On complete biorthogonal systems

Abstract:

Fundamental to the study of bases in a separable Hilbert space $H$ is the notion of a biorthogonal system. Two sequences $\left \{ {{f_n}} \right \}$ and $\left \{ {{g_n}} \right \}$ of elements from $H$ are said to be biorthogonal if $({f_n},{g_m}) = {\delta _{nm}}$. A complete sequence that possesses a biorthogonal sequence is called exact. Despite the symmetry of the definition of biorthogonality, simple examples show that $\{ {f_n}\}$ may be exact while $\{ {g_n}\}$ fails to be exact. For sequences of complex exponentials in ${L^2}( - \pi ,\pi )$, the situation is dramatically different—if the sequence $\{ {e^{i{\lambda _n}t}}\}$ is exact, then its biorthogonal sequence is also exact.