Geodesic
The space of almost periodic functions with the Hausdorff metric
Sbornik. Mathematics, Tome 81 (1995) no. 2, pp. 321-341 Cet article a éte moissonné depuis la source Math-Net.Ru

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The function space $\mathbf{H}$ obtained as the completion of the space $\mathbf{B}$ of real-valued uniformly almost periodic functions (a.p.) (Bohr a.p. functions) with respect to the Hausdorff metric is considered. Elements of the space $\mathbf{H}$ are called $H$-a.p. functions. Analogs of the theorems of Lyusternik (a criterion for compactness of a function family), Bochner (a criterion for almost periodicity), and Bohr (on representation of a.p. functions as diagonals of limit periodic functions) are obtained. The relationship between the space $\mathbf{H}$ and the space of $N$-a.p. functions is studied. In particular, it is shown that a continuous function in $\mathbf{H}$ may not belong to $\mathbf{B}$, but it is always an $N$-a.p. function. At the same time, the sum and the product of two continuous $H$-a.p. functions are not, in general, in $\mathbf{H}$ (but they are $N$-a.p. functions). Due to the coincidence of the topologies on $\mathbf{B}$ generated by the uniform and the Hausdorff metrics, the indicated space, in spite of its nonlinearity, is closer to the space $\mathbf{B}$ than the corresponding completions of $\mathbf{B}$ with respect to integral metrics. 
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     title = {The space of almost periodic functions with {the~Hausdorff} metric},
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}
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A. P. Petukhov. The space of almost periodic functions with the Hausdorff metric. Sbornik. Mathematics, Tome 81 (1995) no. 2, pp. 321-341. http://geodesic.mathdoc.fr/item/SM_1995_81_2_a3/

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