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

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The result of the inner product must be greater than zero if the vector is not zero.

\mathrm{v \neq 0 \Rightarrow \lang{v,v}\rang > 0}

This condition for a inner product can be related to the length of a vector(from2) and is very useful when trying to understand abstract vector spaces.

Example 1. Let

E = \mathbb{R}_2[x]

(from4) and the mapping

E \times E \mapsto \mathbb{R}

is defined by:

\lang{P, G}\rang = \int^{+\infin}_0P(x)Q(x)e^{-x}dx

How can we show

\lang{P, G}\rang

is an inner product over

E

? If you are just understanding the inner product as ‘dot product’ which is called ‘standard inner product’, it might be difficult to see how we can interpret as an inner product. One of the conditions(from1) which I just mentioned in this document is particularly challenging to apply.

\mathrm{P \neq 0 \Rightarrow \lang{P,P}\rang > 0}

Check the ‘custom’ inner product

\lang{P, G}\rang

permits the condition.

\lang{P, P}\rang = \int^{+\infin}_0P(x)P(x)e^{-x}dx

= \int^{+\infin}_0(P(x))^2e^{-x}dx > 0

e^{-x}

is greater than

0

over

[0, +\infin)

and

(P(x))^2

is greater than

0

P(x) \neq 0

over

[0, +\infin)

\lang{P, G}\rang

could be an inner product.

Example 2. Let

E

be the vector space of the continuous functions on

[0, 1]

, together with the inner product.

\lang{f, g}\rang = \int^1_0f(t)g(t)dt

Consider the linear subspace

F \subset E

, how can we show this?

F^\perp =\{0_E\}

Let’s rewrite the

F^\perp

(from3).

F^\perp =\{g \in E | \lang{f,g}\rang=0, {\forall}f \in F\}

Then we might see:

If f(x)=0f(x) = 0f(x)=0 over [0,1][0, 1][0,1], ggg could be anything (include 0E0_E0E​)

If f(x)≠0f(x) ≠ 0f(x)=0 over [0,1][0, 1][0,1], ggg must be 0E0_E0E​

To permit both conditions 1 and 2,

F^\perp

should be:

F^\perp =\{0_E\}

parse me : 언젠가 이 글에 쓰이면 좋을 것 같은 재료들.

None

from : 과거의 어떤 생각이 이 생각을 만들었는가?

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

b3.4_5.1_1.1____1. title:
The length of a vector can be expressed as the dot product of itself.

supplementary : 어떤 새로운 생각이 이 문서에 작성된 생각을 뒷받침하는가?

None

opposite : 어떤 새로운 생각이 이 문서에 작성된 생각과 대조되는가?

None

to : 이 문서에 작성된 생각이 어떤 생각으로 발전되고 이어지는가?

None

참고 : 레퍼런스

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