Regarding a Basis for Infinite Dimensional Vector Spaces In my linear algebra class, during the discussion of vector spaces, our instructor mentioned infinite dimensional spaces, including the polynomial space over Q and the space of all continuous functions over the interval [0,1]. Our teacher also warned that although Q[x] has an infinite basis, the actual elements of the vector space can be described as linear combinations of a finite subset of the basis. My question is this: is this always the case? Or are some vectors described only by a sort of "infinite linear combination" of their basis elements. Also, what do the basis elements of functional spaces, such as the one mentioned above, look like? And in what way are they defined?

A basis for $\mathbb{Q}[x]$ looks like: $\{1,x,x^2,\ldots\}$
For a basis of $C[0,1]$, you might have a look at The Faber–Schauder system which is an example of a  Schauder basis. Wikipedia gives a sufficient explanation of both. 
Note that, assuming the axiom of choice, every vector space has a basis. The proof of which invokes Zorn's Lemma, (which is equivalent to $\mathsf{AC}$). In fact, the statement "Every vector space has a basis" is equivalent to the axiom of choice.  
I would also like to direct you to Martin's answer in this question: Does $\mathbb{R}^\mathbb{R}$ have a basis?  He does a great job of massaging this topic. 
