e0 crAo DUC vA. oao TAo DAI HQC HUE xi THr ruYiN srNH sAU DAr M6n thi: GiAi tfch . (dd,nh cho Cao hpc) Hg ud, t€n th{ s'inh: Sd b6,o danh,: HOC NAM 2OO7 I. Thdi g'ian ld,m bd,i: 1-80 phrit Tim mibn h6i tu crla chu6i nbm Er+l '[rj]; KhAo s6t suh6i tu dbu cria chu6i hb; Er#' Tfnh tich ph6,n ff I I @'+v2)ardv, J Jo' trong d6 D: {(r, y) € R.2l2ar < 12 *92 <2br},0 ( a 1b. fl. 1" U t (:-'t z , '! i t \ tr€n mibn (0, +oo). a. b. c. II. Cho X Ib tQp hqp gbm tdt cd, c6c tA,p con compact khdc 0 crla IR., a. V6i moi r € R, d{,t d(r,A) : inf{l* -yl , A € A}. Chfrng minh rXng, v6i moi z € IR., A eX, tbn t4i ro€ A sao cho lr - nol - d(r,A). b. Goi d: X x X + R 1A, 6nh xa dusc x6c dinh nhu sau: d,(A,B) :- inf{d : A c Bt, B C At}, trong d.6, A5 - inf{r € R : d,(n,A) < d}. Chfrng minh rB,ng d Ih, mQt metric tr6n X. III. Kf hiQu X - Cp,zl th kh6ng gian dinh chudn cr{,c hhm s6 li6n tuc tr6n 10, 2] v,ii chudn ll"ll : mil({|"(t)l : t € lo, 2l} vh kh6ng gian con Y - {r e X : r(0) - 0} cda X. Cho r{,nh xa A : X -> Y. fr n Ar xdc dinh bdi: Ar(t)- ['*(s)ds; telo,z]. Jo Chfrng minh rXng A Ib to6n trl tuy6n tfnh Ii€n tuc tli X vho Y. Tfnh llAll. Anh xq Ac6 phd,i th, mQt toir,n anh hay kh6ng? Cho kh6ng gian Hilbert phfrc If vi, tQp hqp {O"ln € N} c H thod md,n lld"ll :1 v6i moi n € N vA, sao cho vdi mgi / e H, ta c6: - lt ttt2 -i,/r ], tt2 ' ll/ll- : k\LQn)l Chirng minh rH,ng: a. {d"ln € N} th, m6t co sd truc chudn cta I{. b. Day (d",),ex h6i tu y6u ddn 0. 0'. b. IV. Ghi chri: C6,n b6 coi, thi, kh6ng gi,d,i, thfuh gi, th€m. VIETMATHS.NET . inf{l* -yl , A € A}. Chfrng minh rXng, v6i moi z € IR., A eX, tbn t4i ro€ A sao cho lr - nol - d(r,A). b. Goi d: X x X + R 1A, 6nh xa dusc x6c dinh nhu sau: d,(A,B) :- inf{d. @'+v2)ardv, J Jo' trong d6 D: {(r, y) € R.2l2ar < 12 *92 <2br},0 ( a 1b. fl. 1& quot; U t ( :-& apos;t z , '! i t tr€n mibn (0, +oo). a. b. c. II. Cho. HUE xi THr ruYiN srNH sAU DAr M6n thi: GiAi tfch . (dd,nh cho Cao hpc) Hg ud, t€n th{ s'inh: Sd b6,o danh,: HOC NAM 2OO7 I. Thdi g'ian ld,m bd,i: 1- 8 0 phrit Tim mibn