22: Continuous, Normed Vectors Spaces Map

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definition of continuous, normed vectors spaces map

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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 normed vectors spaces map continuous at point.

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Target Context

• The reader will have a definition of continuous, normed vectors spaces map.

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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:
$F_1$: $\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure
$F_2$: $\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure
$V_1$: $\in \{\text{ the normed vectors spaces over }{F}_1\}$
$V_2$: $\in \{\text{ the normed vectors spaces over }{F}_2\}$
$\ast f$: $:V_1\to V_2$
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Conditions:
$\mathrm{\forall }p\in V_1(f\in \{\text{ the maps from }{V}_1\text{ to }{V}_2\text{ continuous at }p\})$
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2: Natural Language Description

For $\mathbb{R}$ or $\mathbb{C}$ with the canonical field structure, $F_1$, $\mathbb{R}$ or $\mathbb{C}$ with the canonical field structure, $F_2$, any normed vectors space over $F_1$, $V_1$, and any normed vectors space over $F_2$, $V_2$, any map, $f:V_1\to V_2$, such that for each $p\in V_1$, $f$ is continuous at $p$

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References

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