What if the basis not countable, then what? I'm second year physics & mathematics student, and self-studying Abstract linear algebra from the book Linear Algebra by Werner Greub.
In the mean time I have come across several times to the notion of countable basis.I know what can or can not do if a some set is countable/uncountable, but while studying linear algebra I do not exactly know what I couldn't do if the basis of space is not countable ? 
I generally worked either on abstract (finite / infinite) spaces or finite abstract spaces, and while in the abstract case, we have never assumed that the basis is countable, so I'm not sure what I would lose if I have a space of infinite dimension whose basis is uncountable.
tl;dr
If our vector space has a basis that is not countable, then which properties would be lost or gained compared to the case where we have space having countable basis (finite or infinite).
 A: For an infinite dimensional vector space we must distinguish form an Hamel (algebraic) basis and a Schauder basis.
The first concept is defined for any vector space and the axiom of choice guarantees that  any vector space have a Hamel basis, but this basis might be uncountable.   
In such a basis any vector can be expressed as finite linear combination of elements of the basis.
The simpler case is the space of real numbers  $\mathbb {R}$ considered as a vector space over the rationals $\mathbb{Q}$ that has an infinite  non-countable basis.  In this case we cannot have a  '' construction'' that enumerate all the elements of the basis.
A Schauder basis can be defined only for a topological vector space, where we can define the convergence of a series. In this case a vector can be expressed as  a series ( an ''infinite'' linear combination) of elements of the Schauder basis.
A simple example of a space that has  a non countable Hamel basis and a countable Schauder  basis is the space of the functions $L^p[0,\pi]$ over $\mathbb{C}$ where the set of functions $\{e^{nix}\: n \in \mathbb{Z}\}$ is a Schauder basis.
A: A vector space $W$ of countably infinite dimension is never isomorphic to the dual $V^\star$ of some other vector space $V$.
To see this take a basis $B$ of such a hypothetical $V$. Clearly $B$ must be infinite. But then we can find an uncountable subset $Q\subset \mathcal P(B)$ of the powerset of $B$ such that for each $p,q\in Q$ the intersection $p\cap q$ is finite (see here). Then the uncountable family $\{\chi_q\mid q\in Q\}\subset V^\star$ (where $\chi_q(b)=1$ iff $b\in q$ and $\chi_q(b)=0$ otherwise) is linearly independent (see here for more details). Hence $V^\star$ ist not of countable dimension.
