The space of almost periodic functions with the Hausdorff metric

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.