Being $\mathbb{T}$ the unit torus and $\mathbb{K}$ the real or complex numbers, I know that the dual of the space of integrable functions $L^1(\mathbb{T}, \mathbb{K})$ can be identified with the space of essentially bounded functions $L^\infty(\mathbb{T}, \mathbb{K})$. Also, that the space of continuous functions $C(\mathbb{T},\mathbb{K})$ can be identified with the space of measures of bounded variation, $M(\mathbb{T})$.
These results of duality hold for domains different than $\mathbb{T}$? For example, can I take $\mathbb{T}^n$ or even something more general, any compact space, and the corresponding result still holds? If so, any good reference where I can see the proofs of those?
The duality $(L^1(X,\mu))^\ast\cong L^\infty(X,\mu)$ holds for every $\sigma$-finite measure space $(X,\mu)$ and $C(X)^\ast\cong M(X)$ holds for every compact space $X$. In particular, both hold for the $n$-torus (in the first case you have to choose a measure, for example the Haar measure).
Proofs can be found in various places. For example, they are included as appendices B and C in Conway's Course in Functional Analysis.