587: Limit of Normed Vectors Spaces Map at Point

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definition of limit of normed vectors spaces map at point

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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
• 3: Note

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Starting Context

• The reader knows a definition of normed vectors space.
• The reader knows a definition of map.

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Target Context

• The reader will have a definition of limit of normed vectors spaces map at point.

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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
$V_1$: $\in \{\text{ the normed vectors spaces over }{F}_1\}$
$V_2$: $\in \{\text{ the normed vectors spaces over }{F}_2\}$
$f$: $:V_1\to V_2$
$p$: $\in V_1$
$\ast li{m}_{p}f$: $\in V_2$
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Conditions:
$\mathrm{\forall }ϵ\in \mathbb{R}\text{ such that }0<ϵ(\mathrm{\exists }\delta \in \mathbb{R}\text{ such that }0<\delta (\mathrm{\forall }{p}^{\prime }\in V_1\text{ such that }\Vert p^{\prime }-p\Vert <\delta (\Vert f(p^{\prime })-li{m}_{p}f\Vert <ϵ)))$.
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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 $F_1$, $V_1$, any normed vectors space over $F_2$, $V_2$, any map, $f:V_1\to V_2$, and any point, $p\in V_1$, $li{m}_{p}f\in V_2$ such that for each $ϵ\in \mathbb{R}$ such that $0<ϵ$, there is a $\delta \in \mathbb{R}$ such that $0<\delta$ and for each $p^{\prime }\in V_1$ such that $\Vert p^{\prime }-p\Vert <\delta$, $\Vert f(p^{\prime })-li{m}_{p}f\Vert <ϵ$

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3: Note

$li{m}_{p}f$ does not necessarily exist.

There cannot be multiple limits, because supposing that a $li{m}_{p}f$ exist, for each $l\in V_2$ such that $l\ne li{m}_{p}f$, for each $p^{\prime }\in V_1$, $\Vert l-li{m}_{p}f\Vert =\Vert l-f(p^{\prime })+f(p^{\prime })-li{m}_{p}f\Vert \le \Vert l-f(p^{\prime })\Vert +\Vert f(p^{\prime })-li{m}_{p}f\Vert$, which implies that $\Vert l-li{m}_{p}f\Vert -\Vert f(p^{\prime })-li{m}_{p}f\Vert \le \Vert l-f(p^{\prime })\Vert$, but $0<\Vert l-li{m}_{p}f\Vert$, and for a $\delta$, for each $p^{\prime }\in V_1$ such that $\Vert p^{\prime }-p\Vert <\delta$, $\Vert f(p^{\prime })-li{m}_{p}f\Vert <1/2\Vert l-li{m}_{p}f\Vert$, which implies that $1/2\Vert l-li{m}_{p}f\Vert <\Vert l-f(p^{\prime })\Vert$, and for $ϵ=1/2\Vert l-li{m}_{p}f\Vert$, for whatever ${\delta }^{\prime }$, there is a $p^{\prime }$ such that $\Vert p^{\prime }-p\Vert <min(\delta ,{\delta }^{\prime })$ and $ϵ<\Vert l-f(p^{\prime })\Vert$, which implies that $l$ is not any limit.

$li{m}_{p}f$ does not necessarily equal $f(p)$.

If $li{m}_{p}f$ exists and equals $f(p)$, $f$ is continuous at $p$.

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References

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