Type, cotype and K-convexity
Type,cotype and K-convexity
Bernard Maurey
Laboratoire d’Analyse et Math′e matiques Appliqu′e es,UMR8050
Universit′e de Marne la Vall′e e
Boulevard Descartes,Cit′e Descartes,Champs sur Marne 77454Marne la Vall′e e Cedex2,FRANCE
1The pre-history of type and cotype,as I remember it
At the end of the sixties,Pietsch[Pi]promoted the notion of p-summing operators between Banach spaces,which extends to all values of p∈[1,+∞) the study of some classes of operators introduced by Grothendieck[Gro],under di?erent names,for the special values p=1,2.In an important paper devoted to p-summing operators,Lindenstrauss and Pe l czy′n ski[LP]gave a second birth to what we know in Banach space theory as the Grothendieck theorem; one formulation of it states that every operator from 1to 2is1-summing; another formulation is the famous Grothendieck’s inequality.Around1969, L.Schwartz introduced radonifying maps,a notion that turned out to be closely related to p-summing maps.A special case of this notion deals with the Wiener measure and with linear maps from a Hilbert space H to a Banach space X,that transform the canonical cylindrical Gaussian measure of H into a true Radon probability measure on X(see L.Gross[Gr1,Gr2]for another viewpoint on this subject).L.Schwartz organized a seminar at the Ecole Polytechnique in Paris([Sem],1969–70)about these topics.This is one of the reasons why Paris,and especially the Ecole Polytechnique,became one of the places where the subject of type and cotype was developed.
Type and cotype conditions appeared?rst in the framework of p-summing operators,or more precisely in connection with the factorization through L p, p>1,of operators with values in L1(in this paper,operator means bounded linear operator).In the spring of1972I saw the preprint of the paper[Ro] by H.Rosenthal;this paper played an essential role for me;it contains sev-eral ideas that I later used and developed in[Ma2].Two of these ideas taken from[Ro]are the factorization conditions and the notion of stable type p.By Pietsch’s factorization theorem,which extends some factorization results due to Grothendieck[Gro],every q-summing operator from C(K)to a Banach space factors through the natural injection C(K)→L q(K,μ),for some prob-ability measureμon K.Rosenthal dualizes this fact,and shows that given T:X→L1linear such that T?is q-summing,then T factors through a mul-tiplication operator M f:L p→L1by a function f∈L q(1/p+1/q=1;let us simply write L r for L r(K,μ),0 ( i |T(x i)|p)1/p dμ≤C( i x i p)1/p, for some C and every?nite sequence(x i)?X,then T factors as T=M f?T1 for some f∈L q.The proof of the factorization theorem is just an application of the Hahn-Banach separation theorem,either directly as in[Ma1],or by going back to Pietsch’s factorization as in [Ro].One gets in this way a function f ∈L q such that f q ≤1and |T (x )/f |p dμ≤C p x p for every x ∈X .The above operator T 1is then de?ned by T 1(x )=T (x )/f ∈L p for every x ∈X .Next,it is shown in [Ro]that a simple norm condition on X ,that happens to be true for X =L s when 2≥s >p >1,easily implies the above factorization condition,as soon as T :X →L 1is bounded (and linear).This condition on a Banach space X is of the form i f i (t )x i dt ≤K ( i x i p )1/p , where K is a constant depending only upon X ,(f i )is a sequence of L 1-normalized p -stable variables,and (x i )an arbitrary sequence in X .This con-dition was called stable type p in [Ma1,Ma2];it was used in [Ro](without this name)for the injection of X ?L 1to L 1,and in the general case in [Ma2].For example,since a Hilbert space has type 2,we obtain in this way that every bounded linear map from a Hilbert space to L 1factors through a multipli-cation M f :L 2→L 1,a statement dual to one of the results of [Gro]:every operator from a C (K )-space to a Hilbert space is 2-summing.By trace duality,this yields that every operator from 1to 2is 2-summing;we may call this the easy Grothendieck theorem .The same proof shows that every operator from a C (K )-space,to a space X such that the dual X ?has type 2,is 2-summing:this result appeared for the ?rst time in a paper by Dubinsky,Pe l czy′n ski and Rosenthal [DPR]. It is obvious to generalize to operators from X to L r the condition that gives a factorization through a multiplication operator L p →L r (0 [Ma1,Ma2]).In particular,some of the results obtained for 0 A ?rst relation between these topics and ?nite dimensional geometry comes from the paper [Ro];there,a delicate quantitative Lemma (Lemma 6from [Ro])shows that when the injection from a subspace X ?L 1to L 1does not factor through any L p ,p >1,then X must contain complemented almost iso-metric copies of n 1for every n ≥1,proving thus that every re?exive subspace of L 1embeds in some L p ,p >1(the main result of [Ro]).This Lemma was ex-tended in [Ma2]to a general Banach space X as follows:when there exists an operator T :X →L p that does not factor through any L p +ε,ε>0,then the injections n 1→ n p ,n ∈N ,uniformly factor through X .In particular,when there exists an operator T :X →L 1that does not factor through any L 1+ε,ε>0,then X contains uniformly isomorphic and complemented copies of n 1,for every n ≥1.This gives a new (bizarre)proof of Grothendieck’s theorem:since n 1is not uniformly complemented in c 0,the preceding statement implies that every bounded linear map from c0to L1factors through L1+ε,and it reduces Grothendieck’s theorem to a much easier variant.It is a model for a list of reduction results,for example this sort of extension of the Grothendieck theorem:every operator from a cotype2space X to any Banach space,which is2-summing,is already1-summing(see[Ma2];as we have just said,when X=L1,this is the information that one needs in order to pass from the easy Grothendieck theorem to the real one).This line of results displayed inter-esting connections between some simple?nite dimensional phenomenons and analytic facts about Banach spaces. In the same years,Ho?man-J?rgensen[HJ1]proved general results about se-ries of vector valued independent random variables,that are in the spirit of Kahane’s inequalities for vector valued Rademacher series;he also de?ned Rademacher type-p and showed connections to the law of large numbers in [HJ2].The notion of type2(with a di?erent name)appeared?rst in[DPR], and it was shown in this article that stable type2and Rademacher type2are identical.The results from[HJ1]imply that stable type p and Rademacher type p are closely related for every p∈(1,2]:stable type p implies Rademacher type p,and Rademacher type p implies stable type p?εfor everyε>https://www.wendangku.net/doc/8613980326.html,ter on,it has been universally admitted that Rademacher type is easier to work with,and the notion of stable type p essentially disappeared,except for p=2, because2-stable type and cotype express interesting properties of Gaussian probability measures on a Banach space.With Rademacher type p(we say simply type p in what follows),several points are simpli?ed;it is obvious that type p implies type r for r≤p,and the opposite for cotype;the results for L r spaces are easier to formulate,and simple to prove using Khintchine’s inequal-ity:L r has type r and cotype2when1≤r≤2and type2and cotype r when 2≤r<+∞.Clearly,L r does not have type r+ε,ε>0when1≤r≤2,and does not have cotype r?εwhen2≤r≤+∞.This suggested that one could possibly read some geometrical information about X from the limit values of p and q that give type p or cotype q for X. The?rst attempts to relate type,cotype to the fact that X contains almost isometric copies of some classical spaces concerned n ∞and n 1 .The?rst re- sult[MP1]gave the equivalence between non-trivial cotype for X and the fact that X does not contain n ∞uniformly;today,the proof in[MP1]looks a bit ridiculous by its complication.It was presented at the Conference at Oberwol-fach,October73;at the same meeting,James presented a much deeper result, namely his solution of the“re?exive vs B-convex”problem(see below).This was perhaps the beginning of what was later called“Local theory”.For the relation between the absence of n 1s in a Banach space X and other properties of this X,the?rst steps are due to Beck,Giesy and James,several years before this story[Be,G1,J1];Beck showed the relevance to the law of large numbers in Banach spaces of the fact that X does not contain copies of n 1s.Beck and Giesy de?ned B-convex Banach spaces as follows:the Banach space X is B-convex if for some n>1andε>0,and for all norm one vectors(x i)n i=1 in X, at least one choice of signs gives n i=1 ±x i ≤n(1?ε).Giesy proved several Banach space?avoured results about B-convexity,for example that X?and X??are B-convex when X is B-convex.James[J1]also worked on this class, which he called uniformly non n 1;in this paper[J1],he conjectured that B- convex spaces can be renormed to be uniformly convex,and must therefore be re?exive(and he disproved this conjecture in1973,as we have said above). Shortly after the result for cotype and n ∞,Pisier proved the type and n 1 case[P1];he developed the submultiplicativity method for the type constants, which was important for the following paper[MP2].Pisier’s result showed that the class of B-convex spaces coincides with the class of spaces X that have type p for some p>1.Then Pisier and I started to work on the relations between the limit values for the type or cotype of X,and the existence of subspaces of X that look somewhat like n p .Our?rst approach to the results of [MP2]was to strengthen the Dvoretzky-Rogers factorization[DR]for a Banach space X,using information on the limits of type and cotype;it just happened that the beautiful result of Krivine[Kr2](see section4)appeared during the preparation of[MP2]and allowed us to prove a much more satisfactory result. In the?rst version of[MP2],we proved that when X has type p?εbut not p+εfor everyε>0,then the injections n 1→ n p factor almost isometrically through a subspace of X for all n≥1,which means that we can?nd norm one vectors x1,...,x n in X such that ( n i=1 |a i|p)1/p≤(1+ε) n i=1 a i x i ≤(1+ε) n i=1 |a i| for all scalars(a i);the second inequality is of course obvious.When p<2, this is a strengthening of the Dvoretzky-Rogers Lemma which says that the above statement holds in every Banach space when p=2.Krivine’s theorem appeared shortly after the?rst version of[MP2]was written;fortunately, Studia Math was so slow to publish at that time that we were able to modify our article in the form which is known as Maurey-Pisier or Maurey-Pisier-Krivine theorem.I will call it here MP+K theorem,to emphasize the fact that these three persons did not work together on this particular paper. Kwapie′n was visiting Paris in1971and72,just before all this started,and he played a signi?cant role in the mathematical education of some of the young French;he gave several seminar talks that had a serious impact on us;he read and found the mistakes in several false“new proofs”that I had for the Grothendieck theorem,and he was the?rst person who checked the eventually correct proof of that I gave in[Ma2].His result in[Kw]had a great in?uence on the subject of type and cotype;it appeared actually before the de?nitions of type and cotype were given,but it is nice to formulate it as follows:if X has both type2and cotype2,then X is isomorphic to a Hilbert space.This is one of the?rst isomorphic characterizations of the Hilbert space.Some time later,I used in[Ma3]a small modi?cation of Kwapie′n’s argument and showed that every bounded linear operator from a subspace X0of a type2space X to a cotype2space Y factors through a Hilbert space,and extends to an operator from the whole space X to Y.In particular,every cotype2subspace X0of a type2space X is Hilbertian and complemented in X.This was a generalization of a well known result due to Kadec and Pe l czy′n ski[KP],that Hilbertian subspaces of L p,2≤p<+∞,are complemented. Super-properties appeared in the work of James on super-re?exivity(see[J2] and[J3],and section2below);ultraproduct methods[DK]give more insight on super-properties:a property is a super-property when it passes to ultra-powers.Super-re?exivity is obviously a super-property,and B-convexity is an-other super-property;James showed that super-re?exive spaces are B-convex. Deciding whether B-convex and super-re?exive spaces are the same class,as was conjectured by James in[J1],remained a di?cult problem for some time, and was?nally solved by James,who constructed a non-re?exive B-convex space([J4],improved in[J5]);before this,Brunel and Sucheston[BS1,BS2] had tried to prove that B-convex spaces were re?exive,and a part of their attempt introduced an important concept,that of spreading model,which will be used here in sections4and5.From this point on,there were two clearly distinct settings:super-re?exive spaces are those that can be renormed to be uniformly convex(En?o[En]);they have martingale type p(the basis for Pisier’s renorming theorem[P2]),and the class of B-convex or type-p spaces, p>1,is strictly larger.However,contrary to the general case,type and uni-form convexity are strongly related for lattices(see Johnson[Jo],and[LT, 1.f]).In a lattice X with non-trivial cotype,it is possible to prove Khintchine- type inequalities.Given(x i)n i=1in X,these inequalities permit to replace the estimate of a Rademacher average n i=1 εi(t)x i in L2(X)by an estimate of the square function( n i=1 |x i|2)1/2in X.This kind of“functional calculus” for lattices was developed by Krivine in[Kr1],where he obtained interesting formulations of the Grothendieck theorem,relating operators between lattices and the square function(see also[LT,1.f.14]). Early signs of a tendency to move from abstract Banach spaces to the study of C?-algebras and operator spaces also came in this framework.N.Tomczak [To]proved that the Schatten classes have the same type or cotype properties than the L p spaces.Pisier[P3]generalized Grothendieck’s theorem to C?-algebras;the result was revisited by Haagerup[Ha]and was the start for many further exchanges between them.Several other factorization results related to Grothendieck’s theorem were proved in those years,see[P8]. The?rst really striking application of cotype as a classi?cation tool appears in the results of Figiel,Lindenstrauss and Milman[FLM].They showed that Dvoretzky’s theorem takes a very strong form in cotype2spaces:if X has cotype2,there exists a constant c>0such that for every integer n,every n-dimensional subspace of X contains a further subspace X0such that dim X0= m≥c n and d(X0, m 2)≤2.This result makes use of a certain fundamental formula k=[η(τ)n M2 r /b2] proved in[FLM,Theorem2.6],relating the dimension k of(1+τ)-spherical sections of an n-dimensional normed space to some integral invariant M r.This formula appears already–with a di?erent normalization–as equation(14)in [Mi1].It gives the spectacular consequence above when using cotype2in an appropriate way;actually,[FLM]quantify the dimension of spherical sections in terms of the cotype q property,for every q≥2,and the previous result for cotype2is a special case.Another approach to the problem of spherical sections,the notion of volume ratio developed by Szarek and Tomczak[ST], also singles out the special behaviour of cotype2spaces.This approach is based on the work of Szarek[Sz],who introduced volume arguments in a new proof of the results of Kaˇs in[Kˇs]about n 1;of course Szarek need not mention cotype2when working with the explicit norm of n 1!The fact that cotype2 spaces have a uniformly bounded volume ratio was proved later by Bourgain and Milman[BMi],and this motivated the introduction of weak cotype2by Milman and Pisier([MP],see also Chapter10of Pisier’s book[P9]). Type is a nice tool for estimating the behaviour of the entropy of a convex hull; a simple observation of mine,written in[P5],was used in entropy problems by Carl[Ca].This observation states that in a Banach space X with type p>1, every point x from the convex hull of a subset A of the unit ball B X can be approximated by a convex combination of n points of A,with an error of order n?1/q(with q conjugate to p).Lemma9below is in the spirit of this result. Type and cotype have some simple stability properties;for example,the dual of a type p space has cotype q for the conjugate exponent,but the converse is false as shown by the pair( 1, ∞).The two young and ignorant authors of [MP2]left open a nice intriguing conjecture:is it possible to dualize cotype when we have some non-trivial type?It is clear that what is needed is the boundedness of the Rademacher projection on L2(X).Spaces such that the Rademacher projection is bounded were called K-convex in[MP2](was it because K was the?rst available letter after J for J-convex,a notion due to James and named by Brunel and Sucheston[BS2],or to acknowledge the importance of Kwapie′n’s work on Rademacher averages?)It was conjectured in[MP2]that every space with type r>1should be K-convex,which would imply that the dual X?of a space X with cotype q and some non-trivial type should be of type p,with1/p+1/q=1.Six years later,Pisier proved what I consider the most beautiful result in this area,making use of Kato’s theorem on holomorphic semi-groups(see[P6]and section6of this article): every B-convex space is K-convex. Although very beautiful,the preceding theorem is not the one that has been most useful for local theory.The most useful is another result obtained earlier by Pisier[P4],on the way to the general theorem above.This result asserts that the K-convexity constant of X is bounded by C(1+ln d X),where d X is the distance from X to the Hilbert space of the appropriate dimension (see Theorem13below).In particular,the K-convexity constant is bounded by C(1+ln n)for any n-dimensional normed space.The quantitative?nite-dimensional K-convexity,together with the notion of -norm,leads to a power-ful tool for geometric estimates(Theorem3.11in[P9];this theorem appeared ?rst in[FT]).These results play an important role in the QS-theorem of Mil-man([Mi2],see also[P9]). 2Super-properties Several of the properties P that are de?ned for a Banach space X are expressed in the following way:suppose that a number N P(E)is associated to every?nite dimensional normed space E,in such a way that N P(F)tends to N P(E)when the Banach-Mazur distance d(F,E)between F and E tends to1;the most common such dependence is when N P(F)≤d(F,E)N P(E).We then say that the Banach space X satis?es property P when N P(X)=sup E N P(E)<+∞, where the supremum is extended to all?nite dimensional subspaces E of X. Clearly,the fact that such a property P holds for X only depends upon the family F(X)of all?nite dimensional normed spaces F such that for every ε>0,there exists E?X for which d(F,E)<1+ε.After James[J2],we say that Y is?nitely representable in X when F(Y)?F(X);for instance,L p is ?nitely representable in p,and it is known that X??is?nitely representable in X for every Banach space X(local re?exivity).A property P of Banach spaces is called super-property if we know that whenever a Banach space X has P,then every Banach space Y?nitely representable in X has P.Clearly, every property P expressed by N P(X)<+∞as above is a super-property. Super-properties were de?ned by James in[J2]. Type and cotype are such properties.Let us recall a few de?nitions and facts that are developed in[JL].Let(εi)+∞ i=1denote the sequence of Rademacher functions on[0,1],or any independent sequence of centered Bernoulli random variables.Let p∈[1,+∞).We say that X has type p when there exists a constant T such that 1 0 n i=1 εi(t)x i 2dt 1/2 ≤T( n i=1 x i p)1/p, for every n≥1and every sequence(x i)n i=1?X;we denote by T p(X)the smallest constant T with this property;obviously,every normed space X has type1with T1(X)= 1.On the other hand,it follows from Khintchine’s inequalities that no non-zero normed space has type p when p>2.Saying that X has type p is obviously equivalent to the fact that the family of?nite dimensional subspaces E of X satis?es sup E T p(E)<+∞,thus having type p is a super-property.We say that X has cotype q when there exists a constant C q(X)such that ( n i=1 x i q)1/q≤C q(X) 1 n i=1 εi(t)x i 2dt 1/2 for every n≥1and every sequence(x i)n i=1?X;again,this is equivalent to the fact that sup E C q(E)<+∞,and cotype is therefore another super-property. In both de?nitions of type and cotype,the choice of the L2norm for the Rademacher averages is irrelevant(except for the exact value of the constants); this follows from Kahane’s inequalities(see[Ka,Chapter II,Th.4]),which state that for every q<∞,there exists a constant K q such that 1 0 n i=1 εi(t)x i q dt 1/q ≤K q 1 n i=1 εi(t)x i dt for every n≥1and every family(x i)n i=1 of vectors in a Banach space. It is easy to show that when X has type or cotype,then the same holds for the space L2(X)of X-valued square integrable functions.This fact is used below in section5and section6. 3Ultrapowers and some operator lemmas In the next section about Krivine’s theorem,we use a classical fact for op-erators on a complex Banach space X:ifλis a boundary point of the spec-trum Sp(T)of T∈L(X),thenλis an approximate eigenvalue for T,which means that there exists a sequence(x n)?X of norm one vectors such that lim n(T(x n)?λx n)=0.We shall need a slightly less classical fact about com-muting operators,which is very easy to obtain using the notion of ultrapower (Lemma1below).We shall?rst recall a few facts about ultrapower techniques. These techniques became popular in Banach space theory after the paper by Dacunha-Castelle and Krivine[DK];approximately at the same time,similar objects were introduced for C?-algebras[Ja].The limit spaces used by James [J2]in his study of super-re?exivity,the spreading models of Brunel-Sucheston [BS1],belong to the same family of tools which make possible to construct an abstract space from di?erent pieces taken at di?erent places. Suppose that U is a non-trivial ultra?lter on N.If X is a Banach space,we consider in X∞:= ∞(X)the closed subspace K U of all sequences y=(y n)∈X∞such that lim n→U y n =0,and we let X U be the quotient space X∞/K U. LetπU denote the quotient map from X∞to X U.If x=(x n)andξ=πU(x), then ξ =lim n→U x n .We have a canonical isometry i X,U from X to X U that sends x∈X to the class of the constant sequence x=(x n)where x n=x for every https://www.wendangku.net/doc/8613980326.html,ing this isometric embedding we shall consider that X?X U. The crucial fact is here:suppose thatη1,...,η ∈X U are represented by sequences y j=(y j,n)n≥0∈X∞,for j=1,..., ,and that we have a?nite number of inequality relations (R)a i< x i+ j=1 b i,jηj < c i,i=1,...,k, where a i,c i∈R,x i∈X,(b i,j)is a matrix of scalars.Let us say that a property depending upon n∈N is true when n is U-large if the set A?N of those n for which the property holds belongs to U;then we can say that when n is U-large,we have in X (R n)a i< x i+ j=1 b i,j y j,n < c i,i=1,...,k. This implies that X U is?nitely representable in X(and slightly more:if E is any?nite dimensional subspace of X U,we can?nd a(1+ε)-isomorphism T from E into X such that T(x)=x for every x∈E∩X).We see that F belongs to F(X)if and only if F is isometric to a subspace of X U.Every super-property of X passes to X U,for example type or cotype. Suppose now that T is a bounded linear operator on X.We de?ne T∞on X∞in the obvious way, T∞(x)=(T(x n)), whenever x=(x n)∈X∞.It is clear that K U is stable under T∞,so that T∞induces a bounded linear map T U on X U.It is easy to check that T→T U is an isometric homomorphism of unital Banach algebras from L(X)to L(X U). Using the above principle(R)?(R n),we see that if x=(x n)∈X∞and ifξ=π(x)∈X U,then this vectorξsatis?es T U(ξ)=λξif and only if lim n→U(T(x n)?λx n)=0;in particular,if X is complex,for every boundary pointλof the spectrum of T we can?nd a sequence(x n)?X of norm one vectors such that lim n(T(x n)?λx n)=0,which shows that the eigenspace ker(T U?λI)is not trivial. Lemma1Suppose that X is a complex Banach space,and that S,T are com-muting bounded linear operators on X.If(x n)?X is a sequence of norm one vectors such that T(x n)?λx n tends to0,we can?ndμ∈C and a norm one vector x∈X such that T(x)～λx and S(x)～μx. PROOF.We know that Xλ=ker(T U?λI)is not{0},and S U commutes with T U,therefore Xλis stable under S U.Ifμis a boundary point of the spectrum of the restriction of S U to Xλ,we can?nd a norm one vectorξin Xλsuch that S U(ξ)～μξ.Bringing backξto X—using(R)?(R n),with η1=ξ,η2=T U(ξ)andη3=S U(ξ)—we obtain for everyε>0a norm one vector x∈X such that T(x)?λx <εand S(x)?μx <ε. Let X be a complex Banach space,and let T be an into isomorphism from X into X,with x ≤C T(x) for every x∈X.For every integer n≥1,we may de?ne K n as the smallest constant for which x ≤K n T n(x) for every x∈X.It is clear that K m+n≤K m K n,so that r=lim n K1/n n exists by a standard lemma.Also,K n≤C n and K n T n(x) ≤K n T n x yield that0< T ?1≤r≤C. Lemma2There existsλ∈C with|λ|=r and a sequence(x n)of norm one vectors in X such that lim n(T(x n)?λ?1x n)=0. PROOF.We introduce an operator S of which r will be the spectral radius; this S acts as a sort of inverse for T U.For every x∈X,let N(x)denote the supremum of k such that x belongs to the range of T k(this value N(x) may be+∞).Let Z0be the subspace of X U consisting of allξthat have a representative x=(x n)such that lim U N(x n)=+∞.It is obvious that Z0 is stable under T U;let Z be the closure in X U of Z0,and let T Z denote the restriction of T U to Z. Whenξ∈Z0,we see thatξ=T Z(η)for some(unique)η:indeed,if x=(x n) belongs to the class ofξand lim U N(x n)=+∞,we have that N(x n)≥1 when n is U-large,which means that A={n:N(x n)≥1}∈U;hence for every n∈A we have x n=T(y n)for some y n∈X;if we let y n=0for n/∈A, then y=(y n)satis?es lim n→U N(y n)=+∞(because N(y n)≥N(x n)?1); ifη=π(y),thenηbelongs to Z and T Z(η)=ξ;clearly η ≤C ξ .This shows that T Z is invertible in L(Z). Let S=T?1 Z .It is quite clear that S n ≤K n,so that the spectral radius ρ(S)=lim k S k 1/k of S satis?esρ(S)≤r;we shall see that r=ρ(S).Let us ?x k ≥2and ε>0.For n large,we know that K nk >(r ?ε)nk ,thus we can ?nd a vector x n ∈X such that x n >(r ?ε)nk T nk (x n ) .Let h be a large integer,but small compared to n ,say h ?1<√n ≤h for example.If we had T jk (x n ) ≤(r ?2ε)k T jk +k (x n ) =(r ?2ε)k T k (T jk (x n )) for every j =h,...,n ?1,it would follow that (r ?ε)nk T nk (x n ) < x n ≤C hk (r ?2ε)nk ?hk T nk (x n ) , which is impossible when n is large.For every n ≥n 0,and for some j such that √n ≤j 1= y n >(r ?2ε)k T k (y n ) ,and this vector satis?es N (y n )≥k √n .If y =(y n )and η=π(y )we get η∈Z and S k (η) >(r ?2ε)k η .It follows that the spectral radius of S is larger than r ?2ε,hence equal to r . Let λ∈Sp(S )be such that |λ|=r .It follows from the “boundary of the spectrum lemma”that we can ?nd a norm one vector ξ∈Z such that S (ξ)～λξ,or T Z (ξ)～λ?1ξ;bringing back ξto X in the usual way gives a norm one vector x for which T (x )～λ?1x ,as was to be proved. 4Krivine’s theorem See [MS,Chapter 12]or [BLi,Chapter 12]for a more precise presentation of the results of this section.I prefer here to tell a pleasant story,rather than being too technical.Roughly speaking,Krivine’s theorem says that every Banach space X contains (1+ε)-isomorphs of n p ,for some p ∈[1,+∞]and every n ≥1,or in other words it says that some p (or c 0,when p =+∞)is ?nitely representable in X .More precise statements tell us that,given a basic sequence in X ,or simply a sequence (x n )with no Cauchy subsequence,there exists p ∈[1,+∞]such that for every n ≥1and ε>0,we can ?nd blocks of the given sequence that are (1+ε)-equivalent to the unit vector basis of n p .It is sometimes useful to be more speci?c,and to predict what values of p can be realized,starting from some norm invariants of the sequence (x n ).This will be the case in the next section about type,cotype and the MP+K theorem.The proofs of Krivine’s theorem are usually divided into two steps:the ?rst step replaces the given sequence by one that has some minimal regularity;this step uses only subsequences,or just di?erences of two vectors from the original sequence (as opposed to the second step,that requires clever long blockings).The argument is due to Brunel and Sucheston:given a sequence with no Cauchy subsequence,and using Ramsey’s theorem,we may ?nd a subsequence which is asymptotically invariant under spreading,see[BS1],and also[Go]; alternatively,this can be achieved by general abstract arguments involving iterated ultrapowers,usual in model theory where a somewhat parent notion of indiscernible sequence is de?ned.Given a Banach space X and a space Y of scalar sequences,we say that Y is a spreading model for X if there exists a normalized sequence(x n)?X,with no Cauchy subsequence,such that k j=1 a j e j Y=lim k j=1 a j x n j X for every k≥1and all scalars(a j)k j=1;the limit is taken when n1→∞and n1 The second part of this?rst step,also due to Brunel and Sucheston,is to observe that the di?erences(e2j+1?e2j)are suppression-unconditional in Y (see below for a de?nition);further,the di?erences are bounded away from zero because the sequence(x n)had no Cauchy subsequence;this implies that we can?nd2-unconditional?nite sequences(z i)k i=1in X,with k as large as we wish,whose vectors z i are di?erences z i=x n 2i ?x n 2i?1 of two suitable vectors from the given sequence(x n).The spreading model Y is?nitely representable in X,in a special way:any?nite sequence(y k)of blocks of the basis in Y can be sent to blocks from the sequence(x n)in X.We shall therefore present the rest of the proof of Krivine’s theorem assuming that we start from this situation,replacing the original space X by a spreading model X ,which is (block)?nitely representable in X and has a nice basis.The real thing is to prove Krivine’s theorem for X . Let X be a Banach space with a basis(e n)n≥0;we say that this basis is a suppression-unconditional basis when for every x∈X,the norm does not increase if we replace one of the non-zero coordinates of x by0;this yields that the basis is unconditional,with unconditionality constant≤2(in the real case).Let X be a Banach space with a suppression-unconditional basis (e n)n≥0;we say that the norm is invariant under spreading if for every integer k≥0and all n0 k j=0 a j e n j = k j=0 a j e j for all scalars(a j).Let x= k j=0 a j e j be a vector with?nite support in X; we say that y is a copy of x if y= k j=0a j e n j for some n0 x= a j e j and y= b j e j,we write x x appear before those of y,that is max{j:a j=0} After the preliminary work has been done,the heart of Krivine’s result is the following Theorem 3.The arguments of Brunel-Sucheston imply that for every Banach space X ,we can ?nd a space X 0with a suppression-unconditional basis,invariant under spreading,such that X 0is ?nitely representable in X ;if X 0contains k p ,then X will also.We shall therefore assume that X is a Banach space with a suppression-unconditional basis (e n )n ≥0,and a norm invariant under spreading.For every integer n ≥1,let R n be the smallest constant and S n be the largest constant such that for every x ∈X ,we have S n x ≤ n i =1x i ≤R n x whenever x 1 Theorem 3Let X be a Banach space with a suppression-unconditional basis (e n )n ≥0,and a norm invariant under spreading;suppose that p ≥1is de?ned by the equation (a ):21/p =lim sup n (R 2n )1/n or (b ):21/p =lim inf n (S 2n )1/n .For every k ≥1and ε>0it is possible to ?nd k successive blocks x 1<... PROOF.Let I be the set of rational numbers r such that 0≤r <1,let (f r )r ∈I be the standard unit vector basis for R (I ),and let us de?ne a norm on the linear span Y 0of (f r )r ∈I as follows:if r 0 k j =0 a j f r j Y = k j =0a j e j X for all scalar coe?cients (a j ).If Y 0is real,we complexify it in any reasonable way,for example x +iy =sup θ sin(θ)x +cos(θ)y , which preserves invariance under spreading and unconditionality.Let Y be the completion of Y 0;it is clear that (f r )is a suppression-unconditional basis for Y ,invariant under spreading in the new context.We say that y ∈Y is a copy of y = r ∈I a r f r if y = r ∈I a r f φ(r )for some increasing map φfrom I into itself.What we mean by successive copies of a given vector in Y is clear.It is also clear that Y is ?nitely representable in X ,and a ?nite-dimensional subspace of Y generated by successive copies of some vector in Y can be approximated by a subspace of X ,generated by successive copies of some vector in X . We can now relate the behaviour of sums of copies of vectors in X to the properties of some linear operators de?ned on this space Y .Indeed,we may de?ne a doubling operator D on Y by the formula ?y ∈Y,D (y )= 0≤r<1/2y (2r )f r + 1/2≤r<1y (2r ?1)f r , or D (y )(r )=y (2r mod 1),considering y as a function I →C .For every y ∈Y 0,the vector D (y )is the sum of two copies y 1 before Lemma 2),therefore if 21/p =lim n S 1/n 2n ,we know by Lemma 2that we can again ?nd λ∈C with |λ|=21/p and a norm one vector z ∈Y 0such that D (z )～λz .Using unconditionality,we get D (|z |)～|λ||z |.In both cases (a)and (b)we found a norm one vector y =α|z |∈Y 0(with 1/2≤α≤2)such that D (y )～21/p y .Reproducing y in X gives a norm one vector x ∈X such that,when x 1 j =1a j x j p ～k j =1a p j , provided all coe?cients are of the form a j =2?k j /p ,for some integer k j such that 0≤k j ≤n ,and j a p j =1(if K =max k j ,replace each a j x j by 2 K ?k j copies of 2?K/p x ;this gives 2K copies of 2?K/p x ,which we may group two by two again,obtaining after K steps a single copy of the vector x ). This is not quite enough,and we also introduce an operator T on Y which reproduces three times every vector y ∈Y , T (y )= 0≤r<1/3y (3r )f r + 1/3≤r<2/3y (3r ?1)f r + 2/3≤r<1y (3r ?2)f r . It is clear that DT =T D is the operator that replaces every vector x by six copies of x ;the commutation property and Lemma 1enable us to ?nd a norm one vector z such that D (z )～21/p z and T (z )～μz ;then T (|z |)～|μ||z |,so that we may assume that z and μare real and ≥0.Some simple lattice arguments (involving comparisons of the norms of sums of respectively 2h ,3i and 2j copies of z when 2h <3i <2j )show that necessarily μ=31/p .If D (z )?21/p z and T (z )?31/p z are small enough,and if z 1 (K )(1+ε)?p/2k j =1a p j ≤ k j =1a j x j p ≤(1+ε)p/2k j =1a p j , for all non-negative scalars (a j ).Everything would be ?ne if the basis in X was 1-unconditional,but it is not so:what we get so far is a sequence x 1,...,x k which is 2(1+ε)-equivalent to the k p -basis in the real case,and 4(1+ε)in the complex case,for every k ≥2:if v = a j x j ,the p -norm of the coe?cients is dominated by ( v + p + v ? p )1/p ≤21/p v ,using ?rst (K )then suppression unconditionality;in the other direction use v ≤21?1/p ( v + p + v ? p )1/p .Suppose p <∞for simplicity;if k =m 2and if we form new blocks y 1,...,y m in X of the form y i =m ?1/p m j =1(?1)j x m (i ?1)+j ,then (y 1,...,y m )is still a sequence of successive copies of some y ∈X ,hence invariant under spreading,5-equivalent to the m p basis (say),but the unconditional constant is improved to something arbitrarily close to 1as m grows (in the complex case,a similar trick using a primitive root of unity does the required job).We build a limit space X 1from the sequence ([y (m )1,...,y (m )m ])m ,by setting n j =1a j e j X 1=lim m →U n ∧m j =1a j y (m ) j for every n ≥1and all scalars (a j ).This space X 1is ?nitely representable in X ,with a 1-unconditional basis,invariant under spreading,and 5-equivalent to the p basis.In X1we have clearly21/p=lim sup n(R2n)1/n.Applying the above construction to X1gives new blocks x1,...,x k that satisfy(K )in X1: this?nishes the proof,since the basis in X1is1-unconditional. The proof above is due to Lemberg[Le],who was Krivine’s PhD student in the years’80.The fundamental facts are still the same as in the original paper [Kr2],but the details in[Kr2]are harder to https://www.wendangku.net/doc/8613980326.html,bining the arguments of Brunel-Sucheston and the preceding Theorem,we obtain one of the usual forms of Krivine’s theorem. Corollary4Suppose that X is a real or complex Banach space,and(x n)a bounded sequence in X with no Cauchy subsequence.For some p∈[1,+∞], for every k≥1andε>0it is possible to?nd k successive blocks of the given sequence that are(1+ε)-equivalent to the unit vector basis of k p . Our next Corollary is expressed in a slightly unnatural way,but suitable for the next section. Corollary5Suppose r,s≥1are given.If for someκ>0and for every n≥2,a Banach space X contains a normalized suppression-unconditional sequence y(n)=(y(n)1,...,y(n) n )such that i∈C y(n)i ≥κ|C|1/r for every subset C?{1,...,n},or such that i∈C y(n)i ≤κ|C|1/s for every subset C?{1,...,n},then for some p≤r(or p≥s)and for every k≥1,ε>0,it is possible when n≥N(k,ε)to form k successive blocks of the given sequence y(n)that are(1+ε)-equivalent to the unit vector basis of k p . PROOF.We construct as we did before a limit space X from the long sequences as https://www.wendangku.net/doc/8613980326.html,ing Brunel-Sucheston principle,we may select from our long sequences(y(n)i)some(?nite)subsequences z(n)1,...,z(n) k n that are almost indiscernible,and have a length k n tending to∞with n;then we de?ne a norm on c00(the space of?nitely supported scalar sequences)by m i=1 c i e i X =lim n→U m∧k n i=1 c i z(n)i where(e i)i≥0denotes the unit vector basis of c00.Notice that when n is U-large,the length k n exceeds m;this yields that(e i)is normalized in X .We obtain a space X with a normalized suppression-unconditional basis and a norm invariant under spreading.In the?rst case,we get for every n≥1 κn1/r≤ n i=1 e i X ≤R n e1 X =R n, and similarly in the second case we obtain that S n≤κn1/s.We know from Theorem3that we may get k p in X ,with p such that21/p=lim n(R2n)1/n or 21/p=lim n(S2n)1/n,thus p≤r in the?rst case and p≥s in the second. 5Type,cotype and n p s.The MP+K theorem Let X be a Banach space.We denote by p X the supremum of all p such that X has type p,and by q X we denote the in?mum of all q such that X has cotype q.It is clear using Khintchine’s inequality that p X≤2≤q X,already when X=R. Theorem6Let X be an in?nite dimensional Banach space;for every integer k≥1andε>0,the space X contains(1+ε)-isomorphs of k p X and of k q X . For the type case and1 in X of k p is given there as a function of the stable type p constant ST p(X) of the normed space X. PROOF.If p X=2we may use Dvoretzky’s theorem[D2].Assume p X<2 and choose r such that p X 1 n i=1 εi(t)x i r dt≤?(n)r n i=1 x i r for every family x1,...,x n of n vectors in X.It is clear that?is non-decreasing, and tends to+∞since X does not have type r.Suppose that x1,...,x n are chosen in X so that n i=1 x i r=1and 1 n i=1 εi(t)x i r dt> 1999 2000 ?(n)r. We shall use an exhaustion argument inspired by Nikiˇs in’s paper[N2].Let (Bα)α∈I be a maximal family of disjoint subsets of{1,...,n}such that 1 i∈Bα εi(t)x i r dt< 1 2000 i∈Bα x i r. If B denotes the union of these sets Bα,and m denotes the cardinality of I (notice that m 1 i∈B εi(t)x i r dt= α∈I εα(s)( i∈Bα εi(t)x i) r ds dt ≤?(m)r α∈I 1 i∈Bα εi(t)x i r dt≤ ?(m)r 2000 α∈I i∈Bα x i r ≤ ?(n)r 2000 n i=1 x i r= ?(n)r 2000 . Let A denote the complement of B and for every j≥0let A j={k∈A:2?j?1< x k ≤2?j}. Observe that x k ≤1for every k because r i=1 x i r=1,so that the sets (A j)j≥0cover the set A.Let N=max j|A j|denote the maximal cardinality of the sets(A j)j≥0.Then ( 1 i∈A εi(t)x i r dt)1/r≤ +∞ j=0 ( 1 i∈A j εi(t)x i r dt)1/r≤N +∞ j=0 2?j=2N. We obtain 1999 2000 1/r ?(n)< 1 n i=1 εi(t)x i r dt 1/r ≤ ?(n) 20001/r +2N which shows that N is big when?(n)is big.Let j0be such that|A j |=N. By maximality of B we obtain for every non-empty subset C of A j 1 i∈C εi(t)x i r dt≥ 1 2000 i∈C x i r≥ 2?(j0+1)r 2000 |C|. Replacing the vectors(x i)i∈A j0 by normalized vectors(y i),we obtain a nor-malized sequence(y1,y2,...,y m),as long as we wish,such that ( 1 i∈C εi(t)y i r dt)1/r≥κ|C|1/r for every subset C of{1,...,m}(withκ=1 22000?1/r).This inequality re- mains true if we replace the L r(X)norm by the norm of L1(X)andκby someκ >0(use Kahane’s inequalities).For every n≥1,we may thus?nd an unconditional normalized sequence in L1(X),of the form(εj y(n)j)n j=1,with the above property,and since r<2it implies that for some c=c(r,κ )>0,we have n j=1a jεj y(n)j L 1(X) ≥c( n j=1 |a j|2)1/2for all scalars.From Corollary5 follows that for every integer m,we can,when n is large enough,get blocks z1,...,z m∈L1(X)of(εj y(n)j)n j=1that are(1+ε)-equivalent to the unit vector basis of m p for some p≤r,and the 2-norm of the coe?cients in each block z i is bounded by c(r,κ )?1.By Kahane’s inequalities again,all L s(X)norms are equivalent on the span of(εj y(n)j)n j=1,hence the sequence(z1,...,z m)con- sidered in L2(X)is uniformly equivalent to the unit vector basis of m p ;since L2(X)has type s whenever X has type s,we have for every s K?1m1/p≤ 1 m i=1 εi(t)z i L 2(X) dt≤K s m1/s for every m≥1.This yields that s≤p,for every s Starting with a long enough sequence(z i)m i=1and blocking again in the p- sense we may?nd three blocks b1,b2,b3∈L1(X)of some sequence(εj y j), supported on three disjoint intervals J1,J2,J3and such that,lettingω=(εj) and b i(ω)= j∈J i a jεj y j,i=1,2,3, then the three functions b1,b2,b3are(1+ε)-equivalent to the unit vector basis of 3 p in the norm of L1(X),and the coe?cients satisfy j∈J i |a j|2< τ2/12for i=1,2,3and a smallτ>0(use p<2).For every?xed triple (c1,c2,c3)of scalars,this implies by Azuma’s inequality(see[MS,7.4])a strong concentration for the set ofωsuch that c1b1(ω)+c2b2(ω)+c3b3(ω) ～(|c1|p+|c2|p+|c3|p)1/p, and allows us to select a choice ofω=(εj)that works for all(c i)3 i=1,by a standardδ-net argument on the unit sphere of 3 p ;this shows that for most of 第一篇、教你学会看电路图轻松修手机 一、一套完整的主板电路图，是由主板原理图和主板元件位置图组成的。 1.主板原理图，如图： 2.主板元件位置图，如图： 主板元件位置图的作用：是方便用户找到相应元件所在主板的正确位置。而主板原理图是让用户对主板的电路原理有所了解，知道各个芯片的功能，及其线路的连接。 二、相关名词解释 电路图中会涉及到许多英文标识，这些标识主要起到了辅助解图的作用，如果不了解它们，根本不知道他们的作用，也就根本不可能看得懂原理图。所以在这里我们会将主要的英文标识进行解释。希望大家能够背熟记熟，同时希望大家多看电路图，对不懂的英文及时查找记熟。 如图： 以上英文标识在电路图上会灵活出现，比如“扬声器”是“SPEAKER” ，它的缩写就是“SPK”，“正极”是“positive” ，缩写是“P” ，那么如果在图中标记SPKP，那么就证明它是扬声器正极。所以当有英文不明白的时候，可以将它们拆开后再进行理解，请大家灵活运用。 第二节主板元件位置图 一、元件编号 每一个元件在主板元件位置图中，都有一个唯一的编号。这个编号由英文字母和数字共同组成。编号规则可以分成以下几类： 芯片类：以U 为开头，如CPU U101 接口类：以J 为开头，如键盘接口J1202 三极管类：以Q 为开头，如三极管Q1206 二级管类：以D 为开头，如二极管D1102 晶振类：以X 为开头，如26M 晶体X901 电阻类：以R 或VR（压敏电阻）为开头，如电阻R32 VR211 电容类：以C 为开头，如电容C101 电感类：以L 为开头，如电感L1104 侧键类：以S 为开头，如侧键S1201 电池类：以 B 为开头，如备用电池B201 屏蔽罩：以SH 为开头，如屏蔽罩SH1 振动器：以M 为开头，如振子M201 还有一部分标号是主板上的测试点，以TP 为开头。 二、查找元件功能 用户可以根据相应的元件编号去查找主板原理图，从而了解此元件的作用。随便拿块主板作为示例。 如果想了解某一个元件的主要功能（图中红圈内元件） 如图： 第二部分原理篇 第一章手机的功能电路 ETACS、GSM蜂窝手机是一个工作在双工状态下的收发信机。一部移动电话包括无线接收机(Receiver)、发射机(Transmitter)、控制模块(Controller)及人机界面部分(Interface)和电源(Power Supply)。 数字手机从电路可分为，射频与逻辑音频电路两大部分。其中射频电路包含从天线到接收机的解调输出，与发射的I/Q调制到功率放大器输出的电路；逻辑音频包含从接收解调到，接收音频输出、发射话音拾取(送话器电路)到发射I/Q调制器及逻辑电路部分的中央处理单元、数字语音处理及各种存储器电路等。见图1-1所示 从印刷电路板的结构一般分为：逻辑系统、射频系统、电源系统，3个部分。在手机中，这3个部分相互配合，在逻辑控制系统统一指挥下，完成手机的各项功能。 图1-1手机的结构框图 注：双频手机的电路通常是增加一些DCS1800的电路,但其中相当一部分电路是DCS 与GSM通道公用的。 第二章射频系统 射频系统由射频接收和射频发射两部分组成。射频接收电路完成接收信号的滤波、信号放大、解调等功能；射频发射电路主要完成语音基带信号的调制、变频、功率放大等功能。手机要得到GSM系统的服务,首先必须有信号强度指示,能够进入GSM网络。手机电路中不管是射频接收系统还是射频发射系统出现故障,都能导致手机不能进入GSM网络。 对于目前市场上爱立信、三星系列的手机,当射频接收系统没有故障但射频发射系统有故障时,手机有信号强度值指示但不能入网；对于摩托罗拉、诺基亚等其他系列的手机,不管哪一部分有故障均不能入网,也没有信号强度值指示。当用手动搜索网络的方式搜索网络时,如能搜索到网络,说明射频接收部分是正常的；如果不能搜索到网络,首先可以确定射频接收部分有故障。 而射频电路则包含接收机射频处理、发射机射频处理和频率合成单元。 第一节接收机的电路结构 移动通信设备常采用超外差变频接收机，这是因为天线感应接收到的信号十分微弱,而鉴频器要求的输人信号电平较高,且需稳定。放大器的总增益一般需在120dB以上,这么大的放大量,要用多级调谐放大器且要稳定,实际上是很难办得到的，另外高频选频放大器的通带宽度太宽,当频率改变时,多级放大器的所有调谐回路必须跟着改变，而且要做到统一调谐, 一，手机原理图的种类： 手机电路图共分四类：1，方框图；2，整机电原理图；3，元件排列图；4，彩图。 1，方框图： 利用方块形式粗略概述手机的结构与工作原理，方便初学者掌握手机的结构与工作原理，为初学者读懂电原理图打下基础。 2，整机电原理图： 利用电子原件符号清楚表示手机中各元器件的连接和工作原理，方便维修时分析电路原理及故障分析。 3，元件排列图： 利用元件编号在板位图上标明元件所在位置，方便维修时寻找元件在板上的位置。 4，彩图： 即手机照片，方便维修时对照板元件缺损，错位，元件方向。 二，手机电路图的读解原则： 1，读图前要打好电子基础，熟悉各种电子元器件符号，特性和用途；电子元器件在电路中的接法；电路中的电流，电压，电阳之间的关系（欧姆定律）。 2，先读懂方框图，大根了解本机的结构（如那种电源结构，那种时钟结构）；然后按所学的原理去分析原理图。 3，读图时先弄懂直流供电电路，后弄懂交流信号通路。 4，手机电路图是有规律的，一般电源居左下；控制居右下。左射频右逻辑；上收下发中本振。三，手机电路图的读解方法： 1，电源电路读图要点： 1），先了解本机属那种电源结构（分三种）以电源集成为核心。 2），从尾插或电池脚开始，找出电池电压（VBATT，B+）输入线；电池电压一般直接供到电源集成块，充电集成块，功放，背光灯，振铃，振动等电路；也可从上述电路回找。 3），在电源集成块，键盘，内联座处找到开机触发线（ON/OFF或标有开关符号）。 4），在电源集成块上找出各路电压输出线（包括电压走向，电压值多少，是恒定的还是跳变的，在那个单元上可以测到该电压）。 1）VDD--逻辑电压给CPU，字库，暂存等电路（1。8V/2。8V） 2）SYN-VCC（XVCC）时钟电压，使13M电路工作（2。8V） 3）AVCC--音频电压（2。8V） 4）VREF--中频电压（2。8V跳变） 5）3VTX--发射电压（3V跳变） 6）SYN-VCC---频合电压（2。8V） 7)VRTC--实时时钟电压（3V） 8）SIM-VCC--SIM卡电路电压（3V/5V跳变） 9）RST（PURX）--复位信号（0-2。8V） 4），在CPU与电源集成块间找到开机维持线（WD-CP，WATCCH DOG）。 5），从键盘，电源集成块旁边的开关符号到CPU找到关机检测线。 2），充电电路读图要点： 1），以电源集成块或充电集成为核心，找到充电电路。 2），从充电接口（尾插）到电源集成块或充电集成块找出外电输入线 手机供电电路结构和工作原理 一、电池脚的结构和功能。 目前手机电池脚有四脚和三脚两种：（如下图） 正温类负正温负 极度型极极度极 脚脚脚 （图一）（图二） 1、电池正极（VBATT）负责供电。 2、TEMP：电池温度检测该脚检测电池温度；有些机还参与开机，当用电池能开机，夹正负极不能开机时，应把该脚与负极相接。 3、电池类型检测脚（BSI）该脚检测电池是氢电或锂电，有些手机只 认一种电池就是因为该电路，但目前手机电池多为锂电，因此，该脚省去便为三脚。 4、电池负极（GND）即手机公共地。 二、开关机键: 开机触发电压约为2.8-3V(如下图)。 内圆接电池正极外圆接地;电压为0V。 电压为2.8-3V。 触发方式 ①高电平触发：开机键一端接VBAT，另一端接电源触发 脚。 （常用于：展讯、英飞凌、科胜讯芯片平台） ①低电平触发：开机键一端接地，另一端接电源触发脚。 （除以上三种芯片平台以外，基本上都采用低电平触发。如：MTK、AD、TI、飞利浦、杰尔等。） 三星、诺基亚、moto、索爱等都采用低电平触发。 三、手机由电池直接供电的电路。 电池电压一般直接供到电源集成块、充电集成块、功放、背光灯、振铃、振动等电路。在电池线上会并接有滤波电容、电感等元件。该电路常引起发射关机和漏电故障。 四、手机电源供电结构和工作原理。 目前市场上手机电源供电电路结构模式有三种； 1、 使用电源集成块（电源管理器）供电；（目前大部分手机都使用该电路供电） 2、 使用电源集成块（电源管理器）供电电路结构和工作原理：（如下图） 电池电压 逻辑电压(VDD) 复位信号(RST) 射频电压(VREF) VTCXO 26M 13M ON/OFF AFC 开机维持 关机检测 （电源管理器供电开机方框图） 1）该电路特点： 低电平触发电源集成块工作； 把若干个稳压器集为一个整体，使电路更加简单； 把音频集成块和电源集成块为一体。 2）该电路掌握重点: 电 源 管 理 器 CPU 26M 中频 分频 字库 暂存 两小时学会看懂手机电路图 电路图的种类 常见手机维修中的电子电路图有原理图、方框图、元件分布图、装配图和机板图等 (1)原理图 原理图就是用来体现电子电路的工作原理的一种电路图，又被叫做"电原理图"。这种图，由于它直接体现了电子电路的结构和工作原理，所以一般用在设计、分析电路中。分析电路时，通过识别图纸上所画的各种电路元件符号，以及它们之间的连接方式，就可以了解电路的实际工作时情况。原理图又可分为整机原理图，单元部分电路原理图，整机原理图是指手机所有电路集合在一起的分部电路图。 (2)方框图(框图) 方框图是一种用方框和连线来表示电路工作原理和构成概况的电路图。从根本上说，这也是一种原理图，不过在这种图纸中，除了方框和连线，几乎就没有别的符号了。它和上面的原理图主要的区别就在于原理图上详细地绘制了电路的全部的元器件和它们的连接方式，而方框图只是简单地将电路 (3)元件分布图 它是为了进行电路装配而采用的一种图纸，图上的符号往往是电路元件的实物的外形图。我们只要照着图上画的样子，这种电路图一般是供原理和实物对照时使用的。 (4)机板图 机板图的是"印刷电路板图"或"印刷线路板图"，它和元件分布图其实属于同一类的电路图，都是供原理图联系实际电路使用的。 印刷电路板是在一块绝缘板上先覆上一层金属箔，再将电路不需要的金属箔腐蚀掉，剩下的部分金属箔作为电路元器件之间的连接线，然后将电路中的元器件安装在这块绝缘板上，利用板上剩余的金属箔作为元器件之间导电的连线，完成电路的连接。由于铜的导电性能不错，加上相关技术很成熟，所以在制作电路板时，大多用铜。所以，印刷电路板又叫"覆铜板"。但是大家也要注意到：机板图的元件分布往往和原理图中大不一样。这主要是因为，在印刷电路板的设计中，主要考虑所有元件的分布和连接是否合理，要考虑元件体积、散热、抗干扰、抗耦合等等诸多因素，综合这些因素设计出来的印刷电路板，从外观看很难和原理图完全一致；而实际上却能更好地实现电路的功能。 随着科技发展，现在印刷线路板的制作技术已经有了很大的发展；除了单面板、双面板外，还有多面板， 专门找了几个例子，让大家看看。自己也一边学习。 分析一个电源，往往从输入开始着手。220V交流输入，一端经过一个4007半波整流，另一端经过一个10欧的电阻后，由10uF电容滤波。这个10欧的电阻用来做保护的，如果后面出现故障等导致过流，那么这个电阻将被烧断，从而避免引起更大的故障。右边的4007、4700pF电容、82KΩ电阻，构成一个高压吸收电路，当开关管13003关断时，负责吸收线圈上的感应电压，从而防止高压加到开关管13003上而导致击穿。13003为开关管(完整的名应该是MJE13003)，耐压400V，集电极最大电流1.5A，最大集电极功耗为14W，用来控制原边绕组与电源之间的通、断。当原边绕组不停的通断时，就会在开关变压器中形成变化的磁场，从而在次级绕组中产生感应电压。由于图中没有标明绕组的同名端，所以不能看出是正激式还是反激式。 不过，从这个电路的结构来看，可以推测出来，这个电源应该是反激式的。左端的510KΩ为启动电阻，给开关管提供启动用的基极电流。13003下方的10Ω电阻为电流取样电阻，电流经取样后变成电压(其值为10*I)，这电压经二极管4148后，加至三极管C945的基极上。当取样电压大约大于1.4V，即开关管电流大于0.14A时，三极管C945导通，从而将开关管13003的基极电压拉低，从而集电极电流减小，这样就限制了开关的电流，防止电流过大而烧毁(其实这是一个恒流结构，将开关管的最大电流限制在140mA左右)。 变压器左下方的绕组(取样绕组)的感应电压经整流二极管4148整流，22uF电容滤波后形成取样电压。为了分析方便，我们取三极管C945发射极一端为地。那么这取样电压就是负的(-4V左右)，并且输出电压越高时，采样电压越负。取样电压经过6.2V稳压二极管后，加至开关管13003的基极。前面说了，当输出电压越高时，那么取样电压就越负，当负到一定程度后，6.2V稳压二极管被击穿，从而将开关13003的基极电位拉低，这将导致开关管断开或者推迟开关的导通，从而控制了能量输入到变压器中，也就控制了输出电压的升高， EXPRESS WRITTEN PERMISSION OF QUALCOMM. ITS CONTENTS REVEALED IN ANY MANNER TO OTHERS WITHOUT THE NOT TO BE USED, COPIED, REPRODUCED IN WHOLE OR IN PART, NOR U.S.A. San Diego, CA 92121-17145775 Morehouse Drive QUALCOMM Incorporated Sheet # Content 01A. Table of Content 01B. Revision History 01C. Block Diagram 01D. GPIO Map 02. PM8916 Control and MPP/Clock 03. PM8916 Charging 04. PM8916 GPIO/MPP 05. PM8916 Buck converter 06. PM8916 LDO circuits 07. PM8916 CODEC 08. MSM8916 Control 09. MSM8916: EBI 10. MSM8916: GPIO 11. MSM8916: MIPI and RF Control 12. MSM8916: POWER113. MSM8916: POWER214. MSM8916:GND 15. MEMORY:LPDDR3+EMMC 16. Battery Connector 17. Subboard Connector 18. Mic and Receiver 19. EARPHONE 20. LCD interface and backlight 21. Main/Slave Camera and Flash 22. Sensors 23. SIM/TF card 27. WTR1605L TX 28. WTR1605L RX 29. WTR1605L POWER 30. WTR1605L POWER DISTRIBUTION 31. LTE/W/TD/GSM Antenna Switch 32. TRX_Front-High Band QFE234033. TRX_Front-Low Band QFE232034. RESERVED 35. TRX_UMTS_B1/2/5/836. PRX_B34/39/337. PRX_LTE_HB 38. DRX_Antenna Switch 39. QFE1550 DRX Tunner 40. B7/40/41 DRX Switch 41. B39/B1/B3 DRX Switch 42. WTR2605_SECONDARY PATH 43. WTR2605_POWER DISTRIBUTION 44. Antenna_SECONDARY PATH 45. RESERVED 46. RESERVED 47. CDMA BC0(Voice) TRX 48. ET_APT 50. WIFI FEM Sheet # Content 51. GPS/XO DISTRIBUTOR 52. NFC 49. WCN362024. Keypad/LED/Status indicator 25. Touch Interface 26. Test Point/GND/Shields A B C D D C B A 第一篇、教你学会看电路图轻松修手机My id：42409 My name：Aerlant 既然是教程就不能保证100%是原创,难免会引用老师们的宝贵经验，请您别介意哦！ 只要您认真学习完这些教程，就可以正式步入“专业手机维修”行业成为一名优秀的维修员喽！目的很简单，就是让新会员们、新手们，您加入帅虎论坛是正确的。在这里你可以学习到一些实实在在的维修知识，向更高的一个层次迈进、稳步成长。。。 言归正传！有兴趣的朋友往下看，学习一下： 第一节了解电路图 一、一套完整的主板电路图，是由主板原理图和主板元件位置图组成的。 1.主板原理图，如图： 2.主板元件位置图，如图： 主板元件位置图的作用：是方便用户找到相应元件所在主板的正确位置。而主板原理图是让用户对主板的电路原理有所了解，知道各个芯片的功能，及其线路的连接。 二、相关名词解释 电路图中会涉及到许多英文标识，这些标识主要起到了辅助解图的作用，如果不了解它们，根本不知道他们的作用，也就根本不可能看得懂原理图。所以在这里我们会将主要的英文标识进行解释。希望大家能够背熟记熟，同时希望大家多看电路图，对不懂的英文及时查找记熟。 如图： 以上英文标识在电路图上会灵活出现，比如“扬声器”是“SPEAKER” ，它的缩写就是“SPK”，“正极”是“positive” ，缩写是“P” ，那么如果在图中标记SPKP，那么就证明它是扬声器正极。所 以当有英文不明白的时候，可以将它们拆开后再进行理解，请大家灵活运用。 第二节主板元件位置图 一、元件编号 每一个元件在主板元件位置图中，都有一个唯一的编号。这个编号由英文字母和数字共同组成。编号规则可以分成以下几类： 芯片类：以U 为开头，如CPU U101 接口类：以J 为开头，如键盘接口J1202 三极管类：以Q 为开头，如三极管Q1206 二级管类：以D 为开头，如二极管D1102 晶振类：以X 为开头，如26M 晶体X901 电阻类：以R 或VR（压敏电阻）为开头，如电阻R32 VR211 电容类：以C 为开头，如电容C101 电感类：以L 为开头，如电感L1104 侧键类：以S 为开头，如侧键S1201 电池类：以 B 为开头，如备用电池B201 屏蔽罩：以SH 为开头，如屏蔽罩SH1 振动器：以M 为开头，如振子M201 还有一部分标号是主板上的测试点，以TP 为开头。 二、查找元件功能 用户可以根据相应的元件编号去查找主板原理图，从而了解此元件的作用。随便拿块主板作为示例。 第一篇、教你学会看电路图轻松修手机 既然是教程就不能保证100%是原创,难免会引用老师们的宝贵经验，请您别介意哦！ 只要您认真学习完这些教程，就可以正式步入“专业手机维修”行业成为一名优秀的维修员喽！目的很简单，就是让新会员们、新手们，您加入帅虎论坛是正确的。在这里你可以学习到一些实实在在的维修知识，向更高的一个层次迈进、稳步成长。。。 言归正传！有兴趣的朋友往下看，学习一下： 第一节了解电路图 一、一套完整的主板电路图，是由主板原理图和主板元件位置图组成的。 1.主板原理图，如图： 2.主板元件位置图，如图： 主板元件位置图的作用：是方便用户找到相应元件所在主板的正确位置。而主板原理图是让用户对主板的电路原理有所了解，知道各个芯片的功能，及其线路的连接。 二、相关名词解释 电路图中会涉及到许多英文标识，这些标识主要起到了辅助解图的作用，如果不了解它们，根本不知道他们的作用，也就根本不可能看得懂原理图。所以在这里我们会将主要的英文标识进行解释。希望大家能够背熟记熟，同时希望大家多看电路图，对不懂的英文及时查找记熟。 如图： 以上英文标识在电路图上会灵活出现，比如“扬声器”是“SPEAKER” ，它的缩写就是“SPK”，“正极”是“positive” ，缩写是“P” ，那么如果在图中标记SPKP，那么就证明它是扬声器正极。所以当有英文不明白的时候，可以将它们拆开后再进行理解，请大家灵活运用。 第二节主板元件位置图 一、元件编号 每一个元件在主板元件位置图中，都有一个唯一的编号。这个编号由英文字母和数字共同组成。编号规则可以分成以下几类： 芯片类：以U 为开头，如CPU U101 接口类：以J 为开头，如键盘接口J1202 三极管类：以Q 为开头，如三极管Q1206 二级管类：以D 为开头，如二极管D1102 晶振类：以X 为开头，如26M 晶体X901 电阻类：以R 或VR（压敏电阻）为开头，如电阻R32 VR211 简单手机电路图.txt我自横刀向天笑，笑完我就去睡觉。你的手机比话费还便宜。路漫漫其修远兮，不如我们打的吧。 1、方框图： 利用方块形式粗略概述手机的结构与工作原理，方便初学者掌握手机的结构与工作原理，初学者读懂电原理图打下基础 2、整机电原理图： 利用电子元件符号清楚表示手机中各元器件的连接和工作原理，方便维修时分析电路原理及故障 3、元件排列图： 利用元件编号在板位图上标明元件所在方便维修时寻找元件在机板上的位置。 4、彩图： 即手机照片，方便维修时对照机板元件缺损、错位 二、手机电路图的读解原则： 1、读图前先要打好电子基础，熟悉各种电子元件符号、特性和用途；电子元件在电路中的接法 2、先读懂方框图，大概了解本机的结构（如用哪种电源结构、哪种时钟电路）；然后按所学的原理去分析原理图。 3、读图时应先弄懂直流供电电路，后弄懂交流信号通路。 4、手机电路图是有规律的，一般电源居左下；控制居右下。左射频右逻辑；上收下发中本振。 三、手机电路图的读解方法： 1、电源电路读图要点： 1）、先了解本机属哪种电源结构（分三种）;以电源集成块为核心。 2）、从尾插或电池脚开始，找出电池电压（VBATT、B+）输入线；电池电压一般直接供到电源集成块、充电集成块、功放、背光灯、振铃、振动等电路；也可从上述电路往回找。 3）、在电源集成块、键盘、内联座处找到开机触发线（ON/OFF或标有开关符号）。 4）、在电源集成块上找出各路电压输出线（包括电压走向、电压值多少、是恒定的还是跳变的、在哪个元件上可测到该电压）。 1）VDD——逻辑电压给CPU、字库、暂存等电路（1.8V/2.8V） 2）SYN-VCC（XVCC）时钟电压，使13M电路工作（3.8V） 3）AVCC——音频电压(2.8V) 4）VREF—中频电压(2V跳变) 5）3VTX—发射电压(3V跳变) 6）SYN-VCC—频合电压 7）VRTC—实时时钟电压 8)SIM-CC—SIM电路电压（3V/5V跳变） 9）RST（PURX）——复位信号（2.8V） 4）、在CPU与电源集成块间找到开机维持线（WD-CP、WATCCH G）。 5）、从键盘、电源集成块旁边的开关符号到CPU找到关机检测线。 2）、充电电路读图要点： 1）、以电源集成块或充电集成块为核心，找到充电电路。 2）、从充电接口（尾插）到电源集成块或充电出外电输入线 3）、从外电输入线（DC-IN）9到CPU（或电源）找到充电检测线（CHECK）。 4）、从CPU（或电源）到充电集成块找到充电开关控制线（CHARG-ON）。 5）、从充电集成块（或电源）到电池脚（VBATT）找到充电压输出线。教你学会看手机电路图轻松修手机
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