, , ˜ ˜ NGUYÊN HUU ÐIÊN , ˜ ´ N HOC THUÂT NGU TOA ANH - VIÊT , (Ban 2.0) , ´ BAN ÐAI HOC QUÔC ´ GIA HA ` XUÂT ` NÔI NHA Tai ngay!!! Ban co the xoa dong chu nay!!! 16990026032661000000 , ˜ HU ˜,U ÐIÊN NGUYÊN , ˜ ´ N HOC THUÂT NGU TOA ANH - VIÊT , Ban 2.0 , ´ BAN ÐAI HOC QUÔC ´ GIA HA ` XUÂT ` NÔI NHA , ˜ Hu ˜,u Ðiên Nguyên [Muc luc] `,i n´ Lo oi d ¯â`u , ,, , Ðây l` a ban nh´ ap c´ ac thuât an hoc ac ban ¯´ıch khoi d ¯â`u cho c´ ng˜u to´ Muc d , ` ` ´i viê´t b` mo cho c´ ac b´ ao Tâp s´ a ch gô m c´ a c phâ n , ˜ Phâ`n c´ ac thuât ng u , ´ Phâ`n môt sô ch´ u ´y ng˜u ph´ ap ´,c Môt ac d a công thu ¯oc sô´ c´ k´y hiêu v` , `˘ng LaTeX C´ ac k´y hiêu an chuân soan to´ ba , Nh˜ung ´y kiê´n hay vê` viê´t b´ ao tiê´ng anh v` a c´ ach tr`ınh bâ`y ch´ ung , , , , , Ðây chı l` a ban nh´ ap, c` on râ´t nhiê`u nôi ao d¯ây v` a c˜ ung chua ¯ua v` dung chua d ,, d ac ban ¯uo c chon loc, mong c´ cho ´y kiê´n H` a Nôi, ay 13 th´ ang n˘ am 2014 ng` , , ˜ ˜ u Ðiên Nguyên Hu , , ,, ´,u ho,n a tr`ınh bâ`y giao diên Phiên ban 2.0 d¯uo c bô sung râ´t nhiê`u v` d˜ê tra cu ´˘n Ch´ uc c´ ac ban may ma H` a Nôi, ay 24 th´ ang n˘ am 2020 ng` , , ˜ ˜ u Ðiên Nguyên Hu Muc luc `,i n´ Lo oi d ¯â`u Muc luc , ´,i thiêu Chu d ¯ê` Introduction - Gio , , Chu d ¯ê` Acknowlegments - Biê´t on , Chu d y hiêu ¯ê` Notations - K´ , , Chu d ¯ê` Assumptions - Gia thiê´t , Chu d ngh˜ıa ¯ê` Definition - Ðinh , Chu d l´ y ¯ê` Theorem - Ðinh , 6.1 Theorem formulation - Ph´ at biêu d¯inh l´y , , ´i thiêu 6.2 Theorem introductory - Gio ¯inh l´y cua d , , ˜ 6.3 Theorem remarks - Nhung ch´ u ´y cua d¯inh l´y , , ` ´ Chu d ¯ê Proofs - Chung minh ´,ng minh 7.1 Proof Arguments - C´ ac lâp luân chu ´˘t d¯â`u chu ´,ng minh 7.2 Proof Begin - Ba , ´,ng minh 7.3 Proof Conclusion - Kê´t luân cua chu , ´,ng minh d¯iê`u kiên 7.4 Proof Sufficient - Chu d¯u , ´,ng minh d¯iê`u kiên 7.5 Proof Easily - Chu ¯u d ´,ng minh t´ınh châ´t 7.6 Proof Property - Chu , ,´, ´,ng minh Chu d¯ê` Proof steps - C´ ac buo c chu , ´,u Chu d¯ê` References - T` liêu tra cu , ´ ˘c d Chu d y hiêu ¯ê` 10 Môt ¯oc sô´ quy ta k´ , ´ c chung 10.1 C´ ac k´y hiêu công thu 19 21 26 30 37 37 38 39 43 43 46 48 49 50 51 55 65 68 68 10.2 K´y hiêu anh chuyên ng` 72 10.2.1 Logic 72 10.2.2 Set 72 10.2.3 Order 73 10.2.4 Algebra 73 10.2.5 Topology 73 10.2.6 Function 74 10.2.7 Probability , , ´ ˘c ng˜ Chu d u ph´ ap ¯ê` 11 Môt sô´ quy ta , 11.1 Ch´ u ´y mao ac d ¯inh t`u x´ 74 , ˜ Hu ˜,u Ðiên Nguyên 75 75 [Muc luc] MUC LUC , 11.2 Ch´ u ´y mao ac d ¯inh t`u không x´ , , 11.3 Ch´ u ´y bo qua mao t`u , Chu d y hiêu ¯ê` 12 K´ LaTeX , ,, an ban 12.1 K´y hiêu môi tru`ong v˘ ,, 12.2 K´y hiêu an môi tru`ong to´ 12.2.1 K´y hiêu chung , 12.2.2 Ch˜u c´ Hy Lap 12.2.3 Binary Operations and Relations , 12.2.4 Tâp ho p ,, 12.2.5 Quan tam gi´ ac v` a nua m˜ ui tên , , ´c ˘ng thu 12.2.6 Bâ´t d¯a 12.2.7 K´y hiêu ui tên m˜ ,, 12.2.8 To´ an tu to´ an hoc , 12.2.9 Nh˜ung h` am sô´ to´ an hoc , 12.2.10 Nh˜ung k´y hiêu ac kh´ , ,, 12.2.11 Ch˜u c´ môi tru`ong to´ an , ˘ c 12.2.12 Nh˜ung dâ´u ngoa , 12.2.13 Nh˜ung dâ´u châ´m châ´m , , 12.2.14 Lênh biê´n d ¯ôi theo c˜o , T` liêu tham khao , ˜ Hu ˜,u Ðiên Nguyên 76 78 81 81 82 82 82 83 85 85 86 87 88 88 89 90 91 92 93 94 [Muc luc] , ` CHU ÐÊ ´,i thiêu Introduction - Gio We prove that in some families of compact there are no universal elements It is also shown that Some relevant counterexamples are indicated We wish to investigate Our purpose is to It is of interest to know whether We are interested in finding It is natural to try to relate to This work was intended as an attempt to motivate (at motivating) 10 The aim of this paper is to bring together two areas in which 11 we review some of the standard facts on 12 13 14 we have compiled some basic facts we summarize without proofs the relevant material on we give a brief exposition of `˘ng môt ´,ng minh Ta chu sô´ ho compact không tô`n tai c´ a c , ,, ` phân tu to` an thê , `˘ng N´ o c˜ ung chı ra ,, ´ Môt o liên quan d¯uo c , sô th´ı du c´ chı , Ta muô´n khao s´ at , Muc ung ta l` a d¯´ıch cua ch´ ,, Ðiê`u quan tâm d a ¯uo c biê´t m` ´,i viêc Ta quan tâm to , t`ım kiê´m , , ` ´,i Ðiêu tu nhiên l` a thu quan vo ,, , , Công tr`ınh n` ay d u ´y nhu su ¯uoc ch´ , ´˘ng th´ cô´ ga uc d¯ây 10 11 12 13 14 18 we briefly sketch we set up notation and terminology we discuss (study/treat/examine) the case we introduce the notion of 18 19 we develop the theory of 19 15 16 17 , ˜ Hu ˜,u Ðiên Nguyên 15 16 17 , Muc d¯´ıch cua b` b´ ao n` ay l` a kê´t , , ´ m` ho p hai lnhx vu c d¯o a , , ´ ˜ Ta tông quan lai m ôt sô d u kiên , , co ban , , ˜ kê´t ho p môt Ta d¯a sô´ yê´u tô´ co , ban ´˘t không chu ´,ng minh vât Ta t´ om ta châ´t c´ o liên quan , , ´˘n vê` Ta d¯ua giai th´ıch nga ´˘t Ta t´ om ta , Ta d¯ua k´y hiêu a d¯inh ngh˜ıa v` , ,, , Ta thao luân tru`ong ho p , ´,i thiêu Ta gio k´y hiêu cua , , Ta ph´ at triên d¯inh l´y cua , [Chu d¯ê` Introduction] [Muc luc] ˜, To´ Thuât an hoc ngu Anh - Viêt 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 we will look more closely at we will be concerned with it is shown that some of the recent results are it is shown that reviewed in a more general setting, it is shown that some applications are indicated, it is shown that our main results are stated and proved Section contains a brief summary (a discussion) of Section deals with (discusses) the case Section is intended to motivate our investigation of Section is devoted to the study of Section provides a detailed exposition of Section establishes the relation between Section presents some preliminaries We will touch only a few aspects of the theory We will restrict our attention (the discussion/ourselves) to It is not our purpose to study No attempt has been made here to develop It is possible that but we will not develop this point here A more complete theory may be obtained by , ˜ Hu ˜,u Ðiên Nguyên 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 ´,i Ta xem x´et vâ´n d ¯ê` gâ`n vo , ´,i Ta s˜e thu c hiên liên quan vo , `˘ng môt Ðiê`u n` ay chı ra sô´ kê´t , , qua hiên a th`oi ,l` , , Ðiê`u n` ay chı su tông quan , viêc thiê´t lâp tông quan , hon, ´,ng Ðiê`u n` ay cho thâ´y môt sô´ u , ,, dung d¯uo c chı ra, , , , Ðiê`u n` ay chı kê´t qua ch´ınh, cua ,, ˜ d¯uo c ph´ at biêu v` a ch´ ung ta d¯a ´,ng minh chu , ´˘n (môt Ðoan c´ o tông quan nga , , thao luân) cua ´,i tru,`o,ng ho.,p n` Ðoan ay ¯ê` câp , d to (thao luân) , ´,u Ðoan l´ y th´ uc d ¯ây nghiên cu , cua ch´ ung theo ´,u Ðoan d` anh cho viêc nghiên cu , cua , , Ðoan Cung câ´p su mô ta chi tiê´t , cua , Ðoan thiê´t lâp quan gi˜ua Ðoan tr`ınh b` ay môt ¯ê` sô´ vâ´n d ,., ` khoi d ¯âu Ta s˜e d kh´ıa ¯ê` câp d¯ê´n môt v` , canh cua d ¯inh l´y , ´,i han Ta s˜e gio ch´ u ´y cua ch´ ung ta , , , ´ (thao luân ung ta) toi cua ,ch´ , Ðây không phai l` a muc d ¯´ıch cua ´,u ch´ ung ta nghiên cu , , ,, , ˜ lu c d Ta không nô at triên o ¯ê ph´ d¯ây , , д o l` ad o kh a,ng, nhu ¯iê`u c´ , a n˘ ,, ng ta ´od không ph´ at triên d ¯iêm d ¯o ¯ây , , , ` Ðinh o thê nhân ¯u hon c´ , , l´y, d¯ây d d¯uo c t`u , [Chu d¯ê` Introduction] [Muc luc] , , Chu d ¯ê` Introduction - Gio´i thiêu 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 However, this topic exceeds the scope of this paper, However, we will not use this fact in any essential way The basic (main) idea is to apply The basic (main) geometric ingredient is The crucial fact is that the norm satisfies Our proof involves looking at The proof is based on the concept of similar in spirit to The proof is adapted from This idea goes back at least as far as We emphasize that It is worth pointing out that The important point to note here is the form of The advantage of using lies in the fact that The estimate We obtain in the course of proof seems to be of independent interest Our theorem provides a natural and intrinsic characterization of Our proof makes no appeal to Our viewpoint sheds some new light on Our example demonstrates rather strikingly that , ˜ Hu ˜,u Ðiên Nguyên 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 , ,, Tuy nhiên, chu d¯ê` n` ay vuo t qu´ a , , l˜ınh vu c cua b` b´ ao n` ay, , Tuy nhiên, ta s˜e không d` ung d˜u kiên ay moi ach câ`n thiê´t n` c´ , , ´ tu,o,ng co, so, (ch´ınh) l` ´p dung Y aa , , ´ tu,o,ng co, so, (ch´ınh) h`ınh hoc Y , ho p th` anh l` a , , , Yê´u tô´ chu yê´u l` a chuân thoa m˜ an , ´,ng minh cua ch´ ung ta bao Chu h` am xem x´et , ´,ng minh du.,a co, so, kh´ Chu ,, , niêm thâ`n tuong tu tinh , ´,ng minh d¯u,o.,c bô sung thêm Chu , t`u , , ,, ,, Tu tuong n` ay d ¯i nguo c lai ´ıt nhâ´t , l` a xa hon `˘ng Ta nhâ´n manh , `˘ng Ta chı thêm , ,, Ðiêm quan ch´ u ´y o d a ¯ây l` , dang c ua ,, ` , ˘m Thuân lo i d` ung d ¯uo c na , su kiên l` a , ,, ´nh gi´ Su d a ta nhân a ¯a ¯uo c qu´ d , ´ ´ tr`ınh chung minh l` a d¯ôc lâp rât hay , Ðinh l´y cua ch´ ung ta cung câ´p , ˘ môt d a c t´ ınh t u nhiên v` a kh´ ac biêt ¯ , cua , ´,ng minh cua ch´ Chu ung ta không tuân theo , , Theo quan d ung ¯iêm, cua ch´ ´,i thâ´y s´ ang lên d¯iêm mo , ´,ng V´ı du cua ch´ ung ta minh chu , ˘ t ch˜e hon l` cha a , [Chu d¯ê` Introduction] [Muc luc] ˜, To´ Thuât an hoc ngu Anh - Viêt 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 The choice of seems to be the best adapted to our theory The problem is that The main difficulty in carrying out this construction is that In this case the method of breaks down This class is not well adapted to Pointwise convergence presents a more delicate problem The results of this paper were announced without proofs in [8] The detailed proofs will appear in [8] (elsewhere/in a forthcoming publication) For the proofs we refer the reader to [6] It is to be expected that One may conjecture that One may ask whether this is still true if One question still unanswered is whether The affirmative solution would allow one to It would be desirable to but we have not been able to this These results are far from being conclusive This question is at present far from being solved Our method has the disadvantage of not being intrinsic The solution falls short of providing an explicit formula , ˜ Hu ˜,u Ðiên Nguyên 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 ˜u,ng phâ`n câp Chon nh nhât tô´t , ´ nhât cua d ¯inh l´y B` to´ an ngh˜ıa l` a Phâ`n kh´ o nhâ´t ch´ınh nêu , , su xây du ng n` ay l` a ,, ,, , Trong tru`ong ho p n` ay phuong ph´ ap mâ´t t´ ac dung ,, , ´p n` Lo ay không d¯uo c câp nhât tô´t , t`u , , , Hôi tu theo t`ung, d¯iêm thê hiên b` to´ an th´ u vi hon , , C´ ac kê´t qua cua b` b´ ao n` ay ´,ng minh ˜ thông b´ d¯a ao không chu [8] ´,ng minh chi tiê´t s˜e C´ ac chu [8] ´,ng minh ta tham chiê´u ´,i chu Vo ban d¯oc d¯ê´n [6] ,, , , N´ od ¯uo c ch`o d¯o i hy vong , , ,, `˘ng Ngu`oi ta c´ o thê gia thuyê´t , , ,`, ˜ ` Nguoi ta c´ o thê hoi d ay vân ¯iêu n` ´ ´ ng nêu c` on d ¯u , ˜ , Môt câu hoi vân không c´ o câu tra , l`oi l` a , , ,, , L`oi giai t´ıch cu c s˜e cho ph´ep ngu`oi , ta d¯ê , N´ o l` a kh´ at vong nhung ta không , ´ c´ o kha n˘ ang l` am d¯iê`u d ¯o , , ´ ´,i kê´t Nh˜ung kêt qua n` ay c` on xa vo luân , , ´,i Câu hoi n` ay thê hiên on xa mo c` , , , giai quyê´t d¯uo c , ,, Phuong ph´ ap cua ch´ ung ta không , ,, , ˜ trôi ho n nh u ng phu ong ph´ ap kh´ ac , , ´˘n gon L`oi gia cho c´ ach nga công , , ´ c hiên thu , [Chu d¯ê` Introduction] [Muc luc] , , Chu d ¯ê` Introduction - Gio´i thiêu 10 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 What is still lacking is an explicit description of As for prerequisites, the reader is expected to be familiar with The first two chapters of constitute sufficient preparation No preliminary knowledge of is required To facilitate access to the individual topics, the chapters are rendered as self-contained as possible For the convenience of the reader we repeat the relevant material from [7] without proofs, thus making our exposition self-contained The aim of this paper is The purpose of this paper is In this paper we shall be concerned with The paper addresses one of these questions we shall deal with We propose in this paper to desirable The paper deals with this and some closely rolated problems The present section will be devoted to developing a method This paper presents some results concerning The problem to be considered in this paper is that of designing The paper is intended to emphasize , ˜ Hu ˜,u Ðiên Nguyên 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 , ˜ c` Nh˜ung g`ı vân on thiê´u l` a môt mô , , ta r˜ o r` ang cua ´,i d¯iê`u kiên Ðô´i vo tiên quyê´t , , ,`, ´,i nguoi d¯oc am quen vo s˜e phai l` ,, Hai chuong d¯â`u tiên tao nên , , chuân bi d¯â`y d¯u ´,c so, bô l` Không c´ o kiê´n thu a câ`n thiê´t , , Ðê tao d¯iê`u kiên thu,ân lo i cho truy câp ac chu d ¯ê´n c´ ¯ê` riêng d ,, ,, , o nhân , c´ ac chuong d ¯uo c tu xem c´ , thê , ,, , , Cho su tiên ¯oc lo i cua ngu`oi d , ˘ p lai ch´ ung ta la c´ a c t` a i li c´ o liên , quan t`u [7] ´,ng minh , d¯o ´ l` không ,chu am , , cho thê hiên t u l` a m c ua ch´ u ng , Muc n` ay l` a tiêu cua b` , Muc n` ay l` a d¯´ıch cua b` ´,i Trong b` n` ay ta s˜e d¯ê` câp to , B` n` ay b` an vê` môt nh˜ung vâ´n d¯ê` â´y Ch´ ung ta s˜e b` an vê` Trong b` n` ay ta d¯ê` câp d¯ê´n viêc , mô ta ´,i vâ´n d¯ê` d¯o ´ v` B` n` ay x´et to a môt ˘ t sô´ vâ´n d ac c´ o liên quan cha ¯ê` kh´ ´,i n´ ch˜e vo o , ,, , Muc n` a y d anh d ¯uo c d` ¯ê xây du ng , , phuong ph´ ap , B` n` ay tr`ınh bâ`y môt sô´ kê´t qua vê` ,, Vâ´n d¯ê` s˜e d¯uo c d¯ê` câp b` ´ ´ n` ay l` a viêc thiê t kê `˘m nhâ´n manh B` b´ ao nha , [Chu d¯ê` Introduction] [Muc luc] , ˜, ph´ ˘´c ngu Chu d¯ê` 11 Môt ap sô´ quy ta 80 the associativity and commutativity of A the direct sum and direct product the inner and outer factors of f [Note the plural.] , Nhung: a deficit or an excess ,, , ,, ˜ u: Truo´c c´ o tên so hu Minkowski’s inequality, but: the Minkowski inequality Fefferman and Stein’s famous theorem, more usual: the famous Fefferman-Stein theorem , `,, d Trong môt ˘ c biêt a "of": ¯ê` mô ta môt ¯a sô´ mênh d danh tu sau "with" v` an algebra with unit e; an operator with domain H ; a solution with vanishing Cauchy data; a cube with sides parallel to the axes; a domain with smooth boundary; an equation with constant coefficients; a function with compact support; random variables with zero expectation the equation of motion; the velocity of propagation; an element of finite order; a solution of polynomial growth; a ball of radius 1; a function of norm p But: elements of the form f = Let B be a Banach space with a weak symplectic form w Two random variables with a common distribution `, "to have": Sau cum tu F has finite norm F has compact support , Nhung: F has a finite norm not exceeding F has a compact support contained in I , ˜ Hu ˜,u Ðiên Nguyên , ´˘c ng˜u, ph´ [Chu d ap] ¯ê` 11 Quy ta [Muc luc] , ` CHU ÐÊ 12 K´ y hiêu LaTeX , ,`, 12.1 K´ y hiêu môi tru o ng v˘ a n b an $ \$ & \& # \# % \% { \{ } \} ả Đ BB \\P \S \dag \ddag \i \j \t{BB} \dots Ü \"{U} ˙I \.{I} ˜ \~{G} G ˝ \H{A} A Ò \`{O} ˆ \^{C} C ˇ \v{C} C ˚ \r{T} T ´ \'{P} P ˘ \u{M} M ¯ \={N} N E \b{E} S \c{S} F \d{F} â \copyright r \circledR a \textcircled{a} ™ \texttrademark X \checkmark £ \pounds z \maltese • \textbullet \ \textbackslash | \textbar _ \_ – \textendash — \textemdash < \textless > \textgreater ~ \textasciitilde ł \l ^ \textasciicircum Ł \L ¡ \textexclamdown ø \o ¿ \textquestiondown Ø \O ‘ \textquoteleft å \aa ’ \textquoteright Å \AA “ \textquotedblleft ß \ss ” \textquotedblright SS \SS \textvisiblespace ổ \ae \textordmasculine ặ \AE ê \textordfeminine \oe * \textasteriskcentered · \textperiodcentered Œ \OE , ˜ Hu ˜,u Ðiên Nguyên , [Chu d k´y hiêu ¯ê` 12 lênh LATEX] [Muc luc] , Chu d¯ê` 12 K´y hiêu LaTeX 82 ,`, ng to´ an 12.2 K´ y hiêu môi truo 12.2.1 K´ y hiêu chung 6= > ≈ ≡ ∼ = ' ∂ ∞ ∇ ℵ ` ∨ ∧ ∀ ∃ \neq \leqslant \geqslant \approx \equiv \cong \simeq \partial \infty \nabla \aleph \ell \vee \wedge \forall \exists ì ữ / \ ∅ ⊂ ⊃ · © z \pm \mp \times \div \cup \cap \in \notin \setminus \varnothing \subset \supset \cdot \centerdot \copyright \maltese → ⇐⇒ $ £ % & { } _ ả Đ ã o \to \iff \$ \pounds \% \& \{ \} \_ \P \S \ast \dag \ddag \bullet \wr , 12.2.2 Ch˜u c´ Hy Lap α β γ δ λ ω ψ χ ρ κ π φ σ θ \alpha \beta \gamma \delta \lambda \omega \psi \chi \rho \epsilon \kappa \pi \phi \sigma \theta , ˜ Hu ˜,u Ðiên Nguyên υ ξ τ ι η ζ µ ν % ε κ $ ϕ ς ϑ \upsilon \xi \tau \iota \eta \zeta \mu \nu \varrho \varepsilon \varkappa \varpi \varphi \varsigma \vartheta Γ ∆ Λ Ω Π Φ Ψ Σ Θ Υ Ξ z ג k \Gamma \Delta \Lambda \Omega \Pi \Phi \Psi \Sigma \Theta \Upsilon \Xi \digamma \gimel \daleth , [Chu d k´y hiêu ¯ê` 12 lênh LATEX] Γ ∆ Λ Ω Π Φ Ψ Σ Θ Υ Ξ \varGamma \varDelta \varLambda \varOmega \varPi \varPhi \varPsi \varSigma \varTheta \varUpsilon \varXi ℵ i \aleph \beth [Muc luc] ,, an 12.2 K´y hiêu môi tru`ong to´ 83 12.2.3 Binary Operations and Relations ⊕ ⊗ ◦ } ~ q ∪ d t h ] ∨ g Y n o a ` |= / ∵ ∴ & , ˜ Hu ˜,u Ðiên Nguyên \oplus \ominus \otimes \odot \oslash \circ \bigcirc \circledcirc \circleddash \circledast \amalg \cup \Cup \sqcup \leftthreetimes \uplus \vee \curlyvee \veebar \ltimes \rtimes \dashv \vdash \vDash \Vdash \models \Vvdash \bowtie \Join \because \therefore \And > | u \ r ∩ e u i ∧ f Z [ † ‡ t ^ ` _ a • \boxplus \boxminus \boxtimes \boxdot \divideontimes \intercal \dotplus \setminus \smallsetminus \centerdot \diamond \cap \Cap \sqcap \rightthreetimes \backepsilon \wedge \curlywedge \barwedge \doublebarwedge \dagger \ddagger \nvdash \nvDash \nVdash \nVDash \pitchfork \smile \smallsmile \frown \smallfrown \bullet , [Chu d k´y hiêu ¯ê` 12 lênh LATEX] [Muc luc] , Chu d¯ê` 12 K´y hiêu LaTeX 84 ∼ = \cong \ncong \preccurlyeq < \succcurlyeq \curlyeqprec \curlyeqsucc ≺ \prec ⊀ \nprec \preceq \npreceq w \precapprox \precnapprox - \precsim \precnsim \succ \nsucc \succeq \nsucceq v \succapprox \succnapprox \succsim | \mid p \shortmid k \parallel q \shortparallel ∼ \sim ∼ \thicksim ' \simeq v \backsim w \backsimeq ≈ \approx ≈ \thickapprox u \approxeq ≡ \equiv ∝ \propto ∝ \varpropto ( \multimap \succnsim % , ˜ Hu ˜,u Ðiên Nguyên ∦ / = + G ; : l m $ P ⊥ \nmid \nshortmid \nparallel \nshortparallel \nsim \doteq \doteqdot \between \asymp \fallingdotseq \risingdotseq \bumpeq \Bumpeq \circeq \eqcirc \perp , [Chu d k´y hiêu ¯ê` 12 lênh LATEX] [Muc luc] ,, an 12.2 K´y hiêu môi tru`ong to´ 85 , 12.2.4 Tâp ho p ⊂ \subset ⊃ \supset ⊆ \subseteq ⊇ \supseteq j \subseteqq k \supseteqq < \sqsubset = \sqsupset v \sqsubseteq w \sqsupseteq b \Subset c \Supset * \nsubseteq + \nsupseteq ( \subsetneq ) \supsetneq \varsubsetneq ! \varsupsetneq " \nsubseteqq # \nsupseteqq $ \subsetneqq % \supsetneqq & \varsubsetneqq ' \varsupsetneqq ,, 12.2.5 Quan tam gi´ ac v` a nua m˜ ui tên \bigtriangleup \triangleright \ntriangleright B \vartriangleright \rhd \unrhd D \trianglerighteq \ntrianglerighteq I \blacktriangleright , \triangleq * \rightharpoonup + \rightharpoondown \rightleftharpoons \upharpoonright \downharpoonright , ˜ Hu ˜,u Ðiên Nguyên \bigtriangledown / C E J \triangleleft \ntriangleleft \vartriangleleft \lhd \unlhd \trianglelefteq \ntrianglelefteq \blacktriangleleft ( \leftharpoonup ) \leftharpoondown \leftrightharpoons \upharpoonleft \downharpoonleft , [Chu d k´y hiêu ¯ê` 12 lênh LATEX] [Muc luc] , Chu d¯ê` 12 K´y hiêu LaTeX 86 , ´,c ˘ng thu 12.2.6 Bâ´t d¯a < < ≮ \nless > > ≯ \ngtr \leqslant \nleqslant > \geqslant \ngeqslant ≤ \leq \nleq ≥ \geq \ngeq \leqq \nleqq = \geqq \ngeqq \eqslantless \lneqq \eqslantgtr \gneqq \lneq \lvertneqq \gneq \ll \gvertneqq ≪ \lll \gg ≫ \ggg , ˜ Hu ˜,u Ðiên Nguyên \lesssim \lnsim & \gtrsim \gnsim / \lessapprox \lnapprox ' \gtrapprox \gnapprox ≶ \lessgtr l \lessdot ≷ \gtrless m \gtrdot Q \lesseqgtr S \lesseqqgtr R \gtreqless T \gtreqqless , [Chu d k´y hiêu ¯ê` 12 lênh LATEX] [Muc luc] ,, an 12.2 K´y hiêu môi tru`ong to´ 87 12.2.7 K´ y hiêu ui tên m˜ → ⇒ −→ =⇒ ↑ ⇑ ; l m % & ⇐⇒ \rightarrow \Rightarrow \longrightarrow \Longrightarrow \uparrow \Uparrow \nrightarrow \nRightarrow \updownarrow \Updownarrow \nearrow \nwarrow \swarrow \searrow \iff \rightrightarrows \rightleftarrows V \Rrightarrow ,→ \hookrightarrow \rightarrowtail # \looparrowright \twoheadrightarrow y \curvearrowright \circlearrowright 99K \dashrightarrow \Rsh \upuparrows 7→ \mapsto 7−→ \longmapsto ⇒ , ˜ Hu ˜,u Ðiên Nguyên \leftarrow ⇐ \Leftarrow ←− \longleftarrow ⇐= \Longleftarrow ↓ \downarrow ⇓ \Downarrow \nleftarrow : \nLeftarrow = \nleftrightarrow < \nLeftrightarrow ↔ \leftrightarrow ⇔ \Leftrightarrow ←→ \longleftrightarrow ⇐⇒ \Longleftrightarrow ! \leftrightsquigarrow ← \leftleftarrows \leftrightarrows W \Lleftarrow ←- \hookleftarrow \leftarrowtail " \looparrowleft \twoheadleftarrow x \curvearrowleft \circlearrowleft L99 \dashleftarrow \Lsh \downdownarrows \rightsquigarrow ; \leadsto ⇔ , [Chu d k´y hiêu ¯ê` 12 lênh LATEX] [Muc luc] , Chu d¯ê` 12 K´y hiêu LaTeX 88 ,, 12.2.8 To´ an tu to´ an hoc Y Q X P Z H ZZ RR ZZZ Z Z ··· RRRR R ··· ` \coprod \sum [ S \bigcup \ T \bigcap ] U \biguplus \oint G F \bigsqcup \iint _ W \bigvee ^ V \bigwedge M L \iiiint \bigoplus O N \bigotimes \idotsint K J \bigodot \iiint RRR ZZZZ a \int R I \prod R , 12.2.9 Nh˜ung h` am sô´ to´ an hoc arcsin arccos sin \sin arctan cos \cos arg tan \tan m mod n cot \cot m mod n sec \sec m (mod n) csc \csc m (n) ln \ln lim inf dim \dim lim deg \deg lim sup \min lim max \max inj lim inf \inf lim sup \sup −→ proj lim lim ←− , , , ˜ Hu ˜ u Ðiên [Chu d Nguyên ¯ê` 12 \arcsin \arccos \arctan \arg m\mod n m\bmod n m\pmod n m\pod n \liminf \varliminf \limsup \varlimsup \injlim \varinjlim \projlim \varprojlim sinh cosh coth lg log exp hom ker det gcd Pr lim \sinh \cosh \tanh \coth \lg \log \exp \hom \ker \det \gcd \Pr \lim lênh k´y hiêu LATEX] [Muc luc] ,, an 12.2 K´y hiêu môi tru`ong to´ 89 , 12.2.10 Nh˜ung k´ y hiêu ac kh´ = < ℘ > ⊥ ∀ ∃ @ ¬ ∈ ∈ / { ~ } ∇ M O ♦ √ X ] \ [ \Im \Re \wp \top \bot \forall \exists \nexists \neg \in \notin \ni \complement \hbar \hslash \nabla \mho \square \Box \triangle \vartriangle \triangledown \lozenge \surd \checkmark \sharp \natural \flat \prime \backprime \mapstochar , ˜ Hu ˜,u Ðiên Nguyên ` ∂ ð ı k ` a ∞ ∅ ∅ ∠ ] ^ ∫ \ N H ♥ ♦ ♠ ♣ ? F r s \ell \partial \eth \imath \jmath \Bbbk \Finv \Game \infty \emptyset \varnothing \angle \measuredangle \sphericalangle \smallint \backslash \diagdown \diagup \blacksquare \Diamond \blacktriangle \blacktriangledown \blacklozenge \heartsuit \diamondsuit \spadesuit \clubsuit \star \bigstar \circledR \circledS , [Chu d k´y hiêu ¯ê` 12 lênh LATEX] [Muc luc] , Chu d¯ê` 12 K´y hiêu LaTeX 90 , ,`, 12.2.11 Ch˜u c´ môi truo ng to´ an ABCDEF GHIJKLM N OP QRST U V W XY Z abcdef ghijklmnopqrstuvwxyz \mathnormal{ } ABCDEFGHIJKLMN OPQRSTUVWXYZ abcdefghijklmnopqrstuvwxyz 1234567890 \mathrm{ } ABCDEFGHIJKLMN OPQRSTUVWXYZ abcdefghijklmnopqrstuvwxyz 1234567890 \mathsf{ } ABCDEFGHIJKLMN OPQRSTUVWXYZ abcdefghijklmnopqrstuvwxyz 1234567890 \mathbf{ } ABCDEFGHIJKLMN OPQRSTUVWXYZ abcdefghijklmnopqrstuvwxyz 1234567890 \mathit{ } ABCDEFGHIJKLMN OPQRSTUVWXYZ abcdefghijklmnopqrstuvwxyz 1234567890 \mathtt{ } ABCDEFGHIJKLMN OPQRSTUVWXYZ abcdefghijklmnopqrstuvwxyz 1234567890 \mathfrak{ } , ˜ Hu ˜,u Ðiên Nguyên , [Chu d k´y hiêu ¯ê` 12 lênh LATEX] [Muc luc] ,, an 12.2 K´y hiêu môi tru`ong to´ 91 ABCDEF GHIJ KLM N OP QRST U V W XY Z abcdef ghijklmnopqrstuvwxyz \boldsymbol{ } 1234567890 αβπθΦΨΩ ∈ ~∂∇ ← ∞∅2 A BC DE F G H I J K L M N OPQRS T U V W X Y Z , (Ðua v` ao \usepackage{mathrsfs} ,´, truoc \begin{document}) \mathscr{ } ABCDEFGHIJ KLMN OPQRST UVWX YZ \mathcal{ } ABCDEFGHIJKLMN OPQRSTUVWXYZ \mathbb{ } , ˘ c 12.2.12 Nh˜ung dâ´u ngoa ( [ | { h d b ( [ \vert \{ \langle \lceil \lfloor \lgroup ) ] k } i e c ) ] \Vert \} \rangle \rceil \rfloor \rgroup | k z | \ \lvert \lVert \lmoustache \arrowvert \backslash | k { k \rvert \rVert \rmoustache \Arrowvert \bracevert , ˜ Hu ˜,u Ðiên Nguyên , [Chu d k´y hiêu ¯ê` 12 lênh LATEX] [Muc luc]