1076: Linear Map Between Vectors Metric Spaces Induced by Norms Is Continuous iff It Is Bounded

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description/proof of that linear map between vectors metric spaces induced by norms is continuous iff it is bounded

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Topics

About: vectors space
About: metric space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Proof

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Starting Context

• The reader knows a definition of metric induced by norm on real or complex vectors space.
• The reader knows a definition of linear map.
• The reader knows a definition of continuous map.
• The reader knows a definition of bounded map between normed vectors spaces.

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Target Context

• The reader will have a description and a proof of the proposition that any linear map between any vectors metric spaces induced by any norms is continuous if and only if it is bounded.

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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:
$F$: $\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure
$V_1$: $\in \{\text{ the }F\text{ vectors spaces }\}$ with the metric induced by any norm, $\Vert \bullet {\Vert }_1$
$V_2$: $\in \{\text{ the }F\text{ vectors spaces }\}$ with the metric induced by any norm, $\Vert \bullet {\Vert }_2$
$\ast f$: $:V_1\to V_2$, $\in \{\text{ the linear maps }\}$
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Statements:
$f\in \{\text{ the continuous maps }\}$
$⟺$
$f\in \{\text{ the bounded maps }\}$
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2: Proof

Whole Strategy: Step 1: suppose that $f$ is bounded; Step 2: for each $v_1\in V_1$, take any neighborhood of $f(v_1)$, $N_{f(v_1)}\subseteq V_2$, take any open ball around $f(v_1)$, $B_{f(v_1),ϵ}\subseteq V_2$, such that $B_{f(v_1),ϵ}\subseteq N_{f(v_1)}$, take the open ball around $v_1$, $B_{v_1,ϵ/c}\subseteq V_1$, where $c$ is any constant for being bounded, and see that $f(B_{v_1,ϵ/c})\subseteq B_{f(v_1),ϵ}$; Step 3: suppose that $f$ is continuous; Step 4: take the open ball around $0\in V_2$, $B_{0,1}\subseteq V_2$, take an open ball around $0\in V_1$, $B_{0,\delta }\subseteq V_1$, such that $f(B_{0,\delta })\subseteq B_{0,1}$; Step 5: for each $v_1\in V_1$ such that $v_1\ne 0$, evaluate $\Vert f(v_1)\Vert$, using $v_1=(\delta /2)/(\delta /2)|v_1|/|v_1|v_1$.

Step 1:

Let us suppose that $f$ is bounded.

Step 2:

Let $v_1\in V_1$ be any.

Let $N_{f(v_1)}\subseteq V_2$ be any neighborhood of $f(v_1)$.

There is an open ball around $f(v_1)$, $B_{f(v_1),ϵ}\subseteq V_2$, such that $B_{f(v_1),ϵ}\subseteq N_{f(v_1)}$.

As $f$ is bounded, there is a constant, $c\in \mathbb{R}$, such that for each $v_1^{\prime }\in V_1$, $\Vert f(v_1^{\prime })\Vert \le c\Vert v_1^{\prime }\Vert$.

Let us take the open ball around $v_1$, $B_{v_1,ϵ/c}\subseteq V_1$.

For each $v_1^{\prime }\in B_{v_1,ϵ/c}$, $f(v_1^{\prime })=f(v_1^{\prime }-v_1+v_1)=f(v_1^{\prime }-v_1)+f(v_1)$, so, $f(v_1^{\prime })-f(v_1)=f(v_1^{\prime }-v_1)$.

$\Vert f(v_1^{\prime })-f(v_1)\Vert =\Vert f(v_1^{\prime }-v_1)\Vert \le c\Vert v_1^{\prime }-v_1\Vert <cϵ/c=ϵ$.

That means that $f(B_{v_1,ϵ/c})\subseteq B_{f(v_1),ϵ}\subseteq N_{f(v_1)}$.

That means that $f$ is continuous at $v_1$.

As $v_1\in V_1$ is arbitrary, $f$ is continuous.

Step 3:

Let us suppose that $f$ is continuous.

Step 4:

As $f$ is linear, $f(0)=0$.

Let us take the open ball around $f(0)=0\in V_2$, $B_{0,1}\subseteq V_2$.

As $f$ is continuous, there is an open ball around $0\in V_1$, $B_{0,\delta }\subseteq V_1$, such that $f(B_{0,\delta })\subseteq B_{0,1}$.

Step 5:

For each $v_1\in V_1$ such that $v_1\ne 0$, $\Vert f(v_1)\Vert =\Vert f((\delta /2)/(\delta /2)\Vert v_1\Vert /\Vert v_1\Vert v_1)\Vert =\Vert \Vert v_1\Vert /(\delta /2)f((\delta /2)/\Vert v_1\Vert v_1)\Vert =\Vert v_1\Vert /(\delta /2)\Vert f((\delta /2)/\Vert v_1\Vert v_1)\Vert$, but $(\delta /2)/\Vert v_1\Vert v_1\in B_{0,\delta }$, so, $f((\delta /2)/\Vert v_1\Vert v_1)\in B_{0,1}$ and $\Vert f(v_1)\Vert <\Vert v_1\Vert /(\delta /2)1\le 1/(\delta /2)\Vert v_1\Vert$.

So, $c:=1/(\delta /2)$, which does not depend on $v_1$, makes $\Vert f(v_1)\Vert \le c\Vert v_1\Vert$.

So, $f$ is bounded.

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References

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