Difference between revisions of "Inner product"

	Inner product
	[ilmath]\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{F} [/ilmath]; Where [ilmath]V[/ilmath] is a vector space over the field [ilmath]\mathbb{F} [/ilmath]; [ilmath]\mathbb{F} [/ilmath] may be [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath].
relation to other topological spaces
is a	topological space; metric space; normed space;
contains all	(none)
Related objects
Induced norm	[ilmath]\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:V\rightarrow\mathbb{R}_{\ge 0}[/ilmath]; [ilmath]\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle}[/ilmath]; For [ilmath]V[/ilmath] a vector space over [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath]
Induced metric	[ilmath]d_{\langle\cdot,\cdot\rangle}:V\times V\rightarrow\mathbb{R}_{\ge 0}[/ilmath]; [ilmath]d_{\langle\cdot,\cdot\rangle}:(x,y)\mapsto\sqrt{\langle x-y,x-y\rangle}[/ilmath]; ; (As every metric induces a norm) For [ilmath]V[/ilmath] considered as a set;

Latest revision as of 12:57, 19 February 2016

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\rightarrow 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

The most important property by far is that: [ilmath]\forall x\in X[\langle x,x\rangle\in\mathbb{R}_{\ge 0}][/ilmath] - that is [ilmath]\langle x,x\rangle[/ilmath] is real

Proof:

Notice that we (by definition) have [ilmath]\langle x,x\rangle=\overline{\langle x,x\rangle}[/ilmath], so we must have:

[ilmath]a+bj=a-bj[/ilmath] where [ilmath]a+bj:=\langle x,x\rangle[/ilmath], and by equating the real and imaginary parts we see immediately that we have:
- [ilmath]b=-b[/ilmath] and conclude [ilmath]b=0[/ilmath], that is there is no imaginary component.

To complete the proof note that by definition [ilmath]\langle x,x\rangle\ge 0[/ilmath].

Thus [ilmath]\langle x,x\rangle\in\mathbb{R}_{\ge 0}[/ilmath] - as I claimed.

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

Prominent examples

Vector dot product

References

↑ http://en.wikipedia.org/w/index.php?title=Inner_product_space&oldid=651022885
↑ Functional Analysis I - Lecture Notes - Richard Sharp - Sep 2014
↑ ^3.0 ^3.1 ^3.2 ^3.3 ^3.4 Functional Analysis - George Bachman and Lawrence Narici

[1] ttp://en.wikipedia.org/w/index.php?title=Inner_product_space&oldid=651022885

[2] Functional Analysis I - Lecture Notes - Richard Sharp - Sep 2014

[FA-3] 3.0 ^3.1 ^3.2 ^3.3 ^3.4 Functional Analysis - George Bachman and Lawrence Narici

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
+{{:Inner product/Infobox}}
 ==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>)
 Such that:
@@ Line 6: / Line 7: @@
 ** 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]] )
-*: 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>
+==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\rightarrow 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==
-Notice that <math>\langle\cdot,\cdot\rangle</math> is also linear in its second argument as:
+{{Begin Inline Theorem}}
-*<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>
+* '''The most important property by far is that: ''' {{M|\forall x\in X[\langle x,x\rangle\in\mathbb{R}_{\ge 0}]}} - that is '''{{M|\langle x,x\rangle}} is real'''
+{{Begin Inline Proof}}
+'''Proof:'''
+: Notice that we (by definition) have {{M|1=\langle x,x\rangle=\overline{\langle x,x\rangle} }}, so we must have:
+:* {{M|1=a+bj=a-bj}} where {{M|1=a+bj:=\langle x,x\rangle}}, and by equating the real and imaginary parts we see immediately that we have:
+:** {{M|1=b=-b}} and conclude {{M|1=b=0}}, that is there is no imaginary component.
+To complete the proof note that by definition {{M|\langle x,x\rangle\ge 0}}.
+Thus {{M|1=\langle x,x\rangle\in\mathbb{R}_{\ge 0} }} - as I claimed.
+{{End Proof}}{{End Theorem}}
+Notice that <math>\langle\cdot,\cdot\rangle</math> is also linear (ish) in its second argument as:
+{{Begin Inline Theorem}}
+* <math>\langle x,\lambda y+\mu z\rangle =\bar{\lambda}\langle x,y\rangle+\bar{\mu}\langle x,z\rangle</math>
+{{Begin Inline Proof}}
+:<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.
+{{End Proof}}{{End Theorem}}
 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:
+{{Begin Inline Theorem}}
+* {{M|1=\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}} - which isn't worth remembering!
+{{Begin Inline Proof}}
+:'''Proof:'''
+:{{M|1=\langle au+bv,cx+dy\rangle}}
+::{{M|1= =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}}
+: As required
+{{End Proof}}{{End Theorem}}
-==Examples==
+==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!)
+==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]]}}
+==Prominent examples==
 * [[Vector dot product]]
 ==See also==
 * [[Hilbert space]]
+* [[Inner product examples]]
+* [[Inequalities for inner products]]
+* [[Perpendicular]]
 ==References==
 <references/>
+{{Inner product and Hilbert spaces navbox}}
+{{Normed and Banach spaces navbox}}
+{{Metric spaces navbox}}
+{{Topology navbox}}
 {{Definition|Linear Algebra|Functional Analysis}}
+[[Category:Exemplary pages]]

relation to other topological spaces
Inner product
[ilmath]\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{F} [/ilmath] Where [ilmath]V[/ilmath] is a vector space over the field [ilmath]\mathbb{F} [/ilmath] [ilmath]\mathbb{F} [/ilmath] may be [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath].
is a	topological space metric space normed space
contains all	(none)
Related objects
Induced norm	[ilmath]\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:V\rightarrow\mathbb{R}_{\ge 0}[/ilmath] [ilmath]\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle}[/ilmath] For [ilmath]V[/ilmath] a vector space over [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath]
Induced metric	[ilmath]d_{\langle\cdot,\cdot\rangle}:V\times V\rightarrow\mathbb{R}_{\ge 0}[/ilmath] [ilmath]d_{\langle\cdot,\cdot\rangle}:(x,y)\mapsto\sqrt{\langle x-y,x-y\rangle}[/ilmath] (As every metric induces a norm) For [ilmath]V[/ilmath] considered as a set

Difference between revisions of "Inner product"

Latest revision as of 12:57, 19 February 2016

Contents

Definition

Terminology

Properties

Notation

Immediate theorems

Norm induced by

Prominent examples

See also

References

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