1073: Bounded Map Between Normed Vectors Spaces

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definition of bounded map between normed vectors spaces

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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: 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 bounded map between normed vectors spaces.

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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
$V_1$: $\in \{\text{ the }{F}_1\text{ vectors spaces }\}$ with any norm, $\Vert \bullet {\Vert }_1$
$V_2$: $\in \{\text{ the }{F}_2\text{ vectors spaces }\}$ with any norm, $\Vert \bullet {\Vert }_2$
$\ast f$: $:V_1\to V_2$
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Conditions:
$\mathrm{\exists }c\in \mathbb{R}(\mathrm{\forall }{v}_1\in V_1(\Vert f(v_1){\Vert }_2\le c\Vert v_1{\Vert }_1))$
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2: Note

$F_1\ne F_2$ is allowed.

$f$ does not need to be linear.

$f$'s being linear does not guarantee that $f$ is bounded: for each unit vector, $u\in V_1$, $\Vert f(ru){\Vert }_2=\Vert rf(u){\Vert }_2=|r|\Vert f(u){\Vert }_2=|r|\Vert u{\Vert }_1/\Vert u{\Vert }_1\Vert f(u){\Vert }_2=\Vert ru{\Vert }_1\Vert f(u){\Vert }_2\le c_u\Vert ru{\Vert }_1$ where $c_u:=\Vert f(u){\Vert }_2$, but $c_u$ depends on $u$ and there is no guarantee that there is a common $c$ that does not depend on $u$.

Refer to the proposition that any linear map from any finite-dimensional vectors space with the norm induced by any inner product into any normed vectors space is bounded.

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References

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