Definition:Schauder Basis
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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 $X$.
Definition 1
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 $\struct {X, \norm \cdot}$.
Definition 2
We say that $\set {e_n : n \in \N}$ is a Schauder basis for $X$ if and only 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 of the infinite series is understood in $\struct {X, \norm \cdot}$.
Also see
- Results about Schauder bases can be found here.
Source of Name
This entry was named for Juliusz Paweł Schauder.
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.3$: The normed space $\map {CL} {X,Y}$. Operator norm and the normed space $\map {CL} {X, Y}$
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $9.1$: Schauder Bases in Normed Spaces