So here is a 2 part question that I am having a lot of trouble with.

For

each of the following sets determine whether it is a vector space:

(i) $\displaystyle W_1 = \{f(x) \in C(R) | \int_0^{2} x^2 f(x)dx =0\}$

(iii) $\displaystyle W_1 = \{f(x) \in C(R) | \int_0^{2} xf^2 (x)dx =0\}$

Ok so what I know is that in order for it to be a vector space I need to show that all the axioms are true, or to show that it is not a vector space I need to show atleast one axiom fails.

I am kind of confused about how to go about doing this?

For the first one I integrated and got $\displaystyle 4G(2) -4R(2)-2H(2)+2H(0)$

where G(x) is the first integral of f(x), R(x) is the second and H(x) is the third. Im not completely sure wether or not this was neccessary, and if so what to do with this.

Any help would be very much appreciated, thanks!