Let $V_n=${ $p(x)$: $p$ is a polynomial of deg(p)$\leq n$}. Show that $V_n$ is a subspace of $\mathbb{R}^\mathbb{R}$ and find a basis of $V_n$. Let $V_n=${ $p(x)$: $p$ is a polynomial of deg(p)$\leq n$}. Show that $V_n$ is a subspace of $\mathbb{R}^\mathbb{R}$ and find a basis of $V_n$.
I know that we need to show that $V_n$ is closed under addition, scalar multiplication and $0\in V_n$ but I couldn't show. Can you give a hint?
 A: From the question, it seems that you are (implicitly) already given that $\mathbb{R}^\mathbb{R}$ (i.e. the space of all functions from $\mathbb{R}$ to $\mathbb{R}$) is a vector space; $V_n$ is clearly a non-empty subset, so to show that it is a subspace, all we need to apply is the subspace test.
That is, we need to show that,


*

*$\,\,\,\,\,\,0_{V_n} \in V_n \,\,\,$ (where $0_{V_n}$ is the additive identity - see note at end)

*$\,\,\,\,\,\, \forall u, v \in V_n \,\,$ and $\,\,\forall \lambda \in \mathbb{R}, \quad \lambda u + v \in V_n $
This is relatively straightforward:
$$u = a_0 + a_1x + a_2x^2 + ... + a_nx^n,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, v = b_0 + b_1x + b_2x^2 + ... + b_nx^n$$
for some constants $a_0,a_1,...,a_n,b_0,b_1,...,b_n \in \mathbb{R}$. (Note that, the constants here are real as $V_n$ is a subspace of real functions. Also, some, or all, of these constants may well be zero.)
Then,
$$ \lambda u + v = (\lambda a_0 + b_0) + (\lambda a_1 + b_1)x + ... + (\lambda a_n + b_n)x^n$$
which is certainly a polynomial of degree $\leq n$, so is in $V_n$.
Hence, by the subspace test, we conclude that $V_n$ is a subspace of $\mathbb{R}^\mathbb{R}$.
For a proof of the subspace test, see: https://proofwiki.org/wiki/One-Step_Vector_Subspace_Test For more info, on the subspace test, see Google.
Note: Note here that I am counting the zero function as a polynomial of degree zero (this is necessary for the statement in the question to actually be true). There are also some comments to this end on the OP's question.
