Let $L>0$ fixed. Consider the space
$$
\mathcal{P}:=\{f: \mathbb{R} \longrightarrow \mathbb{C} \; ; \; f \: \text{is infinitely differentiable and periodic with period}\: L\}.
$$

For $r \in \mathbb{N}$ consider the Sobolev Space $H^r_{per}([0,L])$. This space can be interpreted as the set of $f \in \mathcal{P}'$ such that
$$f, f' , \cdots, f^{(r)} \in L^2_{per}([0,L]),$$ 
with norm
$$||f||_{H^r_{per}}=\left(\sum_{i=0}^{r}||f^{(i)}||^2_{L^2_{per}} \right)^{\frac{1}{2}}, \; \forall \; f \in H^r_{per}([0,L]), \tag{1}$$
where
$$L^2_{per}([0,L])=\{f: \mathbb{R} \longrightarrow \mathbb{C} \; ; \; f \: \text{is   periodic with period} \; L \; \text{and} \; f|_{[0,L]} \in L^2([0,L])\} $$
with inner product 
$$(f,g)_{L^2_{per}}=\int_0^L f(x) \overline{g(x)}\; dx, \; \forall \; f,g \in L^2_{per}([0,L]).$$

Now, consider
$$
L^2_{per,m}([0,L]):=\left\{ f \in L^2_{per}([0,L]) \; \; \; \int_0^L f(x) \; dx=0\right\}.
$$
Clearly, $L^2_{per,m}([0,L])$ is closed subspace of $L^2_{per}([0,L])$. From the Proposition $3.193$. of  $[1]$ we have, for $r,s \in \mathbb{N}, s \geq r$, that $H^s_{per}([0,L])$ is continuously and densely embedded in $H^r_{per}([0,L])$. Simbolically $H^s_{per}([0,L]) \hookrightarrow H^r_{per}([0,L])$.

**Question**. Is $H^s_{per}([0,L]) \cap L^2_{per,m}([0,L]) \hookrightarrow H^r_{per}([0,L]) \cap L^2_{per,m}([0,L])$ also?

I think that we can write 
$$ 
H^n_{per,m}([0,L]) := H^n_{per}([0,L]) \cap L^2_{per,m}([0,L])=\left\{ f \in H^n_{per}([0,L]) \; \int_0^L f \; dx=0\right\}.
$$ In this way, I think that $H^n_{per,m}([0,L])$ is closed subspace of the $H^n_{per}([0,L])$, and so is Hilbert space (with the induced inner product) for all $n\in \mathbb{N}$. In this sense, I think we would have $H^s_{per,m}([0,L])$ is continuously embedded in $H^r_{per,m}([0,L])$ (with the induced inner product).  But is dense?

$[1]$ IORIO Jr., R. J.; IORIO, V. M.  [*Fourier Analysis and Partial Differential
Equations*](https://www.cambridge.org/core/books/fourier-analysis-and-partial-differential-equations/39312A08B4D4F25F65F39581D229285B). Cambridge University Press, Cambridge, UK, 2001.