Uncountable Basis?

Pick $F$ to be some field, and $X$ to be some uncountable set. Consider now $V$ to be the vector space whose elements are functions $f\colon X\to F$ such that all but finitely many $x\in X$ satisfy $f(x)=0$.

The addition is pointwise addition, and scalar multiplication is pointwise multiplication. One can easily check that this is a vector space.

Now it is not hard to verify that $\{\delta_x\mid x\in X\}$ where $\delta_x$ is the function $$\delta_x(y)=\begin{cases}1 & x=y\\ 0 & x\neq y\end{cases}$$ is a basis for $V$, and it is clearly uncountable ($x\mapsto\delta_x$ is a bijection).

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Perhaps it might be worth adding that using a set theoretic axiom known as The Axiom of Choice, or more often its useful equivalent Zorn's Lemma, we can prove that every vector space has a basis. And then we can conclude under certain conditions that this basis must be uncountable.

However the axiom of choice only assures us the existence of certain objects, it does not supply a description. In particular some naturally occurring vector spaces can be shown to have an uncountable basis using the axiom of choice. And we can show that this use of the axiom of choice is in fact necessary.

If we're mentioning the axiom of choice, perhaps it should also be pointed out that the axiom of choice is in fact equivalent to the statement "Every vector space has a basis".

Consider $\Bbb R$ as a vector space over the rational scalars $\Bbb Q$. As mentioned in Asaf's answer, this vector space must have a basis (at least if you accept the Axiom of Choice). However a countable subset of $\Bbb R$ will give rise to only countably many linear combinations with rational coefficients; therefore we need an uncountable set in order to form a basis.