Schauder basis $L^p(\mathbb{R})$

Question

Let $\{e_{n}(x)\}_{n=0}^{\infty}$ be orthonormal basis of Hilbert space $L^2(\mathbb{R})$. If $\{e_{n}(x)\}_{n=0}^{\infty} \subset L^p(\mathbb{R})$ for some $p\geq 1$, is the $\{e_{n}(x)\}_{n=0}^{\infty}$ Schauder basis for $L^p(\mathbb{R})$?

Any reference?