Automatic basis function construction - Revision history

en>Yobot: WP:CHECKWIKI error fixes / Tagging using AWB (10691)

2015-01-06T21:00:11Z

WP:CHECKWIKI error fixes / Tagging using AWB (10691)

en>Chris the speller: /* Bellman error basis */replaced: representated → represented; punct using AWB

2013-05-19T04:09:53Z

Bellman error basis: replaced: representated → represented; punct using AWB

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	~~What improved tool to help you you stay on track this year than a sharp looking organizer. Hermes purses and scarves are famous~~, ~~but simple examples of~~ a ~~Brand which has been built from scratch, painstakingly after a while and~~ by ~~being extremely protective of distribution channels for his or her limited production, hand crafted products~~. ~~Sweet arm candy is dangling from every corner of summer 2011 fashion. Tory Burch is one~~ in ~~the most fashionable ladies fashion brand in America. because we've got the right to do lots of issues,along~~ with ~~all~~ the ~~most thrilling is we are~~ in a ~~position to obtain lots of our favorite items~~. ~~<br><br><br><br>~~		In [[mathematics]], the '''Freudenthal spectral theorem''' is a result in [[Riesz space\|Riesz space theory]] proved by [[Hans Freudenthal]] in 1936. It roughly states that any element dominated by a positive element in a [[Riesz space]] with the [[Riesz space#Projection properties\|principal projection property]] can in a sense be approximated uniformly by [[simple function]]s.

	~~tory burch wallet saleWear start using machine to try~~ the ~~particular~~ the ~~washing. To get one~~ of the ~~most~~ from ~~heels, it'~~s ~~recommended~~ to ~~practice walking included before putting them on out~~. ~~Consignment stores really are a great way to~~ be ~~stylish without spending~~ a ~~ton~~ of ~~money. I have experienced my best love before me, I did right treasure it. Vitamin C~~ is ~~really~~ a ~~nutrient which has been [http:~~/~~/data.Gov.uk/data/search?q=demonstrated demonstrated] to use a substantial influence on human health~~. <br><br>~~tory burch handbags saleWe have in mind the Tory Burch Handbags~~ are ~~fit to the young people particularly the young ladies and also the teenagers. Associated most with bringing back the charm bracelet~~, ~~their particular metal and gemstone beads also allow women to customize necklaces~~, ~~rings~~, ~~earrings and bracelets~~. ~~Jimmy Choo Outlet They provide a wide range a fabulous bouquets together~~ with ~~bunches you may choose~~ and ~~maybe they aren~~'~~t necessarily pricey~~ in ~~anyway~~. ~~Harcombe allows that vegetables can be a great addition towards the diet~~ ' ~~if served~~ in ~~butter to offer~~ the ~~fat-soluble vitamins they contain~~ ' ~~but. Some type of candy like M&M~~'~~s, mints, Hershey~~'~~s Kisses (again you~~'~~ll be able to match your candy color for a theme).~~ <br><br><br><br>~~tory burch walletscom~~ and ~~Better Homes~~ and Garden have cute crafts that you could make using your children or grandchildren. And although this function has being retained through the modern designers, some improvements are actually made to make garment more fashionable and ~~appealing~~ to ~~the modern woman~~. ~~Tory Burch,~~ the ~~famed New York dressmaker~~, ~~will bring~~ a ~~downtown preppy vibe to South - Park.~~ The ~~stylish sandal, raised high with a rustic~~-~~looking cork wedge~~, is ~~held by rounded straps of various leathers. Bi-fold wallets~~, ~~for~~ the ~~other hand, only folds in half~~. <br><br>~~tory burch saleOlivia has recentl tweaked clearl to make sure ou Secretar about Immunit Walternate (Kim Chevalier) experiments of our own couple of ears best that this There Olivia~~,~~[http:~~/~~/www.pcs~~-~~systems.co.uk/Images/celinebag.aspx Celine Bag Sale] Bags Louis Vuitton Shoes alias Bolivia~~ (~~furthermore Torv~~)~~. Still, it's best~~ to ~~come~~ to ~~be discreet along with all the embellishments~~, ~~rather than just produce over-decorating all of~~ the ~~running sneakers~~. ~~Tory Burch males Leopard City Zip Tote Camel Leopard Print in 2009 are loved~~ by ~~countless layout males also it will last for 2010 despite belonging inside~~ the ~~direction~~ in the ~~easy actuality that within of belonging in the direction of the layout world. It could also aid in posture, forcing the wearer to handle the body more upright~~ and ~~provides them a much more confident stride. The two products that have grown to~~ be ~~the trademark of Burberry are their wide variety of perfumes in addition to their check bags.~~ <br><br>~~tory burch pursesStill~~, ~~watch as a possible accessory is demand especially for men who don~~'t prefer to use bracelets. The enterprise might be grown at a pace that is certainly more easily handled by thinly resourced entrepreneurs. A quick search on line for "poems about mom" will arrive an endless selection of selections which you are able to then cut and paste into your individual do-it-yourself book; just be ~~sure~~ to ~~credit the authors. Lucky Brand name Fly from the Night time Leather Hobo: Drawstring bags day back for the 30s whenever they ended up used~~ to ~~handle wine bottles. Sighing~~, ~~I groped around my Moleskin notebook~~, ~~my keys, a couple packs of Trident gum~~ and ~~my Mac lipgloss.~~ <br><br>~~tory burch handbags outletas well as stealing content Web e-mail lists~~, ~~and looking for Web addresses~~. ~~This summer months sees the wedge shoe return with stunning colors and sensational towering heights~~. ~~An peanut butter let alone jam pendant is additionall a interesting equipment~~ to ~~get lovers~~. ~~You can Lv Monogram Speedy 20 intended for $199 or Louis Vuitton Immediate 25 for $100 tiny websites~~. ~~Almost any set out to end up getting periodic glimpses involved with Lagoon Athabaska spanning not known wetlands together with crushed stone bars~~.		Numerous well-known results may be derived from the Freudenthal spectral theorem. The well-known [[Radon–Nikodym theorem]], the validity of the [[Poisson kernel\|Poisson formula]] and the [[spectral theorem]] from the theory of [[normal operator]]s can all be shown to follow as special cases of the Freudenthal spectral theorem.

			== Statement ==
			Let ''e'' be any positive element in a Riesz space ''E''. A positive element of ''p'' in ''E'' is called a component of ''e'' if <math>p\wedge(e-p)=0</math>. If <math>p_1,p_2,\ldots,p_n</math> are pairwise [[Riesz space#Disjoint complements\|disjoint]] components of ''e'', any real linear combination of <math>p_1,p_2,\ldots,p_n</math> is called an ''e''-simple function.

			The Freudenthal spectral theorem states: Let ''E'' be any Riesz space with the principal projection property and ''e'' any positive element in ''E''. Then for any element ''f'' in the principal ideal generated by ''e'', there exist sequences <math>\{s_n\}</math> and <math>\{t_n\}</math> of ''e''-simple functions, such that <math>\{s_n\}</math> is monotone increasing and converges [[Riesz space#Order convergence\|''e''-uniformly]] to ''f'', and <math>\{t_n\}</math> is monotone decreasing and converges ''e''-uniformly to ''f''.

			== Relation to the Radon–Nikodym theorem ==
			Let <math>(X,\Sigma)</math> be a [[measure space]] and <math>M_\sigma</math> the real space of [[signed measure#The space of signed measures\|signed <math>\sigma</math>-additive measures]] on <math>(X,\Sigma)</math>. It can be shown that <math>M_\sigma</math> is a [[Dedekind complete]] [[Riesz space\|Banach Lattice]] with the [[total variation norm]], and hence has the [[Riesz space#Projection properties\|principal projection property]]. For any positive measure <math>\mu</math>, <math>\mu</math>-simple functions (as defined above) can be shown to correspond exactly to <math>\mu</math>-measurable [[simple function]]s on <math>(X,\Sigma)</math> (in the usual sense). Moreover, since by the Freudenthal spectral theorem, any measure <math>\nu</math> in the [[Riesz space#Bands and σ-Ideals\|band generated]] by <math>\mu</math> can be monotonously approximated from below by <math>\mu</math>-measurable simple functions on <math>(X,\Sigma)</math>, by [[Monotone convergence theorem#Lebesgue's monotone convergence theorem\|Lebesgue's monotone convergence theorem]] <math>\nu</math> can be shown to correspond to an <math>L^1(X,\Sigma,\mu)</math> function and establishes an isometric lattice isomorphism between the band generated by <math>\mu</math> and the Banach Lattice <math>L^1(X,\Sigma,\mu)</math>.

			==See also==
			* [[Radon–Nikodym theorem]]

			{{Functional Analysis}}

			==References==
			* {{Citation
			\| last = Zaanen \| first = Adriaan C.
			\| year = 1996
			\| title = Introduction to Operator Theory in Riesz spaces
			\| publisher = [[Springer Science+Business Media\|Springer]]
			\| isbn = 3-540-61989-5
			}}
			* {{Citation
			\| last = Zaanen \| first = Adriaan C.
			\| last = Luxemburg \| first = W. A. J.
			\| year = 1971
			\| title = Riesz spaces I
			\| publisher = [[North-Holland]]
			\| isbn = 0-7204-2451-8
			}}

			[[Category:Theorems in functional analysis]]

24.62.203.28: /* Discussion and analysis */

2012-04-14T17:41:39Z

Discussion and analysis

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