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Some Gr\"{u}ss type inequalities in semi-inner product modules over$C^*$-algebras and $H^*$-algebras for $n$-tuples of vectors are established.Also we give their natural applications for the approximation of the discreteFourier and the Melin transforms in such modules.

Some Gr\"{u}ss type inequalities for $n$-tuples of vectors in semi-inner modules