Bipolyar teorema - Bipolar theorem

Yilda matematika, bipolyar teorema a teorema yilda funktsional tahlil bipolyarni tavsiflovchi (ya'ni qutbli to'plamning qutbidan). Yilda qavariq tahlil, bipolyar teorema a ga ishora qiladi zarur va etarli shartlar a konus unga teng bo'lish ikki qutbli. Bipolyar teoremani maxsus holat sifatida ko'rish mumkin Fenxel-Moro teoremasi.^[1]:76–77

Dastlabki bosqichlar

Aytaylik X a topologik vektor maydoni (TVS) bilan doimiy er-xotin bo'shliq ${ displaystyle X ^ { prime}}$ va ruxsat bering ${ displaystyle left langle x, x ^ { prime} right rangle: = x ^ { prime} (x)}$ Barcha uchun x ∈ X va ${ displaystyle x ^ { prime} in X ^ { prime}}$. The qavariq korpus to'plamning A, ko (bilan belgilanadiA), eng kichigi qavariq o'rnatilgan o'z ichiga olgan A. The qavariq muvozanatli korpus to'plamning A eng kichigi qavariq muvozanatli o'z ichiga olgan to'plam A.

The qutbli kichik to'plam A ning X quyidagicha aniqlanadi:

${ displaystyle A ^ { circ}: = left {x ^ { prime} in X ^ { prime}: sup _ {a in A} left | left langle a, x ^ { prime} right rangle right | leq 1 right }}$

esa prepolar kichik to'plam B ning ${ displaystyle X ^ { prime}}$ bu:

${ displaystyle {} ^ { circ} B: = left {x in X: sup _ {x ^ { prime} in B} left | left langle x, x ^ { prime } o'ng rangle o'ng | leq 1 o'ng }}$.

The ikki qutbli kichik to'plam A ning X, ko'pincha tomonidan belgilanadi A^∘∘ to'plam

${ displaystyle A ^ { circ circ}: = {} ^ { circ} chap (A ^ { circ} o'ng) = chap {x in X: sup _ {x ^ { prime} in A ^ { circ}} chap | chap langle x, x ^ { prime} right rangle right | leq 1 right }}$.

Funktsional tahlilda bayonot

Ruxsat bering ${ displaystyle sigma chap (X, X ^ { prime} o'ng)}$ ni belgilang zaif topologiya kuni X (ya'ni eng zaif TVS topologiyasi yoqilgan X barcha chiziqli funktsiyalarni ${ displaystyle X ^ { prime}}$ davomiy).

Bipolyar teorema:^[2] Ichki to'plamning bipolyar qismi A ning X ga teng ${ displaystyle sigma chap (X, X ^ { prime} o'ng)}$- yopilishi qavariq muvozanatli korpus ning A.

Qavariq tahlildagi bayonot

Bipolyar teorema:^[1]:54[3] Har qanday kishi uchun bo'sh emas konus A ba'zilarida chiziqli bo'shliq X, bipolyar to'plam A^∘∘ tomonidan berilgan:

${ displaystyle A ^ { circ circ} = operatorname {cl} left ( operatorname {co} left {ra: r geq 0, a in A right } right)}$.

Maxsus ish

Ichki to‘plam C ning X bo'sh emas yopiq qavariq konus agar va faqat agar C⁺⁺ = C^∘∘ = C qachon C⁺⁺ = (C⁺)⁺, qayerda A⁺ to'plamning musbat dual konusini bildiradi A.^[3][4]Yoki umuman olganda, agar C bo'sh bo'lmagan konveks konus bo'lib, bipolyar konus tomonidan beriladi

C^∘∘ = cl (C).

Bilan bog'liqlik Fenxel-Moro teoremasi

Ruxsat bering

${ displaystyle f (x): = delta left (x | C right) = { begin {case} 0 & x in C infty & { text {aks holda}} end {case}}}$

bo'lishi ko'rsatkich funktsiyasi konus uchun C. Keyin qavariq konjugat,

${ displaystyle f ^ {*} (x ^ {*}) = delta (x ^ {*} | C ^ { circ}) = delta ^ {*} (x ^ {*} | C) = sup _ {x in C} langle x ^ {*}, x rangle}$

bo'ladi qo'llab-quvvatlash funktsiyasi uchun Cva ${ displaystyle f ^ {**} (x) = delta (x | C ^ { circ circ})}$. Shuning uchun, C = C^∘∘ agar va faqat agar f = f^**.^[1]:54[4]

Shuningdek qarang

• Ikki tomonlama tizim
• Polar to'plami

Adabiyotlar

Bibliografiya

• Narici, Lourens; Bekenshteyn, Edvard (2011). Topologik vektor bo'shliqlari. Sof va amaliy matematik (Ikkinchi nashr). Boka Raton, FL: CRC Press. ISBN  978-1584888666. OCLC  144216834.
• Shefer, Helmut H.; Volf, Manfred P. (1999). Topologik vektor bo'shliqlari. GTM. 8 (Ikkinchi nashr). Nyu-York, NY: Springer Nyu-York Imprint Springer. ISBN  978-1-4612-7155-0. OCLC  840278135.
• Triv, Fransua (2006) [1967]. Topologik vektor bo'shliqlari, tarqalishi va yadrolari. Mineola, N.Y .: Dover nashrlari. ISBN  978-0-486-45352-1. OCLC  853623322.