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forum.math.uoa.gr • Προβολή θέματος - Θέμα Συναρτησιακής Ανάλυσης

forum.math.uoa.gr

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Ημερομηνία 22 Σεπ 2017, 02:46

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Δημιουργία νέου θέματος Απάντηση στο θέμα  [ 4 δημοσιεύσεις ] 
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 Θέμα δημοσίευσης: Θέμα Συναρτησιακής Ανάλυσης
ΔημοσίευσηΔημοσιεύτηκε: 20 Μάιος 2017, 17:14 
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Εγγραφή: 10 Μαρ 2016, 14:08
Δημοσ.: 32
Για [unparseable or potentially dangerous latex formula] θέτουμε

[unparseable or potentially dangerous latex formula]


i. Δείξτε ότι ο [unparseable or potentially dangerous latex formula] είναι γραμμικός υπόχωρος του [unparseable or potentially dangerous latex formula]

ii. Δείξτε ότι ο [unparseable or potentially dangerous latex formula] είναι κλειστός αν, και μόνο αν, [unparseable or potentially dangerous latex formula] .


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 Θέμα δημοσίευσης: Re: Θέμα Συναρτησιακής Ανάλυσης
ΔημοσίευσηΔημοσιεύτηκε: 20 Μάιος 2017, 23:31 
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Regular Forumer

Εγγραφή: 05 Φεβ 2008, 03:03
Δημοσ.: 417
Υπάρχει ένα πρόβλημα στον ορισμό του [unparseable or potentially dangerous latex formula]. Αν p>1 και[unparseable or potentially dangerous latex formula], τότε δεν είναι απαραίτητο ότι η f ανήκει στον L^1, οπότε πως ερμηνεύεται το ολοκλήρωμα της f στο (0,\infty);

Μήπως στον ορισμό του [unparseable or potentially dangerous latex formula] πρέπει η f να ανήκει στον [unparseable or potentially dangerous latex formula];

_________________
\emptyset\not=\{\emptyset\}


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 Θέμα δημοσίευσης: Re: Θέμα Συναρτησιακής Ανάλυσης
ΔημοσίευσηΔημοσιεύτηκε: 24 Μάιος 2017, 01:28 
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Regular Forumer

Εγγραφή: 05 Φεβ 2008, 03:03
Δημοσ.: 417
Έστω ότι ορίζουμε [unparseable or potentially dangerous latex formula].

Έστω [unparseable or potentially dangerous latex formula]. Τότε [unparseable or potentially dangerous latex formula], και, αν p>1, [unparseable or potentially dangerous latex formula], επομένως f_n\to 0 στον L^p.

Θέτουμε [unparseable or potentially dangerous latex formula] για x>0 και [unparseable or potentially dangerous latex formula]. Τότε [unparseable or potentially dangerous latex formula] για κάθε n\in\mathbb N, και [unparseable or potentially dangerous latex formula]. Άρα [unparseable or potentially dangerous latex formula]. Όμως, [unparseable or potentially dangerous latex formula] στον L^p, και [unparseable or potentially dangerous latex formula], επομένως ο [unparseable or potentially dangerous latex formula] δεν είναι κλειστός.

_________________
\emptyset\not=\{\emptyset\}


Κορυφή
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 Θέμα δημοσίευσης: Re: Θέμα Συναρτησιακής Ανάλυσης
ΔημοσίευσηΔημοσιεύτηκε: 24 Μάιος 2017, 01:49 
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Regular Forumer

Εγγραφή: 05 Φεβ 2008, 03:03
Δημοσ.: 417
Χρησιμοποιώντας την f_n παραπάνω μπορούμε να δείξουμε ότι ο [unparseable or potentially dangerous latex formula] είναι πυκνός στον L^p για κάθε p>1: αν η [unparseable or potentially dangerous latex formula] έχει φορέα στο [0,M], θέτουμε [unparseable or potentially dangerous latex formula]. Τότε η g_n έχει φορέα στο [unparseable or potentially dangerous latex formula] και ανήκει στον L^p, οπότε ανήκει στον L^1. Επίσης, [unparseable or potentially dangerous latex formula], άρα [unparseable or potentially dangerous latex formula].

Επιπλέον, [unparseable or potentially dangerous latex formula], άρα [unparseable or potentially dangerous latex formula]. Όμως, από την ανισότητα του Holder, [unparseable or potentially dangerous latex formula], άρα [unparseable or potentially dangerous latex formula].

Επομένως, έχουμε δείξει οτι ο [unparseable or potentially dangerous latex formula] είναι πυκνός στο χώρο των συναρτήσεων στον L^p με συμπαγή φορέα. Ο τελευταίος χώρος όμως είναι πυκνός στον L^p, άρα ο [unparseable or potentially dangerous latex formula] είναι πυκνός στον L^p.

_________________
\emptyset\not=\{\emptyset\}


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