/Banach Space General/

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Anonymous

/Banach Space General/ 2016-01-14 10:55:49 Post No. 7782879

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/Banach Space General/ 2016-01-14 10:55:49 Post No. 7782879

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ITT: Discuss anything related to banach spaces or functional analysis in general.

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>>7782879

>lel, i poasted it again!

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>>7782879

How is linear algebra and real analysis is needed to functional analysis?

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>>7782906

Because functional analysis is the study of infinite dimensional vector spaces, many interesting examples of which are function spaces

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>>7782906

Linear algebra is the study of linear objects (vectors) and their transformations. Real analysis is, roughly speaking, the study of infinite real-valued sequences/series and their (linear) transformations. Differentiation and integration are linear but linear algebra is poorly equipped to handle infinite dimensional linear transformations. Boom, functional analysis.

Obligatory call for suicide: kill yourself, my man

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>>7782926

>>7782928

Can functional analysis literally past triple integrals, i.e. surpass three dimensions?

Is there any area of functional analysis of where I can deal with multiple integrals; preferably past four dimensions.

In some physics textbook, five, or six integrals all lined up. I hope it would be the same for functional analysis.

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>>7782932

You meme-loving fuck.

It's applying Fubini's theorem. Rinse and repeat.

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>>7782945

I'm actually serious.

The third sentence, I made sounded like a meme, but it wasn't on purpose. I know that meme is going around, I've literally meant if there's ever an oppourtunity I could deal with nth blah blah dimensional integral? I worked with a 9 fold integral in quantum mechanics I was just wondering if its possible to toy with over four fold integrals in functional analysis; that's all.

Sorry what I was saying may have sounded like a meme, but it wasn't trust me.

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>>7782945

Can you also use something like Fubini if you need to integrate functions that have an infinite dimensional vector space as domain?

For example consider [math]f: l^1 \to \mathbb{R}[/math] with

[math] f((x)) = \sum_{n=1}^\infty |x_n|^n[/math] and [math]A = \{f \in l^1 : \|f\|_{l^1} < 1 \} [/math]

How would someone calculate

[eqn] \int_{A} f(x) dx [/eqn]

?

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>>7782990

Or rather

[math] A = \{x \in l^1 : \|x\|_{l^1} < 1 \} [/math]

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>>7782950

Well there is the Riesz representation theorem that tells you that any positive functional on the space of compactly supported continuous fonctions on a locally compact space X defined a Borel measure on X so nothing stops you from defining integration on arbitrary dimensional spaces (and even spaces that don't have any sort of dimension)

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>>7782990

check your integral again, it doesn't make sense.

x has to go through a subdomain of f's domain

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>>7783000

[math] A \subset l^1 [/math]

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>>7782990

Nah, at least not using the usual Fubini (that tells you about integrating on a finite product of spaces)

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>>7783004

in the integral, you say x goes over A.

And then you use f(x), so is f a function that takes other functions as arguments?

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>>7782999

Thank you, dude.

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what is a hilbert space. wikipedia is shit and im too poor for textbooks

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>>7783095

what's wrong with wikipedia?

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>>7783098

bad at explaining hilbert spaces I mean

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>>7783102

A hilbert space is a complete inner product space.

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>>7783095

Banach Space with an inner product

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>>7783095

>wikipedia

no. go get an actual book. don't google shit just so you can pretend you know what it is.

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>>7783143

Specifically a Banach space whose metric comes from an inner product.

http://www.matapp.unimib.it/~pini/teoria%20misura/Rudin(Real_Complex_Analysis).pdf

http://ruangbacafmipa.staff.ub.ac.id/files/2012/02/Real-and-Complex-Analysis-by-Walter-Rudin.pdf

Here are some text books (actually the same book twice) that goes into detail about the differences and similarities.

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>>7783179

I don't want to use Rudin

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>>7782990

You can construct measures on function spaces for example via stochastic processes. However these measures will not behave like Lebesgue measure. Also look up "path integrals" and the mathematics behind them.

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>>7783256

>With functional analysis you can take an infinite number of integrals by integrating on a function space

and thus was born the fourier transform

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>>7783095

>tfw directly related to David Hilbert

>post-grad

>everyone expects you to be the next memestein

feelsbadman

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What sort of jobs can functional analysts could get?

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>>7784395

Doing math for quantum mechanics.

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>>7783307

>>directly related to David Hilbert

Can I suck your dick?

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>>7783307

tell them that there is nothing to be proud of

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>>7784533

Watch out now, I bet it's infinite dicks in an infinite hotel full of dicks

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>>7783285

Nah bro. Fourier transform is only an integral operator. I meant functional integration.

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>>7783307

Ask those people if they even know what Hilber space is.

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>>7785400

>functional integration

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