Editing Inner product space (section)

===Norm properties {{anchor|Norm}}===<!-- This section is linked from [[Cauchy–Schwarz inequality]] -->

Every inner product space induces a [[Norm (mathematics)|norm]], called its {{em|{{visible anchor|canonical norm}}}}, that is defined by
<math display=block>\|x\| = \sqrt{\langle x, x \rangle}.</math> 
With this norm, every inner product space becomes a [[normed vector space]]. 

So, every general property of normed vector spaces applies to inner product spaces. <!-- In particular, an inner product space is a  [[metric space]], for the distance defined by
<math display=block>d(x, y) = \|y - x\|.</math> -->
In particular, one has the following properties:

{{glossary}}
{{term|[[Absolute homogeneity]]}}{{defn|
<math display=block>\|ax\| = |a| \, \|x\|</math>
for every <math>x \in V</math> and <math>a \in F</math>
(this results from <math>\langle ax, ax \rangle = a\overline a \langle x, x \rangle</math>).
}}
{{term|[[Triangle inequality]]}}{{defn|
<math display=block>\|x + y\| \leq \|x\| + \|y\|</math>
for <math>x, y\in V.</math>
These two properties show that one has indeed a norm.}}

{{term|[[Cauchy–Schwarz inequality]]}}{{defn|
<math display=block>|\langle x, y \rangle| \leq \|x\| \, \|y\|</math>
for every <math>x, y\in V,</math>
with equality if and only if <math>x</math> and <math>y</math> are [[Linearly independent|linearly dependent]].
}}
{{term|[[Parallelogram law]]}}{{defn|
<math display=block>\|x + y\|^2 + \|x - y\|^2 = 2\|x\|^2 + 2\|y\|^2</math>
for every <math>x, y\in V.</math>

The parallelogram law is a necessary and sufficient condition for a norm to be defined by an inner product.
}}
{{term|[[Polarization identity]]}}{{defn|
<math display=block>\|x + y\|^2 = \|x\|^2 + \|y\|^2 + 2\operatorname{Re}\langle x, y \rangle</math>
for every <math>x, y\in V.</math>
The inner product can be retrieved from the norm by the polarization identity, since its imaginary part is the real part of <math>\langle x, iy \rangle.</math>
}}
{{term|[[Ptolemy's inequality]]}}{{defn|
<math display=block>\|x - y\| \, \|z\| ~+~ \|y - z\| \, \|x\| ~\geq~ \|x - z\| \, \|y\|</math>
for every <math>x, y,z\in V.</math>
Ptolemy's inequality is a necessary and sufficient condition for a [[seminorm]] to be the norm defined by an inner product.<ref>{{Cite journal|last=Apostol|first=Tom M.|date=1967|title=Ptolemy's Inequality and the Chordal Metric|url=https://www.tandfonline.com/doi/pdf/10.1080/0025570X.1967.11975804|journal=Mathematics Magazine|volume=40|issue=5|pages=233–235|language=en|doi=10.2307/2688275|jstor=2688275}}</ref>
}}
{{glossary end}}