\begin{definition}[Definition:Schauder Basis/Definition 2]
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 $X$ {{iff}}:
:$(1): \quad$ 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 of the infinite series is understood in $\struct {X, \norm \cdot}$.
\end{definition}