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By identifying functions whose difference has zero norm, he proved that {^£{R), || ||) is actually a Banach space. The space had been studied by many authors in the theory of almost periodic functions and generalized harmonic analysis (e.g., Besicovitch [4], Bohr and Folner [6], Bertrandias [3] and Lau and Lee [10]). In [10], it was shown that the transformation defined in (1.1) can be extended to an isomorphism from ^€{R) onto the space T*(R) of functions with

On the Banach spaces of functions with bounded upper means