Suppose $H$ is a Hilbert space and has a Schauder basis $\{e_{j}\}_{j=1}^{\infty} $. $\{\hat{e_{j}}\}_{j=1}^{\infty}$ is a sequence biorthogonal with $\{e_{j}\}_{j=1}^{\infty}$; that is, $<e_{j},\hat{e_{k}}>=\delta_{jk}, $ where $\delta_{jk}=1$ if $j=k$ and otherwise 0.
Is $\{\hat{e_{j}}\}_{j=1}^{\infty}$ also a Schauder basis? This seems perhaps like a standard exercise but I am having some trouble solving it. I'd be very grateful for any help.