Definition 13.9.1 (Schauder Basis).label Let $E$ be a separable Banach space over $K \in \RC$ and $\seq{x_n}\subset E$, then $\seq{x_n}$ is a Schauder basis of $E$ if for each $x \in E$, there exists a unique $\seq{\lambda_n(x)}\in K^{\natp}$ such that
\[x = \limv{N}\sum_{n = 1}^{N} \lambda_{n}(x) x_{n}\]
The sequence of mappings $\seq{\lambda_n}\subset K^{E}$ is the coefficient forms of $\seq{x_n}$, and the Schauder basis $\seq{x_n}$ is normalised if $\norm{x_n}_{E} = 1$ for all $n \in \natp$.
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