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.