and . (Clousures in . Are A+B closed? What is ?
If you include 0 in the natural numbers then A+B will contain all the trigonometric polynomials and therefore . If you do not allow 0 as a natural number then (the orthogonal complement of the constant functions in ).
To see that A+B is not closed, you only need to find some element such that . For that, notice that every x in A+B must be of the form with (and of course to ensure that ). So for example .