Difference between revisions of "Inner product"

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m (Todo: add trivial properties, a bit more rejigging)
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This leads to the most general form:
 
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}}
 
* {{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}}
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{{Begin Theorem}}
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{{Begin Inline 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}}
 
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}}
 
{{Begin Proof}}

Revision as of 11:42, 10 July 2015

Definition

Given a vector space, [ilmath](V,F)[/ilmath] (where [ilmath]F[/ilmath] is either [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath]), an inner product[1][2][3] 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])

Such that:

  • [math]\langle x,y\rangle = \overline{\langle y, x\rangle}[/math] (where the bar denotes Complex conjugate)
    • Or just [math]\langle x,y\rangle = \langle y,x\rangle[/math] if the inner product is into [ilmath]\mathbb{R} [/ilmath]
  • [math]\langle\lambda x+\mu y,z\rangle = \lambda\langle y,z\rangle + \mu\langle x,z\rangle[/math] ( linearity in first argument )
    This may be alternatively stated as:
    • [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,x\rangle \ge 0[/math] but specifically:
    • [math]\langle x,x\rangle=0\iff x=0[/math]

Properties

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]

From this we may conclude the following:

  • [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]

This leads to the most general form:

  • [ilmath]\langle au+bv,cx+dy\rangle=a\langle u,cx+dy\rangle+b\langle v,cx+dy\rangle[/ilmath][ilmath]=a\overline{\langle cx+dy,u\rangle}+b\overline{\langle cx+dy,v\rangle}[/ilmath][ilmath]=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})[/ilmath][ilmath]=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[/ilmath]

Proof of claim: [ilmath]\langle x,\alpha y+\beta z\rangle=\overline{\alpha}\langle x, y\rangle +\overline{\beta}\langle x,z\rangle[/ilmath] </b>


</div>


Examples

See also

References

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

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