Schauder basis and coordinate functionals

Let $X$ be a real or complex Banach space. A positively indexed sequence $(e_n{)}_{n\ge 1}$ in $X$ means a function from ${\mathbb{N}}_{\ge 1}$ to $X$. Such a sequence is a Schauder basis of $X$ if for every $x\in X$ there is a unique scalar family $a:{\mathbb{N}}_{\ge 1}\to \mathbb{K}$, written $(a_n{)}_{n\ge 1}$, such that

$x={\sum }_{n=1}^{\mathrm{\infty }}{a}_n{e}_n$

in the fixed displayed order. This notation denotes the zero-indexed series from Series and absolute convergence in a normed space whose term at $m\in \mathbb{N}$ is $a_{m+1}{e}_{m+1}$; equivalently, ${\sum }_{n=1}^N{a}_n{e}_n\to x$ in norm. The uniqueness is part of the definition.

For each $n$, the algebraically defined map

$e_n^{\ast }:X⟶\mathbb{K},e_n^{\ast }(x)=a_n,$

is the $n$th coordinate functional. It is linear: uniqueness applied to the expansions of $x+y$ and $\lambda x$ gives $e_n^{\ast }(x+y)=e_n^{\ast }(x)+e_n^{\ast }(y)$ and $e_n^{\ast }(\lambda x)=\lambda e_n^{\ast }(x)$. No continuity is included in this definition; boundedness will be proved later.

• A Schauder basis is ordered. Rearranging a conditional basis expansion can destroy convergence.
• Every basis vector is nonzero, since otherwise the zero vector would have two coefficient sequences.