Why is the set of all polynomial functions of degree at most n a subspace of the set of all functions? I am given that:

Let $F(\mathbb R)$ be the set of all functions $f: \mathbb R \rightarrow \mathbb R$ and let $P_n(\mathbb R)$ be the set of all polynomial functions from $\mathbb R$ to $\mathbb R$ of degree at most $n$. $P_n(\mathbb R)$ is a subspace of $F(\mathbb R)$

However I wouldn't think that the zero function was an element of $P_n(\mathbb R)$ since the zero function is a polynomial function of undefined degree and so does not have degree of at most $n$. And so $P_n(\mathbb R)$ cannot be a subspace of $F(\mathbb R)$
Is it assumed here that the zero function is included in $P_n(\mathbb R)$? If so, why?

$P_n(\Bbb R)$ satisfies the subspace criterion.  Namely, it is closed under addition and scalar multiplication.
Thus we have a vector subspace (of $\Bbb F(\Bbb R)$).
I would think that the zero polynomial  would have degree zero.
Looking at the other answer, the comments, and again at your question, you don't explicitly say "vector space", so I am a little confused.  Why it would help to consider the zero polynomial to have degree $-\infty$ is beyond me.  On second perusal, @Qiaochu Yuan has given a couple of intelligent reasons for doing that.
