39: Norm on Real or Complex Vectors Space

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definition of norm on real or complex vectors space

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Topics

About: vectors space

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The table of contents of this article

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

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Starting Context

• The reader knows a definition of %field name% vectors space.

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Target Context

• The reader will have a definition of norm on real or complex 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
$V$: $\in \{\text{ the vectors spaces over }F\}$
$\ast \Vert \bullet \Vert$: $:V\to \mathbb{R}$ //

Conditions:
$\mathrm{\forall }{v}_1,v_2\in V$, $\mathrm{\forall }r\in F$,
(
1) ($0\le \Vert v_1\Vert$) $\wedge$ ($(0=\Vert v_1\Vert )⟺(v_1=0)$)
$\wedge$
2) $\Vert r{v}_1\Vert =|r|\Vert v_1\Vert$
$\wedge$
3) $\Vert v_1+v_2\Vert \le \Vert v_1\Vert +\Vert v_2\Vert$
)
//

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2: Natural Language Description

For any field, $F\in \{\mathbb{R},\mathbb{C}\}$, and any vectors space, $V$, over $F$, any map, $\Vert \bullet \Vert :V\to \mathbb{R}$, such that for each vectors, $v_1,v_2\in V$, and each scalar, $r\in F$, 1) $\Vert v_1\Vert \ge 0$ with the equality holding if and only if $v_1=0$; 2) $\Vert r{v}_1\Vert =|r|\Vert v_1\Vert$; 3) $\Vert v_1+v_2\Vert \le \Vert v_1\Vert +\Vert v_2\Vert$

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References

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