# Proving abstract cases of subspaces

• Mar 21st 2012, 10:03 AM
nicholasglennon
Proving abstract cases of subspaces
I'm not sure whether this qualifies as advanced since it is only out of a first year university course, but this subforum seems like the best fit. If you think I should be posting elsewhere then just let me know I guess.
V = C^∞(R), W is the set of functions f ∈ V for which lim(x→∞ ) f(x) = 0

I need to prove that W either is or isn't a subspace of V. If anyone could help with this or give me some pointers on proving these more abstract/general cases that would be great because there are a few questions like this that I am struggling with.
• Mar 21st 2012, 10:21 AM
FernandoRevilla
Re: Proving abstract cases of subspaces
Quote:

Originally Posted by nicholasglennon
I need to prove that W either is or isn't a subspace of V.

Using a well known theorem of characterization of subspaces, prove that (i) $\displaystyle 0\in W$ (where $\displaystyle 0$ denotes the zero function), (ii) For all $\displaystyle f,g\in W$ also $\displaystyle f+g\in W$ and (iii) For all $\displaystyle \lambda\in\mathbb{R}$ and for all $\displaystyle f\in W$ also $\displaystyle \lambda f\in W$. Then, $\displaystyle W$ is subspace of $\displaystyle V$.