The size of the set of continuous function of period T I have a naive question. The Fourier series give an injection between continuous function of period $T$ and the set of real valued sequences. But, don't we expect the set of continuous function of periode $T$ to be much bigger than the set of sequences? By this I mean that since "$card(\mathbb{N})<card(\mathbb{R})$" we could expect sets of function from $\mathbb{R}$ to $\mathbb{R}$ to be much larger than the set of functions from $\mathbb{N}$ to $\mathbb{R}$. Can anyone comment on this? 
Thanks
 A: It is true that there are more functions $\mathbb{R}\to\mathbb{R}$ than there are functions $\mathbb{N}\to\mathbb{R}$.  However, the key restriction here is that we are only considering continuous functions $\mathbb{R}\to\mathbb{R}$.  Very few (relatively speaking) functions $\mathbb{R}\to\mathbb{R}$ are continuous, so few that in fact they can inject into the set of functions $\mathbb{N}\to\mathbb{R}$.  A more elementary way to see this than Fourier analysis is to just notice that a continuous function $\mathbb{R}\to\mathbb{R}$ is uniquely determined by its restriction to $\mathbb{Q}$.  So restriction gives an injection from the set of continuous functions $\mathbb{R}\to\mathbb{R}$ to the set of functions $\mathbb{Q}\to\mathbb{R}$, which can be identified with the set of functions $\mathbb{N}\to\mathbb{R}$ since $\mathbb{Q}$ is countable.
For some related discussion (and in particular, proofs that all of these sets actually have the same cardinality as $\mathbb{R}$ itself), see Cardinality of set of real continuous functions.
