13 research outputs found
The Bishop-Phelps-Bollobas theorem on bounded closed convex sets
Doctor๋ณธ ๋ฐ์ฌํ์ ๋
ผ๋ฌธ์์๋ ์ ๋ช
ํ ๋น์-ํ ํ์ค-๋ณผ๋ก๋ฐ์ค ์ ๋ฆฌ๋ฅผ ์ผ๋ฐํ๋ ์์ญ์์ ์ฆ๋ช
ํ๋ค. 2008๋
์ดํ๋ก ์์ฉ์์ ๋น์-ํ ํ์ค-๋ณผ๋ก๋ฐ์ค ์ ๋ฆฌ์ ๊ดํ ๋ง์ ์ฐ๊ตฌ๊ฐ ์ด๋ฃจ์ด์ก์์ง๋ง, ์ ์์ญ์ ๋จ์๊ตฌ๋ก ํ์ ๋์ด ์์๋ค. ๊ทธ๋ฌ๋ ์ ๊ณ ๋ฒํจ์์ ์์ผ๋๋ ๋ณ๋ถ๋ฒ ์ ๋ฆฌ๋ฅผ ํตํด ๋ค์ํ ๋ฐ๋ํ ๊ณต๊ฐ์ ์๋ค์ ๋ํด์ ์ผ๋ฐ์ ์ธ ๋ซํ ์ ๊ณ ๋ณผ๋ก์งํฉ์ผ๋ก ํ์ฅ๋ ์ ์์์ ์ฆ๋ช
ํ์๋ค. ์ด๋ฌํ ์๋์ ๋๊ธฐ๋ ๋น์-ํ ํ์ค ์ ๋ฆฌ์ ๋ผ๋-๋์ฝ๋ฌ ์ ๋ฆฌ๊ฐ ๋์น๋ผ๋ ๋ถ๊ฐฑ์ ๋๋ผ์ด ์ ๋ฆฌ๋ก๋ถํฐ ์์๋์๋ค. ๋ํ ๋
ธ๋ฆ์ ์ทจํ๋ ์ ํ ๋ฒํจ์์ ์๋ก์ด ๋์น๊ด๊ณ๋ฅผ ์ฐพ์๋ด๊ณ ๋ ๋์๊ฐ ์์๊ทธ๋ผ์ ๋ฏธ๋ถ๋ํ์ฌ์์ ์ฌ์ฉํ์ฌ ๋งค๋ํ ๋ฆฝ์์ธ ๋ฐ๋ํ ๊ณต๊ฐ์์์ ๊ฐํ ๋น์-ํ ํ์ค-๋ณผ๋ก๋ฐ์ค ์ ๋ฆฌ๋ฅผ ์ฆ๋ช
ํ๋ค.In this thesis, we deal with the celebrated Bishop-Phelps-Bollobas theorem on the general domain. Though a lot of researches on the Bishop-Phelps-Bollobas property for operators have been conducted since 2008, their domains are restricted on the unit ball of Banach spaces. It is shown that for various pairs of Banach spaces this property can be extended to general bounded closed convex sets. The attempt stems from the Bourgain's remarkable result, i.e. a Banach space X has the Radon-Nikodym property if and only if X satisfies the Bishop-Phelps property. We also give new characterization of norm attaining functionals and strong version of the Bishop-Phelps-Bollobas property for Lipschitz smooth Banach spaces using Azagra's diffeomorphisms