Difference between revisions of "Inner product"

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==Notation==
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Typically, {{M|\langle\cdot,\cdot\rangle}} is the notation for inner products, however I have seen some authors use {{M|\langle a,b\rangle}} to denote the [[Ordered pair|ordered pair]] containing {{M|a}} and {{M|b}}. Also, notably<ref name="FA"/> use {{M|(\cdot,\cdot)}} for an inner product (and {{M|\langle\cdot,\cdot\rangle}} for an ordered pair!)
 
==Examples==
 
==Examples==
 
* [[Vector dot product]]
 
* [[Vector dot product]]

Revision as of 18:43, 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 =\bar{\lambda}\langle x,y\rangle+\bar{\mu}\langle x,z\rangle[/math]

[math]\langle x,\lambda y+\mu z\rangle[/math]
[math]=\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]
As required.

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\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] - which isn't worth remembering!

Proof:
[ilmath]\langle au+bv,cx+dy\rangle[/ilmath]
[ilmath]=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]
As required

Notation

Typically, [ilmath]\langle\cdot,\cdot\rangle[/ilmath] is the notation for inner products, however I have seen some authors use [ilmath]\langle a,b\rangle[/ilmath] to denote the ordered pair containing [ilmath]a[/ilmath] and [ilmath]b[/ilmath]. Also, notably[3] use [ilmath](\cdot,\cdot)[/ilmath] for an inner product (and [ilmath]\langle\cdot,\cdot\rangle[/ilmath] for an ordered pair!)

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. ↑ 3.0 3.1 Functional Analysis - George Bachman and Lawrence Narici
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