How to show $C_p^k([-\ell, \ell])$ is not a Banach space?

Question

I need to show the space $$C_p^k([-\ell, \ell])=\{f\in C^k(\mathbb R; \mathbb C); f(x+2\ell)=f(x), \forall x\in\mathbb R\},$$ is not a banach space with the norms $$\|f\|_p=\left(\int_{-\ell}^{\ell}|f(x)|^p\right)^{1/p}, p\in [1, \infty).$$ Does anyone have an idea?

Answer 1

I can suppose $\ell=\pi$ for $C_p([−\ell,\ell])$ is isomorphic to $C_p([−\pi,\pi])$, right?

Correct. But it's not any harder to deal with general $\ell$ at once. For example, consider $$f_n(x)=(n^{-1}+\sin^2 (\pi x/\ell))^{1/2} \tag1$$ and observe that $f_n\to f$ in $L^p$, where $f(x)=|\sin (\pi x/\ell)|$ is not $C^k$ smooth.