Inner product
Inner product | |
⟨⋅,⋅⟩:V×V→F 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 |
For V a vector space over \mathbb{R} or \mathbb{C} |
Induced metric |
For V considered as a set |
Contents
[hide]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
- This may be alternatively stated as:
- \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
- 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:
- \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:
- \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
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
- 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
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