756: For Map Between Normed Vectors Spaces s.t. Image Norm Divided by Argument Norm Converges to 0 When Argument Norm Nears 0, Image Norm of Map Plus Nonzero Linear Map Divided by Argument Norm Does Not Do So

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description/proof of that for map between normed vectors spaces s.t. image norm divided by argument norm converges to 0 when argument norm nears 0, image norm of map plus nonzero linear map divided by argument norm does not do so

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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: Proof

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Starting Context

• The reader knows a definition of normed vectors space.
• The reader knows a definition of linear map.
• The reader admits the proposition that for any nonzero linear map between any normed vectors spaces, the image norm divided by the argument norm does not converge to 0 when the argument norm nears 0.

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Target Context

• The reader will have a description and a proof of the proposition that for any map between any normed vectors spaces such that the image norm divided by the argument norm converges to 0 when the argument norm nears 0, the image norm of the map plus any nonzero linear map divided by the argument norm does not converge to 0 when the argument norm nears 0.

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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:
$V_1$: $\in \{\text{ the normed vectors spaces }\}$
$V_2$: $\in \{\text{ the normed vectors spaces }\}$
$f_1$: $:V_1\to V_2$
$f_2$: $:V_1\to V_2$, $\in \{\text{ the linear maps }\}$
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Statements:
(
$li{m}_{\Vert v\Vert \to 0}\Vert f_1(v)\Vert /\Vert v\Vert =0$
$\wedge$
$\mathrm{\exists }{v}_0\in V_1(f_2(v_0)\ne 0)$
)
$⟹$
$\Vert (f_1+f_2)(v)\Vert /\Vert v\Vert$ does not converge to 0 when $\Vert v\in V_1\Vert$ nears 0
//

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2: Natural Language Description

For any normed vectors spaces, $V_1,V_2$, any map, $f_1:V_1\to V_2$ such that $li{m}_{\Vert v\Vert \to 0}\Vert f_1(v)\Vert /\Vert v\Vert =0$, and any nonzero linear map, $f_2:V_1\to V_2$, $\Vert (f_1+f_2)(v)\Vert /\Vert v\Vert$ does not converge to 0 when $\Vert v\in V_1\Vert$ nears 0.

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3: Proof

Whole Strategy: Step 1: suppose that $li{m}_{\Vert v\Vert \to 0}\Vert f_1(v)\Vert /\Vert v\Vert =0$ and $\mathrm{\exists }{v}_0\in V_1(f_2(v_0)\ne 0)$ and suppose that $li{m}_{\Vert v\Vert \to 0}\Vert (f_1+f_2)(v)\Vert /\Vert v\Vert =0$, and find a contradiction.

Let us define $f_3:V_1\to V_2:=f_1+f_2$.

Let us suppose that $li{m}_{\Vert v\Vert \to 0}\Vert f_3(v)\Vert /\Vert v\Vert =0$.

$f_3-f_1=f_2$. $\Vert (f_3-f_1)(v)\Vert /\Vert v\Vert =\Vert f_3(v)-f_1(v)\Vert /\Vert v\Vert \le (\Vert f_3(v)\Vert +\Vert f_1(v)\Vert )/\Vert v\Vert =\Vert f_3(v)\Vert /\Vert v\Vert +\Vert f_1(v)\Vert /\Vert v\Vert$, which would converge to $0$ when $\Vert v\Vert$ neared $0$, because the both terms would do so, so, $\Vert (f_3-f_1)(v)\Vert /\Vert v\Vert$ would converge to $0$. That is a contradiction, because $\Vert f_2(v)\Vert /\Vert v\Vert$ does not converge to $0$ when $\Vert v\Vert$ nears $0$, by the proposition that for any nonzero linear map between any normed vectors spaces, the image norm divided by the argument norm does not converge to 0 when the argument norm nears 0.

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References

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