Do monomials form a basis for the vector space of real analytic functions? Does the set ${1, x, x^2...}$ form a basis for the vector space of real analytic functions over the real numbers? It seems obvious that they span, but not obvious that they are independent.
 A: Short answer: NO.
Long answer:
It depends on your use of the word 'basis'. Remember that when working with infinite dimensional vector spaces you can define 2 types of basis. Schauder's basis and Hamel's basis. 
$\textbf{1)}$ Hamel Basis: For any infinite dimensional vector space $V$ a l.i. collection of vectors $ HB = \{ v_1,v_2, \cdots \} $ is named a Hamel basis for $V$ if for every vector $ v \in V $ there is a $\textbf{finite}$ number of elements of $HB$ such that
$$ v = \displaystyle \sum_{i=1}^N c_i v_i $$
where of course $c_i$ belongs to the corresponding field.
$ \textbf{2)} $ Schauder Basis: For any infinite dimensional vector space $V$ a l.i. collection of vectors $ SB = \{ v_1,v_2, \cdots \} $ is named a Sachauder basis for $V$ if for every vector $ v \in V $ there is a subset $\{ u_1, u_2, \cdots \} \subseteq SB$ such that
$$ v = \sum_{i=1}^{\infty} c_i u_i $$
Is the difference clear? In the first one you can only accept $\textbf{finite}$ linear combinations while in the second one you can have a series. 
The thing with the Schauder basis it is that they are often very difficult to work with. Banach himself asked if any separable Banach space had a Schauder basis. Years after this question was solved by Enflo. (No). That is a very disappointing thing for the fans of Schauder's basis because existence is one of the first things you want to ensure.
A: Complementing the other answer where the difference between a vector space basis (Hamel basis) and a Schauder basis was introduced, let me mention that in this article P. Domaǹski and D. Vogt show that the space of real analytic functions does not have a Schauder basis.
Note that the Taylor series expansion might lead to a different series at each point but in order for the monomials to form a basis one would need that have the same expansion at every point.
