Editing Cauchy–Schwarz inequality (section)

== Statement of the inequality ==
The Cauchy–Schwarz inequality states that for all vectors <math>\mathbf{u}</math> and <math>\mathbf{v}</math> of an [[inner product space]]

{{NumBlk|:|<math>\left |\langle \mathbf{u}, \mathbf{v} \rangle\right |^2 \leq \langle \mathbf{u}, \mathbf{u} \rangle \cdot \langle \mathbf{v}, \mathbf{v} \rangle,</math>|{{EquationRef|Cauchy–Schwarz inequality [written using only the inner product]|1}}}}
where <math>\langle \cdot, \cdot \rangle</math> is the [[inner product]]. Examples of inner products include the real and complex [[dot product]]; see the [[Inner product space#Examples|examples in inner product]]. Every inner product gives rise to a Euclidean <math>\ell_2</math> [[Norm (mathematics)|norm]], called the {{em|canonical}} or [[inner product space#Norm|{{em|induced}} {{em|norm}}]], where the norm of a vector <math>\mathbf{u}</math> is denoted and defined by
<math display="block">\|\mathbf{u}\| := \sqrt{\langle \mathbf{u}, \mathbf{u} \rangle},</math> where <math>\langle \mathbf{u}, \mathbf{u} \rangle</math> is always a non-negative real number (even if the inner product is complex-valued). 
By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form in terms of the norm:<ref name="Strang5">{{cite book|last=Strang|first=Gilbert|date=19 July 2005|title=Linear Algebra and its Applications|edition=4th|chapter=3.2|publisher=Cengage Learning|location=Stamford, CT|isbn=978-0030105678|pages=154–155}}</ref><ref name=":0">{{cite book|last1=Hunter|first1=John K.|last2=Nachtergaele|first2=Bruno|year=2001|title=Applied Analysis|publisher=World Scientific|isbn=981-02-4191-7|url=https://books.google.com/books?id=oOYQVeHmNk4C}}</ref>

{{NumBlk|:|<math>|\langle \mathbf{u}, \mathbf{v} \rangle| \leq \|\mathbf{u}\|  \|\mathbf{v}\|.</math>|{{EquationRef|Cauchy–Schwarz inequality - written using norm and inner product|2}}}}

Moreover, the two sides are equal if and only if <math>\mathbf{u}</math> and <math>\mathbf{v}</math> are [[linear independence|linearly dependent]].<ref>{{cite book|last1=Bachmann|first1=George|last2=Narici|first2=Lawrence|last3=Beckenstein|first3=Edward|date=2012-12-06|title=Fourier and Wavelet Analysis|publisher=Springer Science & Business Media|isbn=9781461205050|page=14|url=https://books.google.com/books?id=PkHhBwAAQBAJ}}</ref><ref>{{cite book|last=Hassani|first=Sadri|year=1999|title=Mathematical Physics: A Modern Introduction to Its Foundations|publisher=Springer|isbn=0-387-98579-4|page=29|quote=Equality holds iff <c{{pipe}}c>&nbsp;=&nbsp;0 or {{pipe}}c>&nbsp;=&nbsp;0. From the definition of {{pipe}}c>, we conclude that {{pipe}}a> and {{pipe}}b> must be proportional.}}</ref><ref>{{cite book|last1=Axler|first1=Sheldon|date=2015|title=Linear Algebra Done Right, 3rd Ed.|publisher=Springer International Publishing|isbn=978-3-319-11079-0|page=172|url=https://books.google.com/books?id=CQWwoQEACAAJ|quote=This inequality is an equality if and only if one of ''u'', ''v'' is a scalar multiple of the other.}}</ref>