Dual pairing ($⟨\phi ,v⟩=\phi (v)$)

The dual pairing evaluates a linear functional on a vector:It is bilinear: it is linear in each argument separately. Even over the complex numbers, this algebraic evaluation has no complex conjugation. An inner product instead has two vectors as inputs and, over the complex numbers, is sesquilinear. For $\phi (x,y)=2x-y$ and $v=(3,1)$, the dual pairing equals five. Its value is independent of the basis used to describe its two inputs.

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 $P_n^2=P_n$.