Coordinate functional of a Schauder basis ($e_n^{\ast }$)

The nth coordinate functional of a Schauder basis is the bounded linear functional $e_n^{\ast }({\sum }_k{a}_k{e}_k)=a_n$.

The dual sequence consists of the coordinate functionals $(e_n^{\ast })$ in $X^{\ast }$. It is always a basic sequence, though it need not span the entire dual space.

A Schauder basis is shrinking when its dual sequence is a Schauder basis of the whole dual space. Every Schauder basis of a reflexive Banach space is shrinking.