Difference between revisions of "Inner product"

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* <math>\langle x,x\rangle \ge 0</math> but specifically:
 
* <math>\langle x,x\rangle \ge 0</math> but specifically:
 
** <math>\langle x,x\rangle=0\iff x=0</math>
 
** <math>\langle x,x\rangle=0\iff x=0</math>
−
 
+
==Terminology==
 +
Given a vector space {{M|X}} over either {{M|\mathbb{R} }} or {{M|\mathbb{C} }}, and an inner product {{M|\langle\cdot,\cdot\rangle:X\times X\rangle F}} we call the space {{M|(X,\langle\cdot,\cdot\rangle)}} an:
 +
* ''inner product space'' (or ''i.p.s'' for short)<ref name="FA"/> or sometimes a
 +
* ''pre-[[Hilbert space|hilbert]] space''<ref name="FA"/>
 
==Properties==
 
==Properties==
 
Notice that <math>\langle\cdot,\cdot\rangle</math> is also linear (ish) in its second argument as:
 
Notice that <math>\langle\cdot,\cdot\rangle</math> is also linear (ish) in its second argument as:
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==Notation==
 
==Notation==
 
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!)
 
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!)
 +
 +
==Immediate theorems==
 +
Here {{M|\langle\cdot,\cdot\rangle:X\times X\rightarrow \mathbb{C} }} is an ''inner product''
 +
{{Begin Theorem}}
 +
'''Theorem: ''' if {{M|1=\forall x\in X[\langle x,y\rangle=0]}} then {{M|1=y=0}}
 +
{{Begin Proof}}
 +
: Suppose that {{M|y\ne 0}}, then {{M|\forall x\in X[\langle x,y\rangle=0]}} by hypothesis:
 +
:* {{M|1=\forall x\in X[\langle x,y\rangle =0]}}
 +
: Specifically that means for {{M|y\in X}} we have {{M|1=\langle y,y\rangle=0}}
 +
:* Of course by definition, {{M|\langle y,y\rangle\ge 0}} for {{M|\forall y\in X}}, and specifically
 +
:** {{M|1=\langle x,x\rangle = 0\iff x=0}}
 +
: So we have {{M|1=\langle y,y\rangle =0}} '''contradicting''' that {{M|y\ne 0}}
 +
* We conclude that if {{M|1=\forall x\in X[\langle x,y\rangle=0]}} then we must have {{M|1=y=0}}
 +
*: (As required)
 +
{{End Proof}}{{End Theorem}}
 +
==Norm induced by==
 +
* Given an ''inner product space'' {{M|(X,\langle\cdot,\cdot\rangle)}} we can define a [[Norm|norm]] as follows<ref name="FA"/>:
 +
** {{M|1=\forall x\in X}} the inner product induces the norm {{M|1=\Vert x\Vert:=\sqrt{\langle x,x\rangle} }}
 +
{{Todo|Find out what this is called, eg compared to the [[Norm#Induced metric|metric induced by a norm]]}}
 
==Examples==
 
==Examples==
 
* [[Vector dot product]]
 
* [[Vector dot product]]

Revision as of 19:34, 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]

Terminology

Given a vector space [ilmath]X[/ilmath] over either [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath], and an inner product [ilmath]\langle\cdot,\cdot\rangle:X\times X\rangle F[/ilmath] we call the space [ilmath](X,\langle\cdot,\cdot\rangle)[/ilmath] an:

  • inner product space (or i.p.s for short)[3] or sometimes a
  • pre-hilbert space[3]

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!)

Immediate theorems

Here [ilmath]\langle\cdot,\cdot\rangle:X\times X\rightarrow \mathbb{C} [/ilmath] is an inner product

Theorem: if [ilmath]\forall x\in X[\langle x,y\rangle=0][/ilmath] then [ilmath]y=0[/ilmath]

Suppose that [ilmath]y\ne 0[/ilmath], then by hypothesis:
  • [ilmath]\forall x\in X[\langle x,y\rangle =0][/ilmath]
Specifically that means for [ilmath]y\in X[/ilmath] we have [ilmath]\langle y,y\rangle=0[/ilmath]
  • Of course by definition, [ilmath]\langle y,y\rangle\ge 0[/ilmath] for [ilmath]\forall y\in X[/ilmath], and specifically
    • [ilmath]\langle x,x\rangle = 0\iff x=0[/ilmath]
So we have [ilmath]\langle y,y\rangle =0[/ilmath] contradicting that [ilmath]y\ne 0[/ilmath]
  • We conclude that if [ilmath]\forall x\in X[\langle x,y\rangle=0][/ilmath] then we must have [ilmath]y=0[/ilmath]
    (As required)

Norm induced by

  • Given an inner product space [ilmath](X,\langle\cdot,\cdot\rangle)[/ilmath] we can define a norm as follows[3]:
    • [ilmath]\forall x\in X[/ilmath] the inner product induces the norm [ilmath]\Vert x\Vert:=\sqrt{\langle x,x\rangle}[/ilmath]

TODO: Find out what this is called, eg compared to the metric induced by a norm

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 3.2 3.3 3.4 Functional Analysis - George Bachman and Lawrence Narici
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