Consider the $n$ dimensional torus $\mathbb{T}^n$. The $L^p$ spaces over $\mathbb{T}^n$ is defined as consisting of an equivalence class of functions satisfying:$$\int_{\mathbb{T}^n}|f|^p < \infty. \tag{1}$$For simplicity suppose $n = 1$ so that $\mathbb{T}^1$ can be identified as the unit interval with periodic boundary conditions. Thus in practice (1) is often computed as$$\int_0^1 |f(x)|^p dx < \infty. \tag{2}$$However, it is well known that as a smooth manifold $T^1 \cong S^1$ does not have a global coordinate chart. Thus I am confused on what justifies (2), where one set of coordinates are used to compute the norm over the whole space.
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