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Cauchy–Schwarz tengsizligi

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Cauchy–Schwarz (o'qilishi: Koshi-Shvarz) tengsizligi (ba'zida Cauchy–Bunyakovsky–Schwarz (o'qilishi: Koshi-Bunyakovskiy-Shvarz) tengsizligi) matematikadagi eng muhim va keng qo'llaniladigan tengsizliklardan biri sifatida qaraladi.

Yig'indilar uchun tengsizlik Augustin-Louis Cauchy tomonidan 1821-yilda nashr etilgan. Integrallar uchun mos keladigan tengsizlik Viktor Bunyakovsky tomonidan 1859-yilda va Hermann Schwarz tomonidan 1888-yilda nashr etilgan. Schwarz integral ko'rinishdagi tengsizlikning zamonaviy isbotini keltirgan.

Tengsizlik bayoni[tahrir | manbasini tahrirlash]

Cauchy-Schwarz tengsizligi barcha ichki ko'paytma aniqlangan fazodagi barcha va vektorlar uchunAndoza:NumBlktengsizlikning o'rinli ekanligini ta'kidlaydi, bu yerda ichki ko'paytma hisoblanadi. Ichki ko'paytmalarga misollar haqiqiy va kompleks nuqtali ko'paytmalarni o'z ichiga oladi. Ichki ko'paytma mavzusidagi misollarga qarang. Har bir ichki ko'paytma kanonik yoki keltirilgan norma deb ataladigan normani keltirib chiqaradi, bu yerda vektorning vektor normasi quyidagicha belgilanadi va aniqlanadi:

Bu norma va ichki ko'paytma aniqlovchi shart bilan o'zaro bog'langan bo'lib, bu yerda har doim nomanfiy haqiqiy son bo'ladi (hattoki ichki ko'paytma kompleks qiymatli bo'lsa ham). Yuqoridagi tengsizlikning har ikki tomonining kvadrat ildizini olib, Cauchy-Schwarz tengsizligini uning koʻproq tanish bo'lgan koʻrinishida yozish mumkin:Andoza:NumBlkBundan tashqari, tengsizlikning ikkala tomoni faqat va faqat va lar bir-biriga chiziqli bog'liq bo'lsagina bir-biriga teng bo'ladi.

Maxsus holatlar[tahrir | manbasini tahrirlash]

Sedrakyan lemmasi - musbat haqiqiy sonlar[tahrir | manbasini tahrirlash]

R 2 - Tekislik[tahrir | manbasini tahrirlash]

R n - n -o'lchovli Evklid fazosi[tahrir | manbasini tahrirlash]

C n - n -o'lchovli Kompleks fazo[tahrir | manbasini tahrirlash]

Qo'llanilishi[tahrir | manbasini tahrirlash]

Tahlil[tahrir | manbasini tahrirlash]

Geometriya[tahrir | manbasini tahrirlash]

Ehtimollar nazariyasi[tahrir | manbasini tahrirlash]

Dalillar[tahrir | manbasini tahrirlash]

Haqiqiy ichki ko'paytmali fazolar uchun[tahrir | manbasini tahrirlash]

Nuqtali ko'paytma uchun dalil[tahrir | manbasini tahrirlash]

Ixtiyoriy vektor fazolar uchun[tahrir | manbasini tahrirlash]

Isbot 1[tahrir | manbasini tahrirlash]

Let and so that and Then

This expansion does not require to be non-zero; however, must be non-zero in order to divide both sides by and to deduce the Cauchy-Schwarz inequality from it. Swapping and gives rise to:

and thus

Isbot 2[tahrir | manbasini tahrirlash]

Umumlashtirishlar[tahrir | manbasini tahrirlash]

Yana qarang[tahrir | manbasini tahrirlash]

Manbalar[tahrir | manbasini tahrirlash]

Manbalar[tahrir | manbasini tahrirlash]

  • Aldaz, J. M.; Barza, S.; Fujii, M.; Moslehian, M. S. (2015), „Advances in Operator Cauchy—Schwarz inequalities and their reverses“, Annals of Functional Analysis, 6 (3): 275–295, doi:10.15352/afa/06-3-20
  • Bunyakovsky, Viktor (1859), „Sur quelques inegalités concernant les intégrales aux différences finies“ (PDF), Mem. Acad. Sci. St. Petersbourg, 7 (1): 6
  • Cauchy, A.-L. (1821), „Sur les formules qui résultent de l'emploie du signe et sur > ou <, et sur les moyennes entre plusieurs quantités“, Cours d'Analyse, 1er Partie: Analyse Algébrique 1821; OEuvres Ser.2 III 373-377
  • Dragomir, S. S. (2003), „A survey on Cauchy–Bunyakovsky–Schwarz type discrete inequalities“, Journal of Inequalities in Pure and Applied Mathematics, 4 (3): 142 pp, 2008-07-20da asl nusxadan arxivlandi, qaraldi: 2022-06-19 {{citation}}: More than one of |archivedate= va |archive-date= specified (yordam); More than one of |archiveurl= va |archive-url= specified (yordam)
  • Grinshpan, A. Z. (2005), „General inequalities, consequences, and applications“, Advances in Applied Mathematics, 34 (1): 71–100, doi:10.1016/j.aam.2004.05.001
  • Andoza:Halmos A Hilbert Space Problem Book 1982
  • Kadison, R. V. (1952), „A generalized Schwarz inequality and algebraic invariants for operator algebras“, Annals of Mathematics, 56 (3): 494–503, doi:10.2307/1969657, JSTOR 1969657.
  • Lohwater, Arthur (1982), Introduction to Inequalities, Online e-book in PDF format
  • Paulsen, V. (2003), Completely Bounded Maps and Operator Algebras, Cambridge University Press.
  • Schwarz, H. A. (1888), „Über ein Flächen kleinsten Flächeninhalts betreffendes Problem der Variationsrechnung“ (PDF), Acta Societatis Scientiarum Fennicae, XV: 318
  • Andoza:Springer
  • Steele, J. M. (2004), The Cauchy–Schwarz Master Class, Cambridge University Press, ISBN 0-521-54677-X

Havolalar[tahrir | manbasini tahrirlash]