36: Inner Product on Real or Complex Vectors Space Induces Norm

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description/proof of that inner product on real or complex vectors space induces norm

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Topics

About: normed 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 inner product on real or complex vectors space.
• The reader knows a definition of norm on real or complex vectors space.
• The reader admits the Cauchy-Schwarz inequality for any real or complex inner-producted vectors space.

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Target Context

• The reader will have a description and a proof of the proposition that any inner product on any real or complex vectors space induces a norm.

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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
$V$: $\in \{\text{ the vectors spaces over }F\}$ with any inner product, $⟨\bullet ,\bullet ⟩$
$\Vert \bullet \Vert$: $:V\to \mathbb{R},v\mapsto \sqrt{⟨v,v⟩}$
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Statements:
$\Vert \bullet \Vert \in \{\text{ the norms on }V\}$.
//

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2: Natural Language Description

For any field, $F\in \{\mathbb{R},\mathbb{C}\}$, and any vectors space, $V$, over $F$, with any inner product, $⟨\bullet ,\bullet ⟩$, $\Vert \bullet \Vert :V\to \mathbb{R},v\mapsto \sqrt{⟨v,v⟩}$ is a norm.

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3: Proof

For any $v_1,v_2\in V$ and any $r\in F$, 1) $0\le \Vert v_1\Vert =\sqrt{⟨v_1,v_1⟩}$ with the equality holding if and only if $v_1=0$; 2) $\Vert r{v}_1\Vert =\sqrt{⟨r{v}_1,r{v}_1⟩}=\sqrt{r⟨v_1,r{v}_1⟩}=\sqrt{r\overline{⟨r{v}_1,v_1⟩}}=\sqrt{r\overline{r⟨v_1,v_1⟩}}=|r|\sqrt{⟨v_1,v_1⟩}=|r|\Vert v_1\Vert$; 3) $\Vert v_1+v_2\Vert =\sqrt{⟨v_1+v_2,v_1+v_2⟩}=\sqrt{⟨v_1,v_1⟩+⟨v_1,v_2⟩+⟨v_2,v_1⟩+⟨v_2,v_2⟩}=\sqrt{⟨v_1,v_1⟩+⟨v_1,v_2⟩+\overline{⟨v_2,v_1⟩}+⟨v_2,v_2⟩}\le \sqrt{⟨v_1,v_1⟩+2|⟨v_1,v_2⟩|+⟨v_2,v_2⟩}\le \sqrt{⟨v_1,v_1⟩+2\sqrt{⟨v_1,v_1⟩}\sqrt{⟨v_2,v_2⟩}+⟨v_2,v_2⟩}$, by the Cauchy-Schwarz inequality for any real or complex inner-producted vectors space, $=\sqrt{(\sqrt{⟨v_1,v_1⟩}+\sqrt{⟨v_2,v_2⟩}{)}^2}=\sqrt{⟨v_1,v_1⟩}+\sqrt{⟨v_2,v_2⟩}=\Vert v_1\Vert +\Vert \Vert v_2\Vert$.

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References

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