Proving a vector space has an uncountable basis I am required to prove that a vector space $V=C[0,1]$ (the space of continuous functions over complex field) has an uncountable basis. The approach I took was to show that there is a subspace $W$ of $V$ with an uncountable basis and concluded that the vector space $V$ itself must have an uncountable basis.
Is this approach correct? I'm just unsure about the last step, if you have a subspace with an uncountable basis does it imply that the space itself must have an uncountable basis?
The answer to your general question is yes. If $W$ is a subspace of $V$ and $W$ has an uncountable basis $B\subseteq W$ then $B$ is still linearly independent as a subset of $V$. So $B$ can be extended to a basis for $V$ which means $V$ also has an uncountable basis.
Let W be the set of all polynomials.  Any finite subset must have a polynomial of largest degree so cannot be a basis.