A claim for a Schauder basis on Hilbert spaces

Let $H$ be a separable Hilbert space and assume that $\{e_k\}_{k=1}^{\infty} \subset H$ is a Schauder basis for $H$ such that given any $v\in H$ there holds:$$ v = \sum_{k=1}^{\infty} \langle v,e_k \rangle_H \,e_k.$$Does it follow that $\{e_k\}_{k=1}^{\infty}$ is an orthonormal basis for $H$?