Funkcionalna analiza — разлика између измена

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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

^ acsu.buffalo.edu
^ History of the Mathematical Sciences. ISBN 978-93-86279-16-3. стр. 195.
^ 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. 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.

Literatura

Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker's Guide, , Springer 2007. (3rd изд.). ISBN 978-3-540-32696-0. . Online doi:10.1007/3-540-29587-9 (by subscription)
Bachman, G., Narici, L.: Functional analysis, Academic Press, 1966. (reprint Dover Publications)
Banach S. Theory of Linear Operations Архивирано на сајту Wayback Machine (28. октобар 2021). Volume 38, North-Holland Mathematical Library. 1987. ISBN 978-0-444-70184-8.
Brezis, H.: Analyse Fonctionnelle, Dunod. ISBN 978-2-10-004314-9. or. ISBN 978-2-10-049336-4.
Conway, J. B.: A Course in Functional Analysis, , Springer-Verlag. (2nd изд.). 1994. ISBN 978-0-387-97245-9.
Dunford, N. and Schwartz, J.T.: Linear Operators, General Theory, John Wiley & Sons, and other 3 volumes, includes visualization charts
Edwards, R. E.: Functional Analysis, Theory and Applications, Hold, Rinehart and Winston, 1965.
Eidelman, Yuli, Vitali Milman, and Antonis Tsolomitis: Functional Analysis: An Introduction, American Mathematical Society, 2004.
Friedman, A.: Foundations of Modern Analysis, Dover Publications, Paperback Edition, July 21, 2010
Giles,J.R.: Introduction to the Analysis of Normed Linear Spaces,Cambridge University Press,2000
Hirsch F., Lacombe G. - "Elements of Functional Analysis", Springer 1999.
Hutson, V., Pym, J.S., Cloud M.J.: Applications of Functional Analysis and Operator Theory, , Elsevier Science. (2nd изд.). 2005. ISBN 978-0-444-51790-6.
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Reed, M., Simon, B.: "Functional Analysis", Academic Press 1980.
Riesz, F. and Sz.-Nagy, B.: Functional Analysis, Dover Publications, 1990
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Saxe, Karen: Beginning Functional Analysis, Springer, 2001
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Shilov, Georgi E.: Elementary Functional Analysis, Dover, 1996.
Sobolev, S.L.: Applications of Functional Analysis in Mathematical Physics, AMS, 1963
Vogt, D., Meise, R.: Introduction to Functional Analysis, Oxford University Press, 1997.
Yosida, K.: Functional Analysis, Springer-Verlag, 6th edition, 1980
Arveson, W. (1976), An Invitation to C*-Algebra, Springer-Verlag, ISBN 0-387-90176-0 . An excellent introduction to the subject, accessible for those with a knowledge of basic functional analysis.
Connes, Alain (1994), Non-commutative geometry, ISBN 0-12-185860-X . This book is widely regarded as a source of new research material, providing much supporting intuition, but it is difficult.
Dixmier, Jacques (1969), Les C*-algèbres et leurs représentations, Gauthier-Villars, ISBN 0-7204-0762-1 . This is a somewhat dated reference, but is still considered as a high-quality technical exposition. It is available in English from North Holland press.
Doran, Robert S.; Belfi, Victor A. (1986), Characterizations of C*-algebras: The Gelfand-Naimark Theorems, CRC Press, ISBN 978-0-8247-7569-8 .
Emch, G. (1972), Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, ISBN 0-471-23900-3 . Mathematically rigorous reference which provides extensive physics background.
A.I. Shtern (2001). „C*-algebra”. Ур.: Hazewinkel Michiel. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104.
Sakai, S. (1971), C*-algebras and W*-algebras, Springer, ISBN 3-540-63633-1 .
Segal, Irving (1947), „Irreducible representations of operator algebras”, Bulletin of the American Mathematical Society, 53 (2): 73—88, doi:10.1090/S0002-9904-1947-08742-5  .
Kałuża, Roman (1996). Through a Reporter's Eyes: The Life of Stefan Banach. Translated by Wojbor Andrzej Woyczyński and Ann Kostant. Birkhäuser. ISBN 0-8176-3772-9.
Kosiedowski, Stanisław. „Stefan Banach”. Mój Lwów. Приступљено 20. 5. 2008.
O'Connor, John J.; Robertson, Edmund F. (2000). „Stefan Banach”. MacTutor History of Mathematics archive. University of St. Andrews. Приступљено 19. 8. 2012.
Siegmund-Schultze, Reinhard (2003). Jahnke, Hans Niels, ур. A History of Analysis. American Mathematical Society. ISBN 0-8218-2623-9.
MacCluer, Barbara (2008). Elementary Functional Analysis. Springer. ISBN 0387855289.
Urbanek, Mariusz (april 2002). „Geniusz: gen i już”. Polityka. 8 (2348).
Waksmundzka-Hajnos, Monika (2006). „Wspomnienie o Stefanie Greczku”. Focus. Gdańsk University (11).
Banach, Stefan (1932), Théorie des opérations linéaires (PDF), Monografie Matematyczne, 1, Warszawa: Subwencji Funduszu Kultury Narodowej, Zbl 0005.20901, Архивирано из оригинала (PDF) 11. 1. 2014. г., Приступљено 10. 6. 2016
Beauzamy, Bernard (1985) [1982], Introduction to Banach Spaces and their Geometry (Second revised изд.), North-Holland
Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin: Springer-Verlag, ISBN 978-3-540-13627-9
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Dunford, Nelson; Schwartz, Jacob T. with the assistance of W. G. Bade and R. G. Bartle (1958), Linear Operators. I. General Theory, Pure and Applied Mathematics, 7, New York: Interscience Publishers, Inc., MR 0117523
Lindenstrauss, Joram; Tzafriri, Lior (1977), Classical Banach Spaces I, Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 92, Berlin: Springer-Verlag, ISBN 3-540-08072-4
Megginson, Robert E. (1998), An introduction to Banach space theory, Graduate Texts in Mathematics, 183, New York: Springer-Verlag, стр. xx+596, ISBN 0-387-98431-3
Ryan, Raymond A. (2002), Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics, London: Springer-Verlag, стр. xiv+225, ISBN 1-85233-437-1
Wojtaszczyk, Przemysław (1991), Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, 25, Cambridge: Cambridge University Press, стр. xiv+382, ISBN 0-521-35618-0

Spoljašnje veze

Hazewinkel Michiel, ур. (2001). „Functional analysis”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104.
Topics in Real and Functional Analysis by Gerald Teschl, University of Vienna.
Lecture Notes on Functional Analysis by Yevgeny Vilensky, New York University.
Lecture videos on functional analysis by Greg Morrow Архивирано на сајту Wayback Machine (1. април 2017) from University of Colorado Colorado Springs
An Introduction to Functional Analysis on Coursera by John Cagnol from Ecole Centrale Paris

[1] acsu.buffalo.edu

[2] History of the Mathematical Sciences. ISBN 978-93-86279-16-3. стр. 195.

[3] Jahnke, Hans Niels (2003). A History of Analysis. American Mathematical Society. ISBN 0821826239.

[4] 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.

[5] James, Ioan (2003). Remarkable Mathematicians: From Euler to von Neumann. Cambridge University Press. ISBN 0521520940.

[6] Bowers, Adam; Kalton, Nigel J. (2014). An introductory course in functional analysis. Springer Science & Business Media. стр. 1.

[7] Kadets, Vladimir (2018). A Course in Functional Analysis and Measure Theory [КУРС ФУНКЦИОНАЛЬНОГО АНАЛИЗА]. Springer. стр. xvi.

[8] 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 .

[9] 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.

[10] Carcassi, Gabriele; Maccone, Lorenzo; Aidala, Christine A. (2021-03-16). „Four Postulates of Quantum Mechanics Are Three”. Physical Review Letters. 126 (11): 110402. S2CID 214623241. arXiv:2003.11007 . doi:10.1103/PhysRevLett.126.110402.

[11] 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.

[12] Hazewinkel Michiel, ур. (2001). „Fréchet space”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104.

[13] Doran & Belfi 1986, стр. 5–6, Google Books.

[14] Doran & Belfi 1986, стр. 6, Google Books.

[15] Segal 1947

[16] Blackadar, Bruce (2005). Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. Encyclopaedia of Mathematical Sciences. Springer-Verlag. ISBN 3-540-28486-9.

[17] M. Takesaki, Theory of Operator Algebras I, Springer, 2001.

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@@ Ред 4: / Ред 4: @@
 '''Funkcionalna analiza''' je grana [[математичка анализа|matematičke analize]], čije jezgro je formirano proučavanjem [[векторски простор|vektorskih prostora]] koji imaju neku vrstu granične strukture (npr. [[Inner product space#Definition|unutrašnji proizvod]], [[Norm (mathematics)#Definition|normu]], [[тополошки простор#Definition|topologiju]], itd.) i [[линеарно пресликавање|linearne funkcije]] definisane na tim prostorima i poštovanje tih struktura u odgovarajućem smislu. Istorijski koreni funkcionalne analize leže u proučavanju [[function space|prostora funkcija]], i u formulaciji svojstava transformacija funkcija, poput [[Furijeova transformacija|Furijeove transformacije]], kao transformacije koja definiše [[Непрекидна функција|kontinualne]], [[unitary operator|unitarne]] i slične operatore između funkcijskih prostora. Ova tačka gledišta pokazala se posebno korisnom za proučavanje [[диференцијална једначина|diferencijalnih]] i [[Интегрална једначина|integralnih jednačina]].
-Upotreba reči ''[[Функционал|funkcionalna]]'' potiče iz [[varijacijski račun|računa varijacija]], implicirajući [[Higher-order function|funkciju čiji je argument funkcija]]. Termin je prvi put korišten u [[Жак Адамар|Adamarovoj]] knjizi iz 1910. godine o toj temi. Međutim, generalni koncept ''funkcionalnog'' je ranije uveo, 1887. godine, italijanski matematičar i fizičar [[Vito Volterra|Vito Voltera]].<ref>[http://www.acsu.buffalo.edu/~wlawvere/Volterra.pdf ''acsu.buffalo.edu'']</ref><ref>''History of the Mathematical Sciences''. {{page|year=|isbn=978-93-86279-16-3|pages=195}}</ref> Teoriju nelinearnih funkcija nastavili su njegovi učenici, a posebno [[Maurice René Fréchet|Freše]] i [[Paul Lévy (mathematician)|Levi]]. Adamar je osnovao i modernu školu linearne funkcionalne analize, koju su dalje razvili [[Frigyes Riesz|Ris]] i [[Lwów School of Mathematics|grupa]] [[пољска|poljskih]] matematičara oko [[Stefan Banach|Stefana Banaha]].
+Upotreba reči ''[[Функционал|funkcionalna]]'' potiče iz [[varijacijski račun|računa varijacija]], implicirajući [[Higher-order function|funkciju čiji je argument funkcija]]. Termin je prvi put korišten u [[Жак Адамар|Adamarovoj]] knjizi iz 1910. godine o toj temi. Međutim, generalni koncept ''funkcionalnog'' je ranije uveo, 1887. godine, italijanski matematičar i fizičar [[Vito Volterra|Vito Voltera]].<ref>[http://www.acsu.buffalo.edu/~wlawvere/Volterra.pdf ''acsu.buffalo.edu'']</ref><ref>''History of the Mathematical Sciences''. {{page|year=|isbn=978-93-86279-16-3|pages=195}}</ref> Teoriju nelinearnih funkcija nastavili su njegovi učenici, a posebno [[Maurice René Fréchet|Freše]] i [[Paul Lévy (mathematician)|Levi]]. Adamar je osnovao i modernu školu linearne funkcionalne analize, koju su dalje razvili [[Frigyes Riesz|Ris]] i [[Lwów School of Mathematics|grupa]] [[пољска|poljskih]] matematičara oko [[Stefan Banach|Stefana Banaha]].<ref>{{cite book | title=A History of Analysis | publisher=American Mathematical Society | author=Jahnke, Hans Niels | year=2003 | isbn=0821826239}}</ref><ref>{{Cite book|editor1-first=E.|editor1-last=Jakimowicz|editor2-first=A.|editor2-last=Miranowicz |title=Stefan Banach - Remarkable life, Brilliant mathematics |publisher=Gdańsk University Press and Adam Mickiewicz University Press |year=2007 |isbn=978-83-7326-451-9 |url =http://kielich.amu.edu.pl/Stefan_Banach/e-jakimowicz_miranowicz.html |ref=harv}}</ref><ref>{{cite book | title=Remarkable Mathematicians: From Euler to von Neumann | url=https://archive.org/details/remarkablemathem0000jame | url-access=registration | publisher=Cambridge University Press | author=[[Ioan James|James, Ioan]] | year=2003 | isbn=0521520940}}</ref>
-U savremenim uvodnim tekstovima funkcionalne analize, subjekt se posmatra kao proučavanje vektorskih prostora koji imaju topologiju, posebno [[Димензија векторског простора|beskonačno-dimenzionalnih prostora]]. Nasuprot tome, [[linearna algebra]] se uglavnom bavi konačno-dimenzionalnim prostorima, i ne koristi topologiju. Važan deo funkcionalne analize je proširenje teorije [[Мера (математика)|mere]], [[integral|integracije]] i [[вероватноћа|verovatnoc]]́e na beskonačno dimenzionalne prostore, takođe poznate kao '''beskonačno dimenzionalna analiza'''.
+U savremenim uvodnim tekstovima funkcionalne analize, subjekt se posmatra kao proučavanje vektorskih prostora koji imaju topologiju, posebno [[Димензија векторског простора|beskonačno-dimenzionalnih prostora]].<ref>{{Cite book| last1=Bowers|first1=Adam|title=An introductory course in functional analysis|last2=Kalton|first2=Nigel J.| publisher=[[Springer Science & Business Media]]|year=2014|pages=1}}</ref><ref>{{Cite book| last=Kadets| first=Vladimir| title=A Course in Functional Analysis and Measure Theory|publisher=[[Springer Publishing|Springer]] | year=2018|pages=xvi|trans-title=КУРС ФУНКЦИОНАЛЬНОГО АНАЛИЗА}}</ref> Nasuprot tome, [[linearna algebra]] se uglavnom bavi konačno-dimenzionalnim prostorima, i ne koristi topologiju. Važan deo funkcionalne analize je proširenje teorije [[Мера (математика)|mere]], [[integral|integracije]] i [[вероватноћа|verovatnoc]]́e na beskonačno dimenzionalne prostore, takođe poznate kao '''beskonačno dimenzionalna analiza'''.
 == Normirani vektorski prostori ==
-Osnovna i istorijski prva klasa prostora proučavana u funkcionalnoj analizi su [[complete space|potpuno]] [[normed vector space|normirani vektorski prostori]] nad [[реалан број|realnim]] ili [[комплексан број|kompleksnim brojevima]]. Takvi prostori se zovu [[Banach space|Banahovi prostori]]. Važan primer je [[Hilbertov prostor]], gde norma nastaje iz unutrašnjeg proizvoda. Ovi prostori su od fundamentalnog značaja u mnogim oblastima, uključujući [[mathematical formulation of quantum mechanics|matematičku formulaciju kvantne mehanike]].
+Osnovna i istorijski prva klasa prostora proučavana u funkcionalnoj analizi su [[complete space|potpuno]] [[normed vector space|normirani vektorski prostori]] nad [[реалан број|realnim]] ili [[комплексан број|kompleksnim brojevima]]. Takvi prostori se zovu [[Banach space|Banahovi prostori]]. Važan primer je [[Hilbertov prostor]], gde norma nastaje iz unutrašnjeg proizvoda. Ovi prostori su od fundamentalnog značaja u mnogim oblastima, uključujući [[mathematical formulation of quantum mechanics|matematičku formulaciju kvantne mehanike]].<ref>{{Cite journal | last1 = Dirac | first1 = P. A. M. | title = The Fundamental Equations of Quantum Mechanics | doi = 10.1098/rspa.1925.0150 | journal = Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | volume = 109 | issue = 752 | pages = 642–653 | year = 1925 |bibcode = 1925RSPSA.109..642D | doi-access = free }}</ref><ref>{{Cite book |last=Cohen-Tannoudji |first=Claude |url=https://www.worldcat.org/oclc/1159410161 |title=Quantum mechanics. Volume 2 |date=2019 |others=Bernard Diu, Franck Laloë, Susan Reid Hemley, Nicole Ostrowsky, D. B. Ostrowsky |isbn=978-3-527-82272-0 |location=Weinheim |oclc=1159410161}}</ref><ref>{{Cite journal |last1=Carcassi |first1=Gabriele |last2=Maccone |first2=Lorenzo |last3=Aidala |first3=Christine A. |date=2021-03-16 |title=Four Postulates of Quantum Mechanics Are Three |url=https://link.aps.org/doi/10.1103/PhysRevLett.126.110402 |journal=Physical Review Letters |volume=126 |issue=11 |pages=110402 |doi=10.1103/PhysRevLett.126.110402|pmid=  |arxiv=2003.11007 |s2cid=214623241 }}</ref>
-Generalnije gledano, funkcionalna analiza obuhvata proučavanje [[Fréchet space|Frešeovih prostora]] i drugih [[topological vector space|topoloških vektorskih prostora]] koji nemaju normu. Važan predmet istraživanja u funkcionalnoj analizi su [[Непрекидна функција|kontinuirani]] [[Линеарно пресликавање|linearni operatori]] definisani na Banahovim i Hilbertovim prostorima. Oni prirodno dovode do definicije [[C*-algebra|C*-algebre]] i drugih [[operator algebra|operatorskih algebri]].
+Generalnije gledano, funkcionalna analiza obuhvata proučavanje [[Fréchet space|Frešeovih prostora]]<ref>{{cite book | last=Berberian | first=Sterling K. | title=Lectures in Functional Analysis and Operator Theory | publisher=Springer | location=New York | year=1974 | series=Graduate Texts in Mathematics | volume=15 | isbn=978-0-387-90081-0 | oclc=878109401 }}</ref><ref>{{springer|title=Fréchet space|id=p/f041380}}</ref> i drugih [[topological vector space|topoloških vektorskih prostora]] koji nemaju normu. Važan predmet istraživanja u funkcionalnoj analizi su [[Непрекидна функција|kontinuirani]] [[Линеарно пресликавање|linearni operatori]] definisani na Banahovim i Hilbertovim prostorima. Oni prirodno dovode do definicije [[C*-algebra|C*-algebre]]<ref>{{harvnb|Doran|Belfi|1986|pp=5–6}}, [https://books.google.com/books?id=6jNbsnJVjMoC&pg=PA5 Google Books].</ref><ref>{{harvnb|Doran|Belfi|1986|p=6}}, [https://books.google.com/books?id=6jNbsnJVjMoC&pg=PA6 Google Books].</ref><ref>{{harvnb|Segal|1947}}</ref>
+i drugih [[operator algebra|operatorskih algebri]].<ref> {{cite book | last = Blackadar | first = Bruce | title = Operator Algebras: Theory of C*-Algebras and von Neumann Algebras | publisher = [[Springer-Verlag]] | series = Encyclopaedia of Mathematical Sciences | year = 2005 | isbn = 3-540-28486-9 }}</ref><ref>M. Takesaki, ''Theory of Operator Algebras I'', Springer, 2001.</ref>
 == Vidi još ==
@@ Ред 51: / Ред 53: @@
 * Vogt, D., Meise, R.: ''Introduction to Functional Analysis'', Oxford University Press, 1997.
 * [[Kōsaku Yosida|Yosida, K.]]: ''Functional Analysis'', Springer-Verlag, 6th edition, 1980
+* {{citation|first=W.|last=Arveson|author-link=William Arveson|title=An Invitation to C*-Algebra|publisher=Springer-Verlag|year=1976|isbn=0-387-90176-0}}. An excellent introduction to the subject, accessible for those with a knowledge of basic [[functional analysis]].
+* {{citation|first=Alain|last=Connes|author-link=Alain Connes|url=https://archive.org/details/noncommutativege0000conn|title=Non-commutative geometry|year=1994|isbn=0-12-185860-X|url-access=registration}}.  This book is widely regarded as a source of new research material, providing much supporting intuition, but it is difficult.
+* {{citation|first=Jacques|last=Dixmier|author-link=Jacques Dixmier|title=Les C*-algèbres et leurs représentations|publisher=Gauthier-Villars|year=1969|isbn=0-7204-0762-1|url-access=registration|url=https://archive.org/details/calgebras0000dixm}}. This is a somewhat dated reference, but is still considered as a high-quality technical exposition. It is available in English from North Holland press.
+* {{citation|last1=Doran|first1=Robert S.|author-link=Robert S. Doran|first2=Victor A.|last2=Belfi|title=Characterizations of C*-algebras: The Gelfand-Naimark Theorems|publisher=CRC Press|year=1986|isbn=978-0-8247-7569-8}}.
+* {{citation|first=G.|last1=Emch|title=Algebraic Methods in Statistical Mechanics and Quantum Field Theory|publisher=Wiley-Interscience|year=1972|isbn=0-471-23900-3}}. Mathematically rigorous reference which provides extensive physics background.
+* {{springer|id=c/c020020|title=C*-algebra|author=A.I. Shtern}}
+* {{citation|first=S.|last=Sakai|author-link=Shoichiro Sakai|title=C*-algebras and W*-algebras|publisher=Springer|year=1971|isbn=3-540-63633-1}}.
+* {{citation|first=Irving|last=Segal|author-link=Irving Segal|title=Irreducible representations of operator algebras|journal=Bulletin of the American Mathematical Society|year=1947|volume=53|pages=73–88|doi=10.1090/S0002-9904-1947-08742-5|issue=2|doi-access=free}}.
+* {{Cite book |first=Roman |last=Kałuża |title=Through a Reporter's Eyes: The Life of Stefan Banach |publisher=Birkhäuser |year=1996 |isbn=0-8176-3772-9
+|url =https://books.google.com/?id=i3ZrfMinChkC&printsec=frontcover&dq=Stefan+Banach#PPP1,M1 |others =Translated by Wojbor Andrzej Woyczyński and Ann Kostant |ref=harv}}
+* {{cite web|last=Kosiedowski|first=Stanisław |title=Stefan Banach|publisher=|year=|work=Mój Lwów|url=http://www.lwow.com.pl/m.htm|accessdate=20. 5. 2008}}
+* {{cite web | url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Banach.html | title=Stefan Banach | publisher=University of St. Andrews | work=MacTutor History of Mathematics archive | year=2000 | accessdate=19. 8. 2012 | last1=O'Connor | last2=Robertson| first1= John J. |first2=Edmund F. |ref=harv}}
+* {{cite book|editor-first=Hans Niels|editor-last=Jahnke|first=Reinhard|last=Siegmund-Schultze|title=A History of Analysis|publisher=American Mathematical Society|year=2003|isbn=0-8218-2623-9|url=https://books.google.com/books?id=CVRZEXFVsZkC&printsec=frontcover&dq=A+History+of+Analysis&source=bl&ots=iczA6zzuXi&sig=R7ow1euyozHBPXr7S5uqEx3s7sU&hl=en&sa=X&ei=jlwxUJSEE6SiywG1ioDICg&ved=0CDMQ6AEwAA#v=onepage&q=A%20History%20of%20Analysis&f=false|ref=harv}}
+* {{cite book | title=Elementary Functional Analysis | publisher=Springer | author=MacCluer,  Barbara | year=2008 | isbn=0387855289}}
+* {{cite journal | url=http://www.lwow.com.pl/g8.htm | title=Geniusz: gen i już | last=Urbanek |first=Mariusz | journal=Polityka |date=april 2002 | volume=8 | issue=2348}}
+* {{cite journal | url=http://kielich.amu.edu.pl/Stefan_Banach/greczek.html | title=Wspomnienie o Stefanie Greczku | last=Waksmundzka-Hajnos |first= Monika | publisher = Gdańsk University| journal=Focus | year=2006 | issue=11 |ref=harv}}
+* {{citation|first=Stefan|last=Banach|authorlink=Stefan Banach|url=http://kielich.amu.edu.pl/Stefan_Banach/pdf/teoria-operacji-fr/banach-teorie-des-operations-lineaires.pdf|title=Théorie des opérations linéaires|publication-place=Warszawa|publisher=Subwencji Funduszu Kultury Narodowej|year=1932|series=Monografie Matematyczne|volume=1|zbl=0005.20901|access-date=10. 6. 2016|archive-url=https://web.archive.org/web/20140111122706/http://kielich.amu.edu.pl/Stefan_Banach/pdf/teoria-operacji-fr/banach-teorie-des-operations-lineaires.pdf|archive-date=11. 1. 2014|url-status=dead}}
+* {{citation
+|author=Beauzamy, Bernard
+|title=Introduction to Banach Spaces and their Geometry
+|year=1985 |origyear=1982
+|edition=Second revised
+|publisher=North-Holland
+}}
+* {{citation|first=Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki|title=Topological vector spaces|series=Elements of mathematics|publisher= Springer-Verlag|publication-place=Berlin|year=1987|isbn=978-3-540-13627-9}}
+* {{citation
+ | last = Carothers | first = Neal L.
+ | title = A short course on Banach space theory
+ | series = London Mathematical Society Student Texts
+ | volume = 64
+ | publisher = Cambridge University Press
+ | location = Cambridge
+ | year = 2005
+ | pages = xii+184
+ | isbn = 0-521-84283-2
+}}
+* {{citation
+ | last = Diestel
+ | first = Joseph
+ | title = Sequences and series in Banach spaces
+ | series = Graduate Texts in Mathematics
+ | volume = 92
+ | publisher = Springer-Verlag
+ | location = New York
+ | year = 1984
+ | pages = [https://archive.org/details/sequencesseriesi0000dies/page/ xii+261]
+ | isbn = 0-387-90859-5
+ | url = https://archive.org/details/sequencesseriesi0000dies/page/
+ }}
+* {{Citation
+ | last1=Dunford | first1=Nelson
+ | last2=Schwartz | first2=Jacob T. with the assistance of W. G. Bade and R. G. Bartle
+ | title=Linear Operators. I. General Theory
+ | publisher=Interscience Publishers, Inc.
+ | location = New York
+ | series = Pure and Applied Mathematics
+ | volume = 7
+ | mr=0117523
+ | year=1958}}
+* {{citation
+ | last1=Lindenstrauss | first1=Joram |author1-link = Joram Lindenstrauss
+ | last2=Tzafriri | first2=Lior
+ | isbn = 3-540-08072-4
+ | location = Berlin
+ | publisher = Springer-Verlag
+ | series = Ergebnisse der Mathematik und ihrer Grenzgebiete
+ | title=Classical Banach Spaces I, Sequence Spaces
+ | volume = 92
+ | year=1977}}
+* {{citation
+ | last = Megginson | first = Robert E. | authorlink = Robert Megginson
+ | title = An introduction to Banach space theory
+ | series = Graduate Texts in Mathematics
+ | volume = 183
+ | publisher = Springer-Verlag
+ | location = New York
+ | year = 1998
+ | pages = xx+596
+ | isbn = 0-387-98431-3
+}}
+* {{citation
+ | last = Ryan |first = Raymond A.
+ | year = 2002
+ | title = Introduction to Tensor Products of Banach Spaces
+ | publisher = Springer-Verlag
+ | series = Springer Monographs in Mathematics
+ | location = London
+ | isbn = 1-85233-437-1
+ | pages = xiv+225
+}}
+* {{citation
+ | last=Wojtaszczyk | first= Przemysław
+ | title = Banach spaces for analysts
+ | series = Cambridge Studies in Advanced Mathematics
+ | volume = 25
+ | publisher = Cambridge University Press
+ | location = Cambridge
+ | year= 1991
+ | pages = xiv+382
+ | ISBN = 0-521-35618-0
+}}
 {{refend}}

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