Funkcionalna analiza

Funkcionalna analiza je grana matematičke analize, čije jezgro je formirano proučavanjem vektorskih prostora koji imaju neku vrstu granične strukture (npr. unutrašnji proizvod, normu, topologiju, itd.) i linearne funkcije definisane na tim prostorima i poštovanje tih struktura u odgovarajućem smislu. Istorijski koreni funkcionalne analize leže u proučavanju prostora funkcija, i u formulaciji svojstava transformacija funkcija, poput Furijeove transformacije, kao transformacije koja definiše kontinualne, unitarne i slične operatore između funkcijskih prostora. Ova tačka gledišta pokazala se posebno korisnom za proučavanje diferencijalnih i integralnih jednačina.

Upotreba reči funkcionalna potiče iz računa varijacija, implicirajući funkciju čiji je argument funkcija. Termin je prvi put korišten u Adamarovoj knjizi iz 1910. godine o toj temi. Međutim, generalni koncept funkcionalnog je ranije uveo, 1887. godine, italijanski matematičar i fizičar Vito Voltera.^[1]^[2] Teoriju nelinearnih funkcija nastavili su njegovi učenici, a posebno Freše i Levi. Adamar je osnovao i modernu školu linearne funkcionalne analize, koju su dalje razvili Ris i grupa poljskih matematičara oko Stefana Banaha.^[3]^[4]^[5]

U savremenim uvodnim tekstovima funkcionalne analize, subjekt se posmatra kao proučavanje vektorskih prostora koji imaju topologiju, posebno beskonačno-dimenzionalnih prostora.^[6]^[7] Nasuprot tome, linearna algebra se uglavnom bavi konačno-dimenzionalnim prostorima, i ne koristi topologiju. Važan deo funkcionalne analize je proširenje teorije mere, integracije i verovatnoće na beskonačno dimenzionalne prostore, takođe poznate kao beskonačno dimenzionalna analiza.

Normirani vektorski prostori

Osnovna i istorijski prva klasa prostora proučavana u funkcionalnoj analizi su potpuno normirani vektorski prostori nad realnim ili kompleksnim brojevima. Takvi prostori se zovu Banahovi prostori. Važan primer je Hilbertov prostor, gde norma nastaje iz unutrašnjeg proizvoda. Ovi prostori su od fundamentalnog značaja u mnogim oblastima, uključujući matematičku formulaciju kvantne mehanike.^[8]^[9]^[10]

Generalnije gledano, funkcionalna analiza obuhvata proučavanje Frešeovih prostora^[11]^[12] i drugih topoloških vektorskih prostora koji nemaju normu. Važan predmet istraživanja u funkcionalnoj analizi su kontinuirani linearni operatori definisani na Banahovim i Hilbertovim prostorima. Oni prirodno dovode do definicije C*-algebre^[13]^[14]^[15]

i drugih operatorskih algebri.^[16]^[17]

Vidi još

Reference

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^ . Grattan-Guiness, Ivor (15. 12. 2003). History of the Mathematical Sciences. Springer. стр. 195. ISBN 978-93-86279-16-3.
^ Jahnke, Hans Niels (2003). A History of Analysis. American Mathematical Society. ISBN 0821826239.
^ Jakimowicz, E.; Miranowicz, A., ур. (2007). Stefan Banach - Remarkable life, Brilliant mathematics. Gdańsk University Press and Adam Mickiewicz University Press. ISBN 978-83-7326-451-9.
^ James, Ioan (2003). Remarkable Mathematicians: From Euler to von Neumann. Cambridge University Press. ISBN 0521520940.
^ Bowers, Adam; Kalton, Nigel J. (2014). An introductory course in functional analysis. Springer Science & Business Media. стр. 1.
^ Kadets, Vladimir (2018). A Course in Functional Analysis and Measure Theory [КУРС ФУНКЦИОНАЛЬНОГО АНАЛИЗА]. Springer. стр. xvi.
^ Dirac, P. A. M. (1925). „The Fundamental Equations of Quantum Mechanics”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 109 (752): 642—653. Bibcode:1925RSPSA.109..642D. doi:10.1098/rspa.1925.0150 .
^ Cohen-Tannoudji, Claude (2019). Quantum mechanics. Volume 2. Bernard Diu, Franck Laloë, Susan Reid Hemley, Nicole Ostrowsky, D. B. Ostrowsky. Weinheim. ISBN 978-3-527-82272-0. OCLC 1159410161.
^ Carcassi, Gabriele; Maccone, Lorenzo; Aidala, Christine A. (2021-03-16). „Four Postulates of Quantum Mechanics Are Three”. Physical Review Letters. 126 (11): 110402. Bibcode:2021PhRvL.126k0402C. PMID 33798366. S2CID 214623241. arXiv:2003.11007 . doi:10.1103/PhysRevLett.126.110402.
^ Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
^ Hazewinkel Michiel, ур. (2001). „Fréchet space”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104.
^ Doran & Belfi 1986, стр. 5–6, Google Books.
^ Doran & Belfi 1986, стр. 6, Google Books.
^ Segal 1947
^ Blackadar, Bruce (2005). Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. Encyclopaedia of Mathematical Sciences. Springer-Verlag. ISBN 3-540-28486-9.
^ M. Takesaki, Theory of Operator Algebras I, Springer, 2001.

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