definition of Hilbert space
Topics
About: metric space
The table of contents of this article
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.
Target Context
- The reader will have a definition of Hilbert space.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure
\( V\): \(\in \{\text{ the } F \text{ vectors spaces }\}\)
\( \langle \bullet, \bullet \rangle\): \(: V \times V \to F\), \(\in \{\text{ the inner products }\}\)
\( \Vert \bullet \Vert\): \(: V \to \mathbb{R}\), \(= \text{ the norm induced by } \langle \bullet, \bullet \rangle\)
\( dist\): \(: V \times V \to \mathbb{R}\), \(= \text{ the metric induced by } \Vert \bullet \Vert\)
\(*(V, dist)\): \(= \text{ the metric space }\)
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Conditions:
\((V, dist) \in \{\text{ the complete metric spaces }\}\)
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