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Structure of Fourier exponents of almost periodic functions and periodicity of almost periodic functions. (English) Zbl 0863.42012
For Banach space valued functions the definition of almost periodicity is introduced using the density of \(\varepsilon\)-almost periods in \(\mathbb R\). This definition is equivalent to the definition given by S. Bochner. The structure of Fourier exponents and of the range of almost periodic functions is studied in the paper and it is shown that the class of continuous periodic Banach space valued functions is not densely distributed in the space of almost periodic functions, the norm being the usual supremum norm for bounded functions defined on \(\mathbb R\).

42A75 Classical almost periodic functions, mean periodic functions
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