Inner product

From Maths
Revision as of 15:23, 31 January 2016 by Alec (Talk | contribs)

Jump to: navigation, search
Inner product
,:V×VF
Where V is a vector space over the field F
F may be R or C.
relation to other topological spaces
is a
contains all

(none)

Related objects
Induced norm
  • ,:VR0
  • \Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle}

For V a vector space over \mathbb{R} or \mathbb{C}

Induced metric
  • d_{\langle\cdot,\cdot\rangle}:V\times V\rightarrow\mathbb{R}_{\ge 0}
  • d_{\langle\cdot,\cdot\rangle}:(x,y)\mapsto\sqrt{\langle x-y,x-y\rangle}
(As every metric induces a norm)

For V considered as a set

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 \langle x,y\rangle = \langle y,x\rangle 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:

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

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

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!


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

Norm induced by

  • Given an inner product space (X,\langle\cdot,\cdot\rangle) 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

See also

References

  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
  3. Jump up to: 3.0 3.1 3.2 3.3 3.4 Functional Analysis - George Bachman and Lawrence Narici