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I am studying Fourier Series and in it, the fact that $C^0 (\mathbb{T})$ or $C^\infty (\mathbb{T})$ is dense(in the $L^p(\mathbb{T})$ norm, i.e. integral restricted to the domain with one period) in $L^p(\mathbb{T})$ where $1\le p <\infty$ and $\mathbb{T}$ denotes the space of functions on the real line with period $1$(or any real number). However, I can't find a reference which states this fact, all I can find is that $C_c (\mathbb{R}^n)$ is dense in $L^p (\mathbb{R}^n)$ but I don't think this fact implies the above.

For instance, in the below proof of the Riemann-Lebesgue lemma for periodic functions, approximation with $C^0(\mathbb{T})$ functions is used.

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    just google "continuous functions dense in L^p" – 2017-02-11
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    @zhw. I can only find continuous functions with compact support are dense in $L^p$ but I need periodic ones for this theorem. – 2017-02-11
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    If you have a continuous function approximating an $L^p$ function within an $\epsilon/2$, then you can modify the values of the function to become periodic, without changing the $L^p$ norm by more than another $\epsilon/2$. – 2017-02-11
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    @TrialAndError How can that be done analytically? We need the value at the endpoints of the function to be the same. I don't see how they can be modified without changing the norm. – 2017-02-11
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    @takecare We just went through this in another thread: Approximate in $L^p,$ then cut it off to make it $0$ at the endpoints, which will not disturb the approximation by much. – 2017-02-11
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    Also, keep in mind that functions which vanish at both endpoints of the interval are already periodic. – 2017-02-11
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    @DisintegratingByParts How can it be periodic? – 2018-12-08

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