Difference between revisions of "Inner product"

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m (Todo: add trivial properties, a bit more rejigging)
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==Definition==
 
==Definition==
−
Given a {{Vector space}} (where {{M|F}} is either {{M|\mathbb{R} }} or {{M|\mathbb{C} }}), an ''inner product''<ref>http://en.wikipedia.org/w/index.php?title=Inner_product_space&oldid=651022885</ref><ref>Functional Analysis I - Lecture Notes - Richard Sharp - Sep 2014</ref> is a map:
+
Given a {{Vector space}} (where {{M|F}} is either {{M|\mathbb{R} }} or {{M|\mathbb{C} }}), an ''inner product''<ref>http://en.wikipedia.org/w/index.php?title=Inner_product_space&oldid=651022885</ref><ref>Functional Analysis I - Lecture Notes - Richard Sharp - Sep 2014</ref><ref name="FA">Functional Analysis - George Bachman and Lawrence Narici</ref> is a map:
 
* <math>\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{R}</math> (or sometimes <math>\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{C}</math>)
 
* <math>\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{R}</math> (or sometimes <math>\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{C}</math>)
 
Such that:
 
Such that:
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** Or just <math>\langle x,y\rangle = \langle y,x\rangle</math> if the inner product is into {{M|\mathbb{R} }}
 
** Or just <math>\langle x,y\rangle = \langle y,x\rangle</math> if the inner product is into {{M|\mathbb{R} }}
 
* <math>\langle\lambda x+\mu y,z\rangle = \lambda\langle y,z\rangle + \mu\langle x,z\rangle</math> ( [[Linear map|linearity in first argument]] )
 
* <math>\langle\lambda x+\mu y,z\rangle = \lambda\langle y,z\rangle + \mu\langle x,z\rangle</math> ( [[Linear map|linearity in first argument]] )
−
*: This may be better stated as:
+
*: This may be alternatively stated as:
−
*:* <math>\langle\lambda x,y\rangle=\lambda\langle x,y\rangle</math> and
+
*:* <math>\langle\lambda x,y\rangle=\lambda\langle x,y\rangle</math> and <math>\langle x+y,z\rangle = \langle x,z\rangle + \langle y,z\rangle</math>
−
*:* <math>\langle x+y,z\rangle = \langle x,z\rangle + \langle y,z\rangle</math>
+
* <math>\langle x,x\rangle \ge 0</math> but specifically:
−
* <math>\langle x,x\rangle \ge 0</math> with <math>\langle x,x\rangle=0\iff x=0</math>
+
** <math>\langle x,x\rangle=0\iff x=0</math>
  
 
==Properties==
 
==Properties==
−
Notice that <math>\langle\cdot,\cdot\rangle</math> is also linear in its second argument as:
+
Notice that <math>\langle\cdot,\cdot\rangle</math> is also linear (ish) in its second argument as:
 
*<math>\langle x,\lambda y+\mu z\rangle = \overline{\langle \lambda y+\mu z, x\rangle}</math><math>=\overline{\lambda\langle y,x\rangle + \mu\langle z,x\rangle}</math><math>=\bar{\lambda}\overline{\langle y,x\rangle}+\bar{\mu}\overline{\langle z,x\rangle}</math><math>=\bar{\lambda}\langle x,y\rangle+\bar{\mu}\langle x,z\rangle</math>
 
*<math>\langle x,\lambda y+\mu z\rangle = \overline{\langle \lambda y+\mu z, x\rangle}</math><math>=\overline{\lambda\langle y,x\rangle + \mu\langle z,x\rangle}</math><math>=\bar{\lambda}\overline{\langle y,x\rangle}+\bar{\mu}\overline{\langle z,x\rangle}</math><math>=\bar{\lambda}\langle x,y\rangle+\bar{\mu}\langle x,z\rangle</math>
  
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* <math>\langle x,\lambda y\rangle = \bar{\lambda}\langle x,y\rangle</math> and
 
* <math>\langle x,\lambda y\rangle = \bar{\lambda}\langle x,y\rangle</math> and
 
* <math>\langle x,y+z\rangle = \langle x,y\rangle + \langle x,z\rangle</math>
 
* <math>\langle x,y+z\rangle = \langle x,y\rangle + \langle x,z\rangle</math>
 +
This leads to the most general form:
 +
* {{M|1=\langle au+bv,cx+dy\rangle=a\langle u,cx+dy\rangle+b\langle v,cx+dy\rangle}}{{M|1= =a\overline{\langle cx+dy,u\rangle}+b\overline{\langle cx+dy,v\rangle} }}{{M|1= =a(\overline{c\langle x,u\rangle} + \overline{d\langle y,u\rangle})+b(\overline{c\langle x,v\rangle}+\overline{d\langle y,v\rangle})}}{{M|1= =a\overline{c}\langle u,x\rangle+a\overline{d}\langle u,y\rangle+b\overline{c}\langle v,x\rangle+b\overline{d}\langle v,y\rangle}}
 +
{{Begin Theorem}}
 +
Proof of claim: {{M|1=\langle x,\alpha y+\beta z\rangle=\overline{\alpha}\langle x, y\rangle +\overline{\beta}\langle x,z\rangle}}
 +
{{Begin Proof}}
 +
 +
{{End Proof}}{{End Theorem}}
  
 
==Examples==
 
==Examples==

Revision as of 11:38, 10 July 2015

Definition

Given a vector space, (V,F) (where F is either R or C), an inner product[1][2][3] is a map:

  • ⟨⋅,⋅⟩:V×V→R
     (or sometimes ⟨⋅,⋅⟩:V×V→C
    )

Such that:

  • ⟨x,y⟩=¯⟨y,x⟩
     (where the bar denotes Complex conjugate)
    • Or just ⟨x,y⟩=⟨y,x⟩
       if the inner product is into R
  • ⟨λx+μy,z⟩=λ⟨y,z⟩+μ⟨x,z⟩
     ( linearity in first argument )
    This may be alternatively stated as:
    • ⟨λx,y⟩=λ⟨x,y⟩
       and ⟨x+y,z⟩=⟨x,z⟩+⟨y,z⟩
  • ⟨x,x⟩≥0
     but specifically:
    • ⟨x,x⟩=0⟺x=0

Properties

Notice that ⟨⋅,⋅⟩

is also linear (ish) in its second argument as:

  • ⟨x,λy+μz⟩=¯⟨λy+μz,x⟩
    =¯λ⟨y,x⟩+μ⟨z,x⟩
    =ˉλ¯⟨y,x⟩+ˉμ¯⟨z,x⟩
    =ˉλ⟨x,y⟩+ˉμ⟨x,z⟩

From this we may conclude the following:

  • ⟨x,λy⟩=ˉλ⟨x,y⟩
     and
  • ⟨x,y+z⟩=⟨x,y⟩+⟨x,z⟩

This leads to the most general form:

  • ⟨au+bv,cx+dy⟩=a⟨u,cx+dy⟩+b⟨v,cx+dy⟩=a¯⟨cx+dy,u⟩+b¯⟨cx+dy,v⟩=a(¯c⟨x,u⟩+¯d⟨y,u⟩)+b(¯c⟨x,v⟩+¯d⟨y,v⟩)=a¯c⟨u,x⟩+a¯d⟨u,y⟩+b¯c⟨v,x⟩+b¯d⟨v,y⟩
[Expand]

Proof of claim: ⟨x,αy+βz⟩=¯α⟨x,y⟩+¯β⟨x,z⟩


Examples

See also

References

  1. Jump up ↑ http://en.wikipedia.org/w/index.php?title=Inner_product_space&oldid=651022885
  2. Jump up ↑ Functional Analysis I - Lecture Notes - Richard Sharp - Sep 2014
  3. Jump up ↑ Functional Analysis - George Bachman and Lawrence Narici
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