Definition:Schauder Basis/Definition 2

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:


 * $(1): \quad$ for each $x \in X$, there exists a sequence $\sequence {\alpha_j}_{j \mathop \in \N}$ in $\Bbb F$ such that:
 * $\ds x = \sum_{j \mathop = 1}^\infty \alpha_j e_j$


 * $(2): \quad$ whenever $\sequence {\alpha_j}_{j \mathop \in \N}$ is a sequence in $\Bbb F$ such that:
 * $\ds \sum_{j \mathop = 1}^\infty \alpha_j e_j = 0$
 * we have $\alpha_j = 0$ for each $j \in \N$

where convergence is understood in $\struct {X, \norm \cdot}$.