Definition:Schauder Basis/Definition 1

Definition

Let $\Bbb F \in \set {\R, \C}$.

Let $\struct {X, \norm \cdot}$ be a normed vector space over $\Bbb F$.

Let $\set {e_n : n \in \N}$ be a countable subset of $X$.

We say that $\set {e_n : n \in \N}$ is a Schauder basis for $X$ if and only if:

for each $x \in X$, there exists a unique sequence $\sequence {\map {\alpha_j} x}_{j \mathop \in \N}$ in $\Bbb F$ such that:

$\ds x = \sum_{j \mathop = 1}^\infty \map {\alpha_j} x e_j$

where convergence of the infinite series is understood in $\struct {X, \norm \cdot}$.