Are the following sets Hamel bases for real functions? We have 2 sets: 
$$A=\{f_a(x)=e^{ax}:a \in \mathbb R\}$$
$$B=\{f_b(x)=|x-b|:b \in \mathbb R\}$$
We can easily prove that they are uncountable linearly independent subsets of the set of continuous real functions $C(\mathbb R)$. I'm wondering if they form a basis for the vector space $C(\mathbb R)$. Can we use that they form a maximal linearly independent subset of $C(\mathbb R)$, as they are both uncountable? Also can we prove it without using the theorem, by finding an explicit way that a given function is a linear combination of those sets?
 A: No, an uncountable linearly independent set need not span, and in particular neither of these do. For instance, it's not hard to show that $e^{x^2}$ is not in the span of either $A$ or $B$. (HINT: consider the growth rates of functions in the spans of $A$ or $B$ . . .)
Although for finite-dimensional vector spaces, any linearly independent set of the same cardinality as some basis is again a basis, this fails for infinite-dimensional sets, roughly because an infinite set can have a proper subset of the same size (e.g. there are as many positive integers as there are integers). An easier example to visualize might be to take $V$ the vector space of all maps $\mathbb{N}\rightarrow \mathbb{R}$ which are equal to $0$ on all but finitely many inputs (algebraically, this is the direct sum of countably infinitely many copies of $\mathbb{R}$). Then the set $\{\delta_{2i}: i\in\mathbb{N}\}$, where $\delta_i$ is defined as $$x\mapsto 1\iff x=i,\quad x\mapsto 0\iff x\not=i,$$ is an infinite linearly independent set which is not a basis (e.g. $\delta_1$ is not in its span).

In fact, there is no "easily-describable" Hamel basis for $\mathcal{C}(\mathbb{R})$; this can be made precise and proved via descriptive set theory. For example, it is consistent with ZF (= set theory without the axiom of choice) that there is no Hamel basis for $\mathcal{C}(\mathbb{R})$. So any Hamel basis has to be so complicated that either it can't be explicitly described in the first place, or proving that it is in fact a basis requires genuinely set-theoretic arguments.
