Thread: Subspace of Infinitely Differentiable Functions on R

1. Subspace of Infinitely Differentiable Functions on R

I have to prove which of the following are subspaces of infinitely differentiable functions on R:
1) {f : f(0) = 0}
2) {f : f'(0) = 0}
3) {f : f(x) + f(-x) = 1}
4) {f : f(x) + f(-x) = f(0)}

I know that a subspace must be closed under addition and scalar multiplication, but I have no idea how to do this. How would you be able to test whether all of these are closed under addition and scalar multiplication?

2. 1) Denote $\displaystyle E$ the vector space of infinitely differentiable functions on $\displaystyle \mathbb{R}$ and $\displaystyle F=\{f\in E:f(0)=0\}$ then:

(a) The zero function $\displaystyle 0:\mathbb{R}\to \mathbb{R},\;0(x)=0$ for all $\displaystyle x\in\mathbb{R}$ is infinitely differentiable on $\displaystyle \mathbb{R}$ and satisfies $\displaystyle 0(0)=0$, so $\displaystyle 0\in F$ .

(b) For all $\displaystyle f,g\in F$ the sum $\displaystyle f+g$ is infinitely differentiable on $\displaystyle \mathbb{R}$ and satisfies:

$\displaystyle (f+g)(0)=f(0)+g(0)=0+0=0$

so, $\displaystyle f+g\in F$.

(c) For all $\displaystyle \lambda\in \mathbb{R}$ and for all $\displaystyle f\in F$ ...

Show your work for the rest.

Fernando Revilla