514: Euclidean Inner Product on Euclidean Vectors Space

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definition of Euclidean inner product on Euclidean vectors space

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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: Note

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Starting Context

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

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Target Context

• The reader will have a definition of Euclidean inner product on Euclidean vectors space.

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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:
${\mathbb{R}}^d$: $=\text{ the Euclidean vectors space }$
$\ast ⟨\bullet ,\bullet ⟩$: $=\in \{\text{ the inner products on }{\mathbb{R}}^d\}$, $:{\mathbb{R}}^d\times {\mathbb{R}}^d\to \mathbb{R},(v_1,v_2)\mapsto \sum _{j=1\sim d}{v}_1^j{v}_2^j$
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Conditions:
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2: Natural Language Description

For the Euclidean vectors space, ${\mathbb{R}}^d$, the inner product, $⟨\bullet ,\bullet ⟩$, $:{\mathbb{R}}^d\times {\mathbb{R}}^d\to \mathbb{R},(v_1,v_2)\mapsto \sum _{j=1\sim d}{v}_1^j{v}_2^j$

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3: Note

It is indeed an inner product: 1) $0\le ⟨v_1,v_1⟩=\sum _{j=1\sim d}{v}_1^j{v}_1^j$ with the equality holding if and only if $v_1=0$; 2) $⟨v_1,v_2⟩=\sum _{j=1\sim d}{v}_1^j{v}_2^j=\sum _{j=1\sim d}{v}_2^j{v}_1^j=⟨v_2,v_1⟩$; 3) $⟨r_1{v}_1+r_2{v}_2,v_3⟩=\sum _{j=1\sim d}(r_1{v}_1^j+r_2{v}_2^j)v_3^j=\sum _{j=1\sim d}{r}_1{v}_1^j{v}_3^j+\sum _{j=1\sim d}{r}_2{v}_2^j{v}_3^j=r_1⟨v_1,v_3⟩+r_2⟨v_2,v_3⟩$.

Although a Euclidean vectors space tends to be implicitly supposed to have the Euclidean inner product, it is not necessarily so.

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References

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