1094: Schauder Basis for Normed Vectors Space

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definition of Schauder basis for normed vectors space

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Topics

About: metric space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of normed vectors space.

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Target Context

• The reader will have a definition of Schauder basis for normed vectors space.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$F$: $\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure
$V$: $\in \{\text{ the normed }F\text{ vectors spaces }\}$ with the metric induced by the norm
$\ast B$: $:\mathbb{N}\setminus \{0\}\to V$
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Conditions:
$\mathrm{\forall }v\in V(!\mathrm{\exists }s:\mathbb{N}\setminus \{0\}\to F(v=\sum _{j\in \mathbb{N}\setminus \{0\}}s(j)B(j)))$
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2: Note

Inevitably, any finite set, $\{B(j_1),...,B(j_n)\}$, is linearly independent, because for $c^{1}B(j_1)+...+c^{n}B(j_n)=0$, there is the $s:j\mapsto 0$ such that $0=\sum _{j\in \mathbb{N}\setminus \{0\}}s(j)B(j)))$, but as $0\in V$, such an $s$ is unique, and so, $c^1=...=c^n=0$, because otherwise, it would constitute another $s^{\prime }$ such that $0=\sum _{j\in \mathbb{N}\setminus \{0\}}{s}^{\prime }(j)B(j)$.

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References

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