Variants of The Polarization Identity The 2019 Stack Overflow Developer Survey Results Are InProving $sumlimits_i=1^k | langle x,v_i rangle langle y,v_irangle| leq |x|cdot |y|$How to motivate the axioms for the inner productNorms on inner product space over $mathbbR$Showing inner product comes from a norm defined using Polarization IdentityInner product identityGeneralization of Cauchy-Schwarz to positive operatorsVector inequality $(langle Ax,y rangle^2) le (langle Ax,xrangle * langle Ay,yrangle)$Very general inner product determinant inequalityProve: $|langle u,v rangle|= |u|cdot |v|iff u=alpha v$Does Cauchy-Schwarz hold for: $ langletextbfu,textbfvrangle ;leq ||textbfu|| cdot ||textbfv|| $
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Variants of The Polarization Identity
The 2019 Stack Overflow Developer Survey Results Are InProving $sumlimits_i=1^k | langle x,v_i rangle langle y,v_irangle| leq |x|cdot |y|$How to motivate the axioms for the inner productNorms on inner product space over $mathbbR$Showing inner product comes from a norm defined using Polarization IdentityInner product identityGeneralization of Cauchy-Schwarz to positive operatorsVector inequality $(langle Ax,y rangle^2) le (langle Ax,xrangle * langle Ay,yrangle)$Very general inner product determinant inequalityProve: $|langle u,v rangle|= |u|cdot |v|iff u=alpha v$Does Cauchy-Schwarz hold for: $ langletextbfu,textbfvrangle ;leq ||textbfu|| cdot ||textbfv|| $
$begingroup$
A problem in Steele's Cauchy Schwarz Master Class asks the reader to prove these "variants of the polarization identity".
Let $langle cdot, cdot rangle$ be a complex inner product and $alpha in mathbbC$ with $alpha^N = 1$ but $alpha^2 neq 1$. Then
$$ langle x,y rangle = frac1N sum_n=0^N-1 left|x+alpha^n y right|^2 alpha^n quad textand quad langle x,y rangle = frac12pi int_-pi^pi left|x+e^ithetay right|^2 e^i theta dtheta.$$
I can prove it, so I'll pose the more important question(s): How does one find these? Is there any intuitive (e.g. geometric) interpretation? Where in mathematics do these come up?
norm inner-product-space
asked Apr 6 at 9:05
KezerKezer
1,405621
$endgroup$
add a comment |
$begingroup$
A problem in Steele's Cauchy Schwarz Master Class asks the reader to prove these "variants of the polarization identity".
Let $langle cdot, cdot rangle$ be a complex inner product and $alpha in mathbbC$ with $alpha^N = 1$ but $alpha^2 neq 1$. Then
$$ langle x,y rangle = frac1N sum_n=0^N-1 left|x+alpha^n y right|^2 alpha^n quad textand quad langle x,y rangle = frac12pi int_-pi^pi left|x+e^ithetay right|^2 e^i theta dtheta.$$
I can prove it, so I'll pose the more important question(s): How does one find these? Is there any intuitive (e.g. geometric) interpretation? Where in mathematics do these come up?
norm inner-product-space
asked Apr 6 at 9:05
KezerKezer
1,405621
$endgroup$
add a comment |
$begingroup$
A problem in Steele's Cauchy Schwarz Master Class asks the reader to prove these "variants of the polarization identity".
Let $langle cdot, cdot rangle$ be a complex inner product and $alpha in mathbbC$ with $alpha^N = 1$ but $alpha^2 neq 1$. Then
$$ langle x,y rangle = frac1N sum_n=0^N-1 left|x+alpha^n y right|^2 alpha^n quad textand quad langle x,y rangle = frac12pi int_-pi^pi left|x+e^ithetay right|^2 e^i theta dtheta.$$
I can prove it, so I'll pose the more important question(s): How does one find these? Is there any intuitive (e.g. geometric) interpretation? Where in mathematics do these come up?
norm inner-product-space
asked Apr 6 at 9:05
KezerKezer
1,405621
$endgroup$
A problem in Steele's Cauchy Schwarz Master Class asks the reader to prove these "variants of the polarization identity".
Let $langle cdot, cdot rangle$ be a complex inner product and $alpha in mathbbC$ with $alpha^N = 1$ but $alpha^2 neq 1$. Then
$$ langle x,y rangle = frac1N sum_n=0^N-1 left|x+alpha^n y right|^2 alpha^n quad textand quad langle x,y rangle = frac12pi int_-pi^pi left|x+e^ithetay right|^2 e^i theta dtheta.$$
I can prove it, so I'll pose the more important question(s): How does one find these? Is there any intuitive (e.g. geometric) interpretation? Where in mathematics do these come up?
norm inner-product-space
norm inner-product-space
asked Apr 6 at 9:05
KezerKezer
1,405621
asked Apr 6 at 9:05
KezerKezer
1,405621
edited Apr 6 at 22:20
Kezer
asked Apr 6 at 9:05
KezerKezer
1,405621
asked Apr 6 at 9:05
KezerKezer
1,405621
asked Apr 6 at 9:05
KezerKezer
1,405621
1,405621
add a comment |
add a comment |
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