How am I suppose to regard this D?
I would like to have some help with the bolded part:
Let n be a positive integer and let E_{n} be the set of mappings f:R->R
that are given by a prescription of the form
f(x)=a_{0}+Σ from k=1 to n (a_{k}cos(kx)+b_{k}sin(kx))
where a_{k}, b_{k}∈R for each k.
Prove that E_{k} is a subspace of Map(R,R).
If f∈E_{n} is the zero mapping, prove that all the coefficients a_{k}, b_{k} must be 0.
[Hint. Proceed by induction. For this, find a prescription for D^{2}f+n^{2}f.]
Deduce that the 2n+1 functions
x->1, x->cos(kx), x->sin(kx) (k=1,...,n)
form a basis for E_{n}.
Thanks in advance for any help you are able to provide.