755: For Nonzero Linear Map Between Normed Vectors Spaces, Image Norm Divided by Argument Norm Does Not Converge to 0 When Argument Norm Nears 0

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description/proof of that for nonzero linear map between normed vectors spaces, image norm divided by argument norm does not converge to 0 when argument norm nears 0

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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.

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Target Context

• The reader will have a description and a proof of 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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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:
$V_1$: $\in \{\text{ the normed vectors spaces }\}$
$V_2$: $\in \{\text{ the normed vectors spaces }\}$
$f$: $:V_1\to V_2$, $\in \{\text{ the linear maps }\}$
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Statements:
$\mathrm{\exists }{v}_0\in V_1(f(v_0)\ne 0)$
$⟹$
$\Vert f(v)\Vert /\Vert v\Vert$ 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, $V_1,V_2$, and any nonzero linear map, $f:V_1\to V_2$, $\Vert f(v)\Vert /\Vert v\Vert$ does not converge to 0 when $\Vert v\in V_1\Vert$ nears 0.

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3: Proof

Whole Strategy: Step 1: choose any vector, $v_0\in V_1$, such that $f(v_0)\ne 0$, take $r{v}_0$ and make $r$ approach 0, and see that it does not converge to 0.

Step 1:

As $f$ is nonzero, there is a vector, $v_0\in V_1$, such that $f(v_0)\ne 0$.

Let us take $v=r{v}_0$. $\Vert f(v)\Vert /\Vert v\Vert =\Vert f(r{v}_0)\Vert /\Vert r{v}_0\Vert =\Vert rf(v_0)\Vert /\Vert r{v}_0\Vert =|r|\Vert f(v_0)\Vert /(|r|\Vert v_0\Vert )=\Vert f(v_0)\Vert /\Vert v_0\Vert \ne 0$, which does not near $0$ when $\Vert r{v}_0\Vert$ nears $0$. Of course, it may be $0$ for a vector, $v_1\in V_1$, but it is not said to converge to $0$ unless it nears $0$ in any way in which $\Vert v\Vert$ nears $0$.

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References

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