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What is a "standard duality argument"?

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Gc

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Jan 8, 2010, 5:24:29 AM1/8/10
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What is a "standard duality argument" in a real analysis? I guess that
this thing have something to do with L_p spaces and their duals?

José Carlos Santos

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Jan 8, 2010, 6:55:23 AM1/8/10
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On 08-01-2010 10:24, Gc wrote:

> What is a "standard duality argument" in a real analysis? I guess that
> this thing have something to do with L_p spaces and their duals?

It might be this: if _f_ belong to L^2 of some space then the L^2-norm
of _f_ is equal to

sup{|integral of f.g| : g in L^2, ||g||_2 = 1}.

Best regards,

Jose Carlos Santos

Gc

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Jan 8, 2010, 7:26:41 AM1/8/10
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Thanks. This is an interesting equality, I will study this more, but
now I think that "standard duality argument" just more or less refers
to the use of Hölder`s inequality :)

Ostap S. B. M. Bender Jr.

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Jan 8, 2010, 7:38:40 AM1/8/10
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Does anybody know if this is in any way related to duality in linear
programming/game theory?

Larry Hammick

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Jan 8, 2010, 3:42:25 PM1/8/10
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"Ostap S. B. M. Bender Jr."
[

Does anybody know if this is in any way related to duality in linear
programming/game theory?
]
Well yes, to the extent that the word "dual" refers to dual vector spaces in
both cases. But in infinite-dimensional spaces in with various additonal
kinds of structure (a topology for example) the word "dual" is accordingly
restricted, loosely speaking. An infinite-dimensional vector space with no
additional structure is /not/ isomorphic to its dual.


David C. Ullrich

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Jan 8, 2010, 3:47:36 PM1/8/10
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On Fri, 8 Jan 2010 02:24:29 -0800 (PST), Gc <gcu...@hotmail.com>
wrote:

>What is a "standard duality argument" in a real analysis? I guess that
>this thing have something to do with L_p spaces and their duals?

Of course this is not a precisely defined mathematical term, it's just
something one inserts to indicate to the reader how to prove
something.

I'd say "standard duality arguments" typically are applications of the
Hahn-Banach theorem.

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