34: Real Inner Product

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A definition of real 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: Definition

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Starting Context

• The reader knows a definition of %field name% vectors space.

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Target Context

• The reader will have a definition of real 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: Definition

For any real vectors space, $V$, any function, $⟨\bullet ,\bullet ⟩:V\times V\to \mathbb{R}$, such that for any vectors, $v_1,v_2,v_3\in V$ and any real numbers, $r_1,r_2\in \mathbb{R}$, 1) $⟨v_1,v_1⟩\ge 0$ with the equality holding if and only if $v_1=0$ 2) $⟨v_1,v_2⟩=⟨v_2,v_1⟩$ 3) $⟨r_1{v}_1+r_2{v}_2,v_3⟩=r_1⟨v_1,v_3⟩+r_2⟨v_2,v_3⟩$

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References

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