# Definition:Schauder Basis

## 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 PaweÅ‚ Schauder.