I'm studying the basics of group theory for physics and to do so I'm using "Group Theory" by Dresselhaus et al. There's a point I really can't understand, that is:
"Suppose that we have a group G with symmetry elements R and symmetry operators $\hat{P}_R$. We denote the irreducible representations by $\Gamma_n$, where $n$ labels the representation. We can then define a set of basis vectors denoted by $|\Gamma_n \alpha\rangle$ . Each vector $|\Gamma_n j\rangle$ of an irreducible representation $\Gamma_n$ is called a component or partner and $j$ labels the component or partner of the representation, so that if we have a two-dimensional representation, then $j = 1, 2$. All partners collectively generate the matrix representation denoted by $D^{(\Gamma_n)}(R)$. These basis vectors relate the symmetry operator $\hat{P}_R$ with its matrix representation $D^{(\Gamma_n)}(R)$ through the relation:
$$\hat{P}_R|\Gamma_n\alpha\rangle = \sum\limits_j{D^{(\Gamma_n)}(R)|\Gamma_n j\rangle}$$
The basis vectors can be abstract vectors; a very important type of basis vector
is a basis function which we define here as a basis vector expressed explicitly
in coordinate space. Wave functions in quantum mechanics, which are basis
functions for symmetry operators, are a special but important example of such
basis functions."
Many times in quantum chemistry/physics I've heard saying "basis" and "complete basis" as they were two distinct things, but I realised that a "basis" of a $n$-dimensional vector space is composed of $n$ linearly independent vectors. So two are the cases:
With "basis" the authors mean a set of lin. indep. vectors, so that the equation holds true by definition. So this should also mean that when they talk about using functions instead of vectors, the basis should be infinite, since no finite number of functions can represent a "basis" for the vectorial space of all the possible functions. Am I right?
With "basis" the authors mean a generic set of vectors, so that the equation cannot hold true in general, but it will only for specific "basis" sets (given a specific group of operators). This would mean that the "basis" is chosen such that the equation holds true for all the operations. Is it correct?
Which of these two cases is the correct one?
Moreover, could you, please, explain me what do the authors mean with "[...]a basis function which we define here as a basis vector expressed explicitly in coordinate space."?
Another, more formal way to think about the different meanings of "basis" is to say that a "non complete basis" (which is formally wrong) can be seen as a basis of a subspace of a certain vectorial space. So, my question, reformulated, is: when the authors say that we define a basis of vectors, what vectorial (sub)space is it referred to (in both vector and function cases)?
Thank you in advance.