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## Homework Statement

Which of the following are subspaces of F[R] = {f |f:R-->R}?

a) U = {f e F[R]|f(-1)f(1)=0

b) V = " |f(1)+f(2)=0

c) S = " |f(x)=f(-x)

d) T = " |f(1)<= 0

## Homework Equations

## The Attempt at a Solution

I got S and V or c) and b), is that correct?

I thought U or a) would be too, but just found out that it is not closed under addition:

Lets say if: f(1)=2 f(-1)=0 whose product is 0, however another possibility is f(1)=0 f(-1)=a number (non-zero), so if you add them it would not be in V (2+0, 0 + a number) whose product would not be zero. Am I correct?

I somehow know whether it is in the subspace or not, but no idea how to write a proof. How do I show it? Do I do as I typed above, like with actual number examples? or do I have to do it with symbols? ex. (f+g)(x) = f(x) + g(x) there closed under addition...like that?

Thanks