Size of function spaces For example, how big is the space $ C^k(\mathbb{R},\mathbb{R}) $ ? How much is, say, $ C^0  $ larger than $ C^1$ ? How can one figure out ?
 A: If you're asking about cardinality, it turns out they both have the same cardinality as $\mathbb{R}$. For $\mathbb{R}^2$ this has been proven many ways. An overkill approach is to use space-filling curves. 
For $C^k(\mathbb{R},\mathbb{R})$, here's a sketch of a proof based on the Cantor-Schroder-Bernstein theorem. Verifying one half of the hypotheses of the theorem is easy: $|C^k(\mathbb{R},\mathbb{R})| \geq |\mathbb{R}|$ because we have an injective function $f : \mathbb{R} \to C^k(\mathbb{R},\mathbb{R})$ defined by $f(x) = y \mapsto x$. In other words, $f$ maps each real number to the corresponding constant function, which is certainly smooth.
The other half is harder. First we certainly have that $|C^k(\mathbb{R},\mathbb{R})| \leq |C(\mathbb{R},\mathbb{R})|$ because we have an inclusion map. 
Showing that $|C(\mathbb{R},\mathbb{R})| \leq |\mathbb{R}|$ is the hard part. The idea of the proof is to show that a continuous function is uniquely defined by its values on $\mathbb{Q}$. In other words, there is an injection $g : C(\mathbb{R},\mathbb{R}) \to \mathbb{R}^{\mathbb{Q}}$, which is given by the restriction map. So $|C(\mathbb{R},\mathbb{R})| \leq |\mathbb{R}^{\mathbb{Q}}| = |\mathbb{R}^{\mathbb{N}}|$ (since $\mathbb{Q}$ is countable).
This last cardinality is the same as that of $\mathbb{R}$. There is a direct proof based on an interlacing argument, similar to how one usually proves that $|\mathbb{N} \times \mathbb{N}| = |\mathbb{N}|$. An "algebraic" proof is:
$$\left | \mathbb{R}^{\mathbb{N}} \right | = \left | (2^{\mathbb{N}})^{\mathbb{N}} \right | = \left | 2^{\mathbb{N} \times \mathbb{N}} \right | = \left | 2^{\mathbb{N}} \right | = \left | \mathbb{R} \right |.$$
A: There are many senses of bigger; here's one that gives an easy, concrete answer: As a vector spaces over $\mathbb{R}$ with the usual notions of addition and scalar multiplication, $C^k(\mathbb{R}, \mathbb{R})$ is infinite-dimensional, whereas $\mathbb{R}^2$ has dimension two.
A: A quite related question: How small is $C^{k+1}([0,1])$ inside $C^{k}([0,1])$. Note that $C^{k}$ is a Banach space with a norm say $\sum_{l=o}^k \sum\sup_t||f^{l}(t)||$. It turns out that $C^{k+1}$ is of first category in $C^{k}$ (http://en.wikipedia.org/wiki/Meagre_set), a classic result.  
