Difference between revisions of "Inner product"

Revision as of 21:24, 11 July 2015

Definition

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

$\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{R}$ (or sometimes $\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{C}$ )

Such that:

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

Terminology

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

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

Properties

[Expand]

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

Proof:

Notice that we (by definition) have

$\langle x,x\rangle=\overline{\langle x,x\rangle}$ , so we must have:

a+bj=a−bj where a+bj:=⟨x,x⟩, and by equating the real and imaginary parts we see immediately that we have:
- $b=-b$ and conclude $b=0$ , that is there is no imaginary component.

To complete the proof note that by definition $\langle x,x\rangle\ge 0$ .

Thus $\langle x,x\rangle\in\mathbb{R}_{\ge 0}$ - as I claimed.

Notice that

$\langle\cdot,\cdot\rangle$ is also linear (ish) in its second argument as:

[Expand]

$\langle x,\lambda y+\mu z\rangle =\bar{\lambda}\langle x,y\rangle+\bar{\mu}\langle x,z\rangle$

$\langle x,\lambda y+\mu z\rangle$

$=\overline{\langle \lambda y+\mu z, x\rangle}$

$=\overline{\lambda\langle y,x\rangle + \mu\langle z,x\rangle}$

$=\bar{\lambda}\overline{\langle y,x\rangle}+\bar{\mu}\overline{\langle z,x\rangle}$

$=\bar{\lambda}\langle x,y\rangle+\bar{\mu}\langle x,z\rangle$

As required.

From this we may conclude the following:

$\langle x,\lambda y\rangle = \bar{\lambda}\langle x,y\rangle$ and
$\langle x,y+z\rangle = \langle x,y\rangle + \langle x,z\rangle$

This leads to the most general form:

[Expand]

$\langle au+bv,cx+dy\rangle=a\overline{c}\langle u,x\rangle+a\overline{d}\langle u,y\rangle+b\overline{c}\langle v,x\rangle+b\overline{d}\langle v,y\rangle$ - which isn't worth remembering!

Proof:

$\langle au+bv,cx+dy\rangle$

$=a\langle u,cx+dy\rangle+b\langle v,cx+dy\rangle$

$=a\overline{\langle cx+dy,u\rangle}+b\overline{\langle cx+dy,v\rangle}$

$=a(\overline{c\langle x,u\rangle} + \overline{d\langle y,u\rangle})+b(\overline{c\langle x,v\rangle}+\overline{d\langle y,v\rangle})$

$=a\overline{c}\langle u,x\rangle+a\overline{d}\langle u,y\rangle+b\overline{c}\langle v,x\rangle+b\overline{d}\langle v,y\rangle$

As required

Notation

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

Immediate theorems

Here $\langle\cdot,\cdot\rangle:X\times X\rightarrow \mathbb{C}$ is an inner product

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

Suppose that

$y\ne 0$ , then by hypothesis:

$\forall x\in X[\langle x,y\rangle =0]$

Specifically that means for

$y\in X$ we have

$\langle y,y\rangle=0$

Of course by definition, ⟨y,y⟩≥0 for ∀y∈X, and specifically
- $\langle x,x\rangle = 0\iff x=0$

So we have

$\langle y,y\rangle =0$ contradicting that

$y\ne 0$

We conclude that if $\forall x\in X[\langle x,y\rangle=0]$ then we must have $y=0$
(As required)

Norm induced by

Given an inner product space (X,⟨⋅,⋅⟩) we can define a norm as follows^[3]:
- $\forall x\in X$ the inner product induces the norm $\Vert x\Vert:=\sqrt{\langle x,x\rangle}$

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

Prominent examples

Vector dot product

References

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

[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

[FA-3] {Jump up to: 3.0} ^3.1 ^3.2 ^3.3 ^3.4 Functional Analysis - George Bachman and Lawrence Narici

[1]

[2]

[3]

@@ Line 16: / Line 16: @@
 ==Properties==
+{{Begin Inline Theorem}}
+* '''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}}
@@ Line 42: / Line 54: @@
 : As required
 {{End Proof}}{{End Theorem}}
 ==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!)

Difference between revisions of "Inner product"

Revision as of 21:24, 11 July 2015

Contents

Definition

Terminology

Properties

Notation

Immediate theorems

Norm induced by

Prominent examples

See also

References

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