Biorthogonal system

A pair of sets $\{a_t\}$ and $\{\xi_t\}$, $t \in T$, of elements of a (topological) vector space $X$ and the dual (topological) space $X^*$, respectively, which satisfies the conditions $$ \xi_t(a_s) = \langle \xi_t, a_s \rangle = 0 $$ if $t \ne s$ and $\ne 0$ if $t=s$ (here, $\langle {\cdot},{\cdot} \rangle$ is the canonical bilinear form coupling $X$ and $X^*$). For instance, a biorthogonal system consists of a Schauder basis and the set formed by the expansion coefficients of $x$ in it. In a Hilbert space $H$ with scalar product $\langle {\cdot},{\cdot} \rangle$ and basis $\{a_t\}$ the set $\{b_s\}$ satisfying the condition $$ \langle a_t, b_s \rangle = \delta_{st} $$ where $\delta_{st} = 1$ if $s = t$ and $\delta_{st} = 0$ if $s \ne t$, is also a basis; it is said to be the basis dual to $\{a_t\}$ and, since $H = H^*$, the sets $\{a_t\}$ and $\{b_s\}$ form a biorthogonal system. In particular, a basis in $H$ is said to be orthonormal if its dual to itself.

However, there also exist biorthogonal systems which do not even form a weak basis; an example is the set of functions $\exp(ikx)$, $k\in\mathbb{Z}$, $x\in\mathbb{R}$, in the space of continuous periodic functions with the norm $\Vert f \Vert = \sup |f(x)|$.