1058: Parallelogram Law on Vectors Space Normed Induced by Inner Product

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description/proof of parallelogram law on vectors space normed induced by inner product

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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: Proof

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Starting Context

• The reader knows a definition of norm induced by inner product on real or complex vectors space.

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Target Context

• The reader will have a description and a proof of the parallelogram law on any vectors space normed induced by any inner product.

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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
$V$: $\in \{\text{ the }F\text{ vectors spaces }\}$, with $\Vert \bullet \Vert :V\to \mathbb{R}$ induced by $⟨\bullet ,\bullet ⟩:V\times V\to F$
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Statements:
$\mathrm{\forall }{v}_1,v_2\in V(\Vert v_1+v_2{\Vert }^2+\Vert v_1-v_2{\Vert }^2=2(\Vert v_1{\Vert }^2+\Vert v_2{\Vert }^2))$
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2: Proof

Whole Strategy: Step 1: expand $\Vert v_1+v_2{\Vert }^2=⟨v_1+v_2,v_1+v_2⟩$ and $\Vert v_1-v_2{\Vert }^2=⟨v_1-v_2,v_1-v_2⟩$, and see that the sum is $2(⟨v_1,v_1⟩+⟨v_1,v_1⟩)=2(\Vert v_1{\Vert }^2+\Vert v_2{\Vert }^2)$.

Step 1:

$\Vert v_1+v_2{\Vert }^2=⟨v_1+v_2,v_1+v_2⟩=⟨v_1,v_1⟩+⟨v_2,v_2⟩+⟨v_1,v_2⟩+⟨v_2,v_1⟩$.

$\Vert v_1-v_2{\Vert }^2=⟨v_1-v_2,v_1-v_2⟩=⟨v_1,v_1⟩+⟨v_2,v_2⟩-⟨v_1,v_2⟩-⟨v_2,v_1⟩$.

So, $\Vert v_1+v_2{\Vert }^2+\Vert v_1-v_2{\Vert }^2=2⟨v_1,v_1⟩+2⟨v_2,v_2⟩=2(⟨v_1,v_1⟩+⟨v_2,v_2⟩)=2(\Vert v_1{\Vert }^2+\Vert v_2{\Vert }^2)$.

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References

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