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#1 2009-02-05 11:11:19

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Cauchy–Schwarz–Bunyakovsky inequality

Today I learned from Introduction to Metric and Topological Spaces by W.A. Sutherland a version of the Cauchy–Schwarz inequality involving integrals. It goes by the funny name of Cauchy–Schwarz–Bunyakovsky inequality. tongue

First, let’s recall the Cauchy–Schwarz inequality. It states

   

for al real numbers

.

Proof:

The inequality is obviously true if

. Hence we may assume that at least one
is not 0.

Then

Treating the LHS as a quadratic in

, we see that its discriminant cannot be positive.

This is the proof given in Introduction to Metric and Topological Spaces. cool

The integral version is as follows.

The proof is similar to the Cauchy–Schwarz case, only this time you start with

Last edited by JaneFairfax (2009-02-06 03:32:01)


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#2 2009-02-05 11:53:01

Daniel123
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Registered: 2007-05-23
Posts: 663

Re: Cauchy–Schwarz–Bunyakovsky inequality

That's really nice smile

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#3 2009-02-06 00:56:02

TheDude
Member
Registered: 2007-10-23
Posts: 361

Re: Cauchy–Schwarz–Bunyakovsky inequality

JaneFairfax wrote:

Treating the LHS as a quadratic in

, we see that its discrimant cannot be positive.

I don't follow this part.  Why can't the discriminant be positive?


Wrap it in bacon

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#4 2009-02-06 01:04:20

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Cauchy–Schwarz–Bunyakovsky inequality

Because then the quadratic equation LHS = 0 would have two distinct real roots and the LHS would be negative between these roots.

Last edited by JaneFairfax (2009-02-06 01:36:09)


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#5 2009-02-06 01:09:12

Daniel123
Member
Registered: 2007-05-23
Posts: 663

Re: Cauchy–Schwarz–Bunyakovsky inequality

TheDude wrote:
JaneFairfax wrote:

Treating the LHS as a quadratic in

, we see that its discrimant cannot be positive.

I don't follow this part.  Why can't the discriminant be positive?

The fact that the quadratic is always greater than or equal to 0 means that it must have at most one real root.

EDIT: In other words, what Jane said. I clicked 'quote' and left the room for too long sad

Last edited by Daniel123 (2009-02-06 01:10:35)

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#6 2009-02-06 04:14:03

TheDude
Member
Registered: 2007-10-23
Posts: 361

Re: Cauchy–Schwarz–Bunyakovsky inequality

I'm an idiot.  Thanks for the explanation.


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#7 2009-02-06 07:01:18

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Cauchy–Schwarz–Bunyakovsky inequality

It goes by the funny name of Cauchy–Schwarz–Bunyakovsky inequality.

That is rather odd, normally I've heard it referred to as the Cauchy-Schwarz special case of Holder's inequality, where Holder's inequality is:

[align=center]

[/align]

Where 1/p + 1/q = 1.  Of course, Cauchy-Schwarz is the special case with p = q = 2.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#8 2009-02-07 09:26:59

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

Re: Cauchy–Schwarz–Bunyakovsky inequality

That's nice. I remember we have a similar proof in Calculus for multi-variable function's Taylor expansion. It adds in lamda as well. Such method is called "adding parameter", which shares the same delicacy as adding a line to solve geometry problems.


X'(y-Xβ)=0

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#9 2010-09-10 16:14:34

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

Re: Cauchy–Schwarz–Bunyakovsky inequality

A good one


X'(y-Xβ)=0

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#10 2010-09-27 01:06:44

tharun reddy
Member
Registered: 2010-09-27
Posts: 2

Re: Cauchy–Schwarz–Bunyakovsky inequality

...hi jane smile.........but you assumed that "lambda" was real but by setting discriminant to less than zero ur making lambda value imaginary:)
smile

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#11 2010-09-27 01:08:05

tharun reddy
Member
Registered: 2010-09-27
Posts: 2

Re: Cauchy–Schwarz–Bunyakovsky inequality

..iam really in need of a solution to this problem please reply soonup

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#12 2010-10-04 22:24:11

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

Re: Cauchy–Schwarz–Bunyakovsky inequality

Jane has already stopped discussing serious topics.
roll


X'(y-Xβ)=0

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#13 2010-10-04 22:40:37

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,636

Re: Cauchy–Schwarz–Bunyakovsky inequality

Hi George,Y;

George you are a good man and I like you.

Jane has done a lot for the forum.
She has solved more than her share of tough problems here.
I know you mean you miss her input, so do I.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#14 2010-11-16 12:56:45

ChiaraM27
Member
Registered: 2010-11-16
Posts: 1

Re: Cauchy–Schwarz–Bunyakovsky inequality

Basically, the roots cannot be real. Consider the quadratic equation as a parabola. If the equation has real roots then it crosses the x axis twice and has negative values between them, but we know that our quadratic function cannot be negative, so the roots have to be imaginary and so the discriminant has to be less than zero.

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#15 2010-11-26 00:46:50

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Cauchy–Schwarz–Bunyakovsky inequality

tharun reddy wrote:

...hi jane smile.........but you assumed that "lambda" was real but by setting discriminant to less than zero ur making lambda value imaginary:)
smile

swear

Last edited by JaneFairfax (2010-12-21 07:37:00)


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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