# Prove that V is a vector space

I've been given the following question. My problem is that I'm not really sure what I'm suppose to do. Can someone help me getting started maybe just give me a theorem I could use.

Consider the set of functions $$V := \{ f: \mathbb{R} \rightarrow \mathbb{C} \, \lvert \, f(-\pi) = f(\pi) = 0 \}$$ (i) Show that $V$ is a vector space with respect to the usual operations of addition and scalar multiplication.

Let $f$ and $g$ be elements of your vector space $V$. Observe that
$$(f+g)(\pi) \;\; =\;\; f(\pi) + g(\pi) \;\; =\;\; 0 \;\; f(-\pi) + g(-\pi) \;\; =\;\; (f+g)(-\pi)$$
hence $f+g \in V$ since it is closed under vector addition. Similarly if $a \in \mathbb{C}$ we have that $af(\pi) = 0 = af(-\pi)$ hence $V$ is closed under scalar multiplication. $V$ is therefore a vector space.
• To prove something is a vector space over a field $K$ you need to verify the $8$ axioms of a vector space. Here you are assuming the set of functions $\mathbb{R}\to \mathbb{C}$ is a vector space, and what you proved is that $V$ is a linear subspace. – Qidi Feb 11 '15 at 23:06