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?

By definition, a basis of a vector space is a linearly independent set such that every vector in the space is a linear combination of elements in the basis. 
In the case of $\mathbb Q[x]$, an obvious basis is given by $\{1,x,x^2,x^3,\ldots\}$.
