Difference between revisions of "Inner product"
m |
m (→Terminology: Typo) |
||
| Line 11: | Line 11: | ||
** <math>\langle x,x\rangle=0\iff x=0</math> | ** <math>\langle x,x\rangle=0\iff x=0</math> | ||
==Terminology== | ==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\ | + | 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 | * ''inner product space'' (or ''i.p.s'' for short)<ref name="FA"/> or sometimes a | ||
* ''pre-[[Hilbert space|hilbert]] space''<ref name="FA"/> | * ''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: | ||
Revision as of 20:51, 11 July 2015
Contents
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]
- This may be alternatively stated as:
- [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:
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]
- Of course by definition, [ilmath]\langle y,y\rangle\ge 0[/ilmath] for [ilmath]\forall y\in X[/ilmath], and specifically
- 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
