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.