 Biorthogonal system

For a normed vector space $X$, a family $(x_i,f_i{)}_{i\in I}$ in $X\times X^{\ast }$ 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 subspace. If its linear span is dense, each prescribed coordinate functional is unique by continuity. For a countable total family, the finite sums $P_{n}x={\sum }_{i\le n}{f}_i(x)x_i$ are bounded projection operators, since biorthogonality gives $P_n^2=P_n$.