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Math Help - Proving abstract cases of subspaces

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    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.
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Proving abstract cases of subspaces

    Quote Originally Posted by nicholasglennon View Post
    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) 0\in W (where 0 denotes the zero function), (ii) For all f,g\in W also f+g\in W and (iii) For all \lambda\in\mathbb{R} and for all f\in W also \lambda f\in W. Then, W is subspace of V.
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