A multiplicative Hankel operator is an operator with matrix representation M(α) = {α(nm)}_n,m=1 ^∞, where α is the generating sequence of M(α). Let M and M₀ denote the spaces of bounded and compact multiplicative Hankel operators, respectively. In this note it is shown that the distance from an operator M(α) ∈ M to the compact operators is minimized by a nonunique compact multiplicative Hankel operator N(β) ∈ M₀. Intimately connected with this result, it is then proven that the bidual of M₀ is isometrically isomorphic to M, M₀** ≅ M. It follows that M₀ is an M-ideal in M. The dual space M₀* is isometrically isomorphic to a projective tensor product with respect to Dirichlet convolution. The stated results are also valid for small Hankel operators on the Hardy space H²(D^d) of a finite polydisk.

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