions of the Hilbert-multi-norm for a Hilbert C∗-module X. We denote these multi-norms by (∥⋅∥Xn:n∈N), (∥⋅∥∗n:n∈N), and (∥⋅∥P(A)n:n∈N). We show that ∥x∥P(A)n≥∥x∥Xn≤∥x∥∗n for each x∈Xn. In the case when X is a Hilbert K(H)-module, for each x∈Xn, we observe that ∥⋅∥P(A)n=∥⋅∥Xn. Furthermore, if H is separable and X is infinite-dimensional, we prove that ∥x∥Xn=∥x∥∗n. Among other things, we show that small and orthogonal decompositions with respect to these multi-norms are equivalent. Several examples are given to support the new findings.

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