Cauchy-Schwarz Formula - Revision history

Lila at 21:19, 28 October 2021

2021-10-28T21:19:18Z

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Lila at 21:04, 28 October 2021

2021-10-28T21:04:00Z

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Lila: /* Statement of the inequality */

2021-10-28T21:00:43Z

Statement of the inequality

Lila: /* Statement of the inequality */

2021-10-28T20:59:51Z

Statement of the inequality

Lila at 20:56, 28 October 2021

2021-10-28T20:56:54Z

Lila: Created page with "The '''Cauchy–Schwarz inequality''' (also called '''Cauchy–Bunyakovsky-Schwarz inequality'''){{cite web|url=https://mathshistory.st-andrews.ac.uk/Biographies/Schwarz/..."

2021-10-25T19:34:20Z

Created page with "The '''Cauchy–Schwarz inequality''' (also called '''Cauchy–Bunyakovsky-Schwarz inequality''')<ref>{{cite web|url=https://mathshistory.st-andrews.ac.uk/Biographies/Schwarz/..."

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 <math>\langle \mathbf{u}, \mathbf{v} \rangle| \leq \|\mathbf{u}\|  \|\mathbf{v}\|.</math>
-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&#124;c>=0 or &#124;c>=0. From the definition of &#124;c>, we conclude that &#124;a> and &#124;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/about/Linear_Algebra_Done_Right.html?id=CQWwoQEACAAJ|quote=This inequality is an equality if and only if one of u, v is a scalar multiple of the other.}}</ref>
+Moreover, the two sides are equal if and only if <math>\mathbf{u}</math> and <math>\mathbf{v}</math> are linearly dependent.
 == Special cases ==

@@ Line 4: / Line 4: @@
 <math>\left|\left\langle \mathbf{u}, \mathbf{v} \right\rangle\right|^2 \leq \langle \mathbf{u}, \mathbf{u} \rangle \cdot \langle \mathbf{v}, \mathbf{v} \rangle,</math>
-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#Some examples|examples in inner product]]. Every inner product gives rise to a [[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:
+where <math>\langle \cdot, \cdot \rangle</math> is the inner product. Examples of inner products include the real and complex dot product; see the examples in inner product. Every inner product gives rise to a norm, called the canonical or induced norm, where the norm of a vector <math>\mathbf{u}</math> is denoted and defined by:
-<math display=block>\left\|\mathbf{u}\right\| := \sqrt{\left\langle \mathbf{u}, \mathbf{u} \right\rangle}</math>
+<math>\left\|\mathbf{u}\right\| := \sqrt{\left\langle \mathbf{u}, \mathbf{u} \right\rangle}</math>
 so that this norm and the inner product are related by the defining condition <math>\left\|\mathbf{u}\right\|^2 = \left\langle \mathbf{u}, \mathbf{u} \right\rangle,</math> where <math>\left\langle \mathbf{u}, \mathbf{u} \right\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:<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>
+By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form:
-{{NumBlk|:|<math>|\langle \mathbf{u}, \mathbf{v} \rangle| \leq \|\mathbf{u}\|  \|\mathbf{v}\|.</math>|{{EquationRef|Cauchy-Schwarz inequality [written using norm and inner product]}}}}
+<math>\langle \mathbf{u}, \mathbf{v} \rangle| \leq \|\mathbf{u}\|  \|\mathbf{v}\|.</math>
 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&#124;c>=0 or &#124;c>=0. From the definition of &#124;c>, we conclude that &#124;a> and &#124;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/about/Linear_Algebra_Done_Right.html?id=CQWwoQEACAAJ|quote=This inequality is an equality if and only if one of u, v is a scalar multiple of the other.}}</ref>

@@ Line 1: / Line 1: @@
 == Statement of the inequality ==
-The Cauchy–Schwarz inequality states that for all vectors <math>u</math> and <math>v</math> of an [[inner product space]] it is true that
+The Cauchy–Schwarz inequality states that for all vectors <math>u</math> and <math>v</math> of an inner product space it is true that
-{{NumBlk|:|<math>\left|\left\langle \mathbf{u}, \mathbf{v} \right\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]}}}}
+<math>\left|\left\langle \mathbf{u}, \mathbf{v} \right\rangle\right|^2 \leq \langle \mathbf{u}, \mathbf{u} \rangle \cdot \langle \mathbf{v}, \mathbf{v} \rangle,</math>
 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#Some examples|examples in inner product]]. Every inner product gives rise to a [[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: