Fourier series on $\mathbb T$ and $S^1$ From my lecture notes: "The notation  $\mathbb T$ will be used for the additive circle and $S^1$ for the multiplicative circle."
What I understand: As a topological group, $S^1$ has the subspace topology of $\mathbb R^2$ and multiplication is defined as $(e^{ia}, e^{ib}) \mapsto e^{i(a + b)}$. 
My guess is that $\mathbb T$ as an additive group should then be something like $(a,b) \mapsto (a + b) \mod 1$. The problem with that is that the space would look like $[0,1)$ but that's not compact. 
But I'm confused: "mod 1" seems to be the same as $\mathbb R / \mathbb Z$ which is $S^1$. But I can't add complex numbers on the unit circle and stay on the unit circle. 
So: What's $\mathbb T$? How are elements in it added?
 A: The map $\mathbb R \to S^1$, $t \mapsto e^{2\pi it}$ descends to a map $\mathbb T \to S^1$ which is an algebraic and topological isomorphism. [Remember that $\mathbb T$ is an algebraic and topological quotient. What is the inverse of this map?] So you can indeed pass freely between the two, and in particular $\mathbb T$ is compact. If you want to see this without reference to $S^1$, note that the restriction of the quotient map $\mathbb R \to \mathbb R/\mathbb Z$ to the compact subspace $[0, 1]$ is still surjective.
A: In my experience the two are used interchangeably, as the two groups are isomorphic, even as topological groups, and this isomorphism (as topological groups) is unique! You are right that "$\bmod 1$" is the same as $\mathbb R/\mathbb Z$, which is best viewed as $[0,1]$ with the endpoints identified. The only differences I can see are one of notation (elements of $\mathbb T$ are added while those in $S^1$ are multiplied) and that $S^1$ is more naturally considered a subset of $\mathbb C$.
