1. Prove that the set of almost periodic functions on R is an algebra, that is, it is a
vector space and the product of two functions in this space lies again in the space.
2. Prove that the fact that the sum of n continuous periodic functions is almost periodic
is equivalent to the following approximation result.
For any positive real numbers t1 , . . . , tn and any ε > 0 there exists L > 0 such that
any interval of length L contains a number t such that |t − nk tk | < ε for some nk ∈ Z and
all k = 1, . . . , n. Hint: use orthonormality of the system {eiλx }λ∈R in the space of almost
periodic functions.
3. Let X be one of the spaces c0 , `p (1 ≤ p < ∞). Define a linear map Pn : X → X by
Pn ((x1 , x2 , . . .)) = (x1 , . . . , xn , 0, 0, . . .).
Show that Pn x → x as n → ∞, moreover, the convegence is uniform on compact sets.
Conclude that any compact operator into X can be approximated by finite rank operators.