I would like to have some help with the bolded part:
Let n be a positive integer and let En be the set of mappings f:R->R
that are given by a prescription of the form
f(x)=a0+Σ from k=1 to n (akcos(kx)+bksin(kx))
where ak, bk∈R for each k.
Prove that Ek is a subspace of Map(R,R).
If f∈En is the zero mapping, prove that all the coefficients ak, bk must be 0.
[Hint. Proceed by induction. For this, find a prescription for D2f+n2f.]
Deduce that the 2n+1 functions
x->1, x->cos(kx), x->sin(kx) (k=1,...,n)
form a basis for En.
Thanks in advance for any help you are able to provide.