Biorthogonal system
= Biorthogonal system
{title2=$f_i(x_j)=\delta_{ij}$}
For a <normed vector space> $X$, a family $(x_i,f_i)_{i\in I}$ in $X\times X^*$ is biorthogonal when the displayed Kronecker-delta equations hold. Its <vectors> are <linearly independent>: applying $f_i$ to a finite linear dependence isolates its $i$th coefficient. No inner product is required, and the family need not span a dense <vector subspace>[subspace]. If its <linear span> is dense, each prescribed coordinate functional is unique by <continuity>. For a countable total family, the finite sums $P_nx=\sum_{i\leq n}f_i(x)x_i$ are bounded <projection operators>, since biorthogonality gives $P_n^2=P_n$.