1047: Hilbert Space

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definition of Hilbert 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

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Starting Context

• The reader knows a definition of norm induced by inner product on real or complex vectors space.
• The reader knows a definition of metric induced by norm on real or complex vectors space.
• The reader knows a definition of complete metric space.

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Target Context

• The reader will have a definition of Hilbert 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
$V$: $\in \{\text{ the }F\text{ vectors spaces }\}$
$⟨\bullet ,\bullet ⟩$: $:V\times V\to F$, $\in \{\text{ the inner products }\}$
$\Vert \bullet \Vert$: $:V\to \mathbb{R}$, $=\text{ the norm induced by }⟨\bullet ,\bullet ⟩$
$dist$: $:V\times V\to \mathbb{R}$, $=\text{ the metric induced by }\Vert \bullet \Vert$
$\ast (V,dist)$: $=\text{ the metric space }$
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Conditions:
$(V,dist)\in \{\text{ the complete metric spaces }\}$
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References

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