Question

Determine if the set is a vector space subspace. If the set is not a subspace identify which of the axioms fails.

Answer 1

there are three things you need to check
is the zero vector there, so for example in question a) does the zero function have x intercepts at multiples of \(\pi\) and the answer is yes, since it is identically zero

Answer 2

second is the sum of two vectors there
i.e. if \(f,g\in C^{\infty}\) with zeros at integer multiples of \(\pi\) does \(f+g\) have zeros at integer multiples of \(\pi\)
again the answer is yes, since by definition \((f+g)(x)=f(x)+g(x)\) so if \(f,g\) is zero at multiples of \(\pi\) does does \(f+g\)

Answer 3

finally are scalar multiples there, i.e. if \(f\) has zeros at integer multiples of \(\pi\) does \(cf\) have zeros at integer multiples of \(\pi\) and again the answer is yes, since multiplying a function by a constant does not change the zeros

Answer 4

repeat for each
b)
is zero an odd function?
is the sum of two odd functions odd?
is a constant multiple of an odd function odd?
etc