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b3.4_5.1_1.1____2.1. [entry] title: Conditions for inner products that generate a real number

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b3.4_5.1_1.1____2. title: A user can define a custom inner product. If a new inner product is defined, the definition of the length of a vector will be changed. But there are certain conditions must be met for the inner product to be considered valid.

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b3.4_5.1_1.1____2.1.1. title: The result of the inner product must be greater than zero if the vector is not zero. This can be related to the length of a vector and is very useful when trying to understand abstract vector spaces.

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User can define a custom inner product.

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b3.4_5.1_1.1____2. title:
A user can define a custom inner product. If a new inner product is defined, the definition of the length of a vector will be changed. But there are certain conditions must be met for the inner product to be considered valid.

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Which meet conditions below. We are used to using the standard inner product, which was defined by physicists, but it can also be considered a custom inner product.

⟨u,v⟩↦R{\lang}u,v{\rang}\mapsto\mathbb{R}⟨u,v⟩↦R

⟨u,v⟩=⟨v,u⟩{\lang}u,v{\rang}= {\lang}v,u{\rang}⟨u,v⟩=⟨v,u⟩

⟨u+v,w⟩=⟨u,w⟩+⟨v,w⟩{\lang}u+v,w{\rang}={\lang}u,w{\rang}+{\lang}v,w{\rang}⟨u+v,w⟩=⟨u,w⟩+⟨v,w⟩

k⟨u,v⟩=⟨ku,v⟩k{\lang}u,v{\rang} = {\lang}ku,v{\rang}k⟨u,v⟩=⟨ku,v⟩

v≠0⇒⟨v,v⟩>0\mathrm{v \neq 0 \Rightarrow \lang{v,v}\rang > 0}v=0⇒⟨v,v⟩>0

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b3.4_5.1_1.1____2.1.1. title:
The result of the inner product must be greater than zero if the vector is not zero. This can be related to the length of a vector and is very useful when trying to understand abstract vector spaces.