Difference between revisions of "Inner product"

Revision as of 17:30, 21 April 2015

Given a vector space, $(V,F)$ (where $F$ is either $\mathbb{R}$ or $\mathbb{C}$ ), an inner product^[1]^[2] 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 better stated as:
- $\langle\lambda x,y\rangle=\lambda\langle x,y\rangle$ and
- $\langle x+y,z\rangle = \langle x,z\rangle + \langle y,z\rangle$
$\langle x,x\rangle \ge 0$ with $\langle x,x\rangle=0\iff x=0$

Notice that

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

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

From this we may conclude the following:

@@ Line 21: / Line 21: @@
 ==Examples==
 * [[Vector dot product]]
+==See also==
+* [[Hilbert space]]
 ==References==