C H A P T E R
PI^ELIMINAnES
In this chapter we collect, for the convenience of the reader, all the known concepts and results i^loh are required in the subsequent chapters. These are divided into sections bearing titles of the topics to which they relate.
1.1. Fitlds with valuation
Let Gi toe €m ordered group, nultiplicatively written. Let, further-f- toe an associative operation on G\ such that the distributive laws
(1.1.1) ac +• ^c z^(gu+&) (1.1.2) -Ca/4fr<^^<= CCOu^t)^ hold for all CL, -6- , C , in Gi . We shall adjoin a zero element to (1.1.3) OJ-O- OCL= 0 for all OU € Gi^ (1.1.4) 0 < (h for all a, & G. - 2 - DEFINITION 1.1.1 (BACHMAN [3], p. 72, Definition 2.2). A valuation of a field K is a map 1 \". K-^ 6i U{oJ, \^ei» G iB an ordered group provided (1.1.5) for a G K , (Cbl = 0 if and only L-f CL = Oj (1.1.6) for CO, ^€ K . 10.^1 = la.1 l-6-( ; (1.1.7) for a.-^GK, lCL+-tl^ l^l+l^l. It is always possible to introduce an additive operation in an ordezed group Gi by defining (1.1.8) OJ-^ (^ = 'Yv^doc. CCb , ^) for a/,^G6t. It is easily seen that this -p is associative and that the distributive laws are satisfied, Further, condition (1.1.7) of Definition 1.1.1 becomes (1.1.9) \ch~{-J^\ <, TVLC^ac Clcol , \^0. The valuation in this case will be called non-archlmediaB. Ve shall assume throughout that DEFINITION 1.1.2 (BACHMAN SO p.77, Definitions 3.1t 3.2). A subgroup H of DEFINITION 1.1.3 (BACHMAN [3l p. 77, Definition 3.3) If I I is a non-archimedian valuation of a field K into ^^U'foj where Gr is an ordered group, then the rank of \ I la the rank of Gj - 3 - PHOPOSITION 1.1.1 (BACHMAN [3] p. 81, Th«oi»m 3.4). An oTiered group G\ of rank 1 is order Isomorphic to a subgroup of the additive group Tl of real numbers. RBMAHK 1.1.1. In riew of Proposition 1.1.1, in consider ing valuations of rank 1, we may assume that the ordered group Gt is a subgroup of the additive group of real numbers with the usual ordering. We shall be concerned with valuations of rank 1 only. A valuation of rank 1 may thus be defined as follows. DEFINITIOH 1,1,4 (BACHMAH [3] p.5, Definition 2,1). A valuation of rank 1 of a field K is a mapping | I : from K into the reals 1\ such that for all CL , ^ e k ^ (1.1.10) \(Xj\7'yO and \ou\-0 if and only if CO = 0 ; (1.1.11) 1 OuL[ - fCLl 1^1 •. (1.1.12) I CL-f ^1 ^ la.l+1^1.; We shall generally omit, for the sake of brevity, mention ing that the valuation is of rank 1 explicitly and simply speak of such a function as a valuation, it being understood that such a map is always into the reals'Bv. K is called a field with a valuation or a valued field. If (1.1.12) is satisfied in the stronger foim (1.1.13) I Cb-+- Qy\ < -ma^c c \ou\ ,1^0, a., ^ e K^ then it is called a non-archimedian valuation. - 4 - We give some examples of non-aichlmedlan valuations. EXAMPLE 1.1.1. Por any field K , we define \0\~0 and \aj\=-\ toT all 0/4^0 in K . This is called the trivial valuation on k . Valuations other than the trivial valuation are called non-trivial valuations. EXAKPLS 1.1.2, Let Q denote the field of rational numbers, c a fixed real number such that 0-cc< I and p a fixed prime number. If x is any rational number other than 0 we can write where CL , -^ , oC are integers, [^ "fa, ^ j't-S- • We define (1.1.14) l^lb= C°^ O/^ct 10((,= 0. It is known (BACHMAH [3l , p.l) that ( (. is a non- archimedian valuation on Q and is called the \:> -adic valuation of Q. EXAMPLE 1,1.3. Let K be any field and KLt]be the ring of polynomials over K . If f>Cfc;i8 an irreducible polynomial in K Lb] then we define (1.1.15) locctM.= , ? where (XCb), -^Cb)Gl It follovnB immediately fiom Definition 1.1.1 that a valuation satisfies the following conditions (1.1.16)1 11 = I • ; (1.1.17)1-11 =^i ^ (1.1.18) 10.^1= I 0.11^1-' ; (1.1.19) |i<5^l-l^i|^ \aj-U. PPOPOSITION 1.1.2 (BACHMAH [?] pp.7» 8). If | | is a non>archiffledian valuation and if Icol > \£>,\ then \(Xy+4>-\~\(l>\ More generally, we hare . COH)ItLABY 1.1.1. For a non-archimedian valuation we have and I a,+ a2_+--- + clw| = I Ob, I -il^ 10.^1^ 10/(1 -for j.= i-.5,-.Ti The Corollary is a consequence of finite induction an Yl using Proposition 1.1.2. * We denote by 1 the unit element of K as also the real number 1, with, of course, no room for confusion. - 6 . The following characterization of a non-aiohlffledlan valuation will be freely used in the sequel. PROPOSITION 1.1.3 (HAMANATHAH [23] , p.204). A Taluation | | on a field K is non-arohimedian if and only If (1.1.20) 1 Til ^ 1 for eveiy Integer yi e K * ThlB shows immediately that COK)LLA!?y 1.1,2 (RAMANATHAN §3] , p. 204). All the valuations of a field of characteristic j=>=|=0 are non- archimedian. We continue to assume that I i is a non-aichimedian valuation and we consider the set Sy of all cu€ K such that I CU| ^ 1 • ^®* ^ denote the set of all cue K such that lal< 1- PHOPOSITIOH 1.1.4 (BACHMAN [3] , p. 8. Theorem 2.3). If I ( is a non-archimedian valuation, then the set V is a zing with unit element and 'P is the luiique maxiaal ideal of V and hence is a prime ideal. DEPIHITIOH 1.1.5. The ring V (defined above) is called the valuation ring associated with the non-archimedian valuation | j • The field \^ is called the associated residue class field. * Throughout we write \r\\ v^ere n is an integer, for Ini/ (1 being the unit element of K ). - 7 - (1.1.11) of Definition 1.1.4 showe that CL -^ (Ctl 1» a homomorphism of the multiplicative group K of non-sseio elements of K into the multiplicative group of positive reals. This homomorphic image is a group | K ( which we call the value group of K (see MMA.HATHAN [23l p,213), PEFINITION 1.1.6. The valuation | j is called discrete or dense according as the value group of K is discrete or dense. HEMAPK 1.1.2. A valuation is either discrete or dense, A valued field K can be viewed as a metrLo space with the distance function P defined by (1.1.21) pccb,t)= |a-^l satisfying (1.1.22) pCa..-6L);^0 and PCa,6.) = 0 if ana only If (X - ^ ', (1.1.23) pca,^)= pc-^.o,): (1.1.24) Pca,^) ^ PCa.,c)-i-pCc,l), Ca,4,c£K) If, in addition* K is a non-arohlmedian valued field then this distance function P can be easily seen to satisfy the following ultrametrlc Inequality (1.1,25), the stronger form of (1.1.24) (1.1.25) ?COL,t) ^ 7na:>c (pCCL, c) , pcc, 6.)) • Thus a valued field K is a metric space. - 8 - THEOREM 1.1.1 (BHJHAT [>] , p.18). Let K be a field with a non-trivial non-archimedian raluation. Then the following conditions axe equivalent. (1.1.26) K is locally-coapact. (1.1.27) The valuation ring V of K is compact, (1.1.28) K is ^ complete metric space. ( | is a discrete valuation and the residue class field is a finite field. The above metric p gives rise to the notion of conver gence in K as follows. If {jJir\,] is a sequence of elements of K then it is said to converge to the element CU in K » i^f for every real number £. > 0 , there exists an integer Nce) such that la^-Cul < ^ for -rv y^ NCe) . In terms of the metric, more explicitly, this means that PCdyu, Cl/)-> 0 , •Xi _^ CO • The sequence •[Q./>^] is said to be a Cauchy sequence if for every real number 6. > 0 there exists an UC&) such that \Ci'y[-(X'Yv\\ PROPOSITION 1.1.5 (LERSKOI [l3l , p. 35, Theorem 1.22). A sequence [ch^] in a non-archimedian valued field K is cauohy if and only if \Clr\+\~ ^til-^ ^ as Ti —/ oo • PKX)F. It i8 obvious that if {0.^,1^8 cauchy then iCL^^,-a^l -^ 0 as y\ -^ 00 . Conversely, let I (Z^^+r Cl^\-^0 as -n -^ cx) . Given E >0, we choose N such that IC^^^.,-(^^\ <£ whenever Ti > N. Ihen for all Yl ,YY) > N . This implies that {(X^] is cauohy. DEPINITION 1.1.7. The field K is called complete with respect to the valuation I I if every cauchy sequence in K has a limit in K • If K is a field with valuation [ | we can introduce GO the notion of convergence of an infinite series 51 Ct-pi in the usual fashion, in terms of convergence of the sequence {^-y^l o^ ^^^ partial sums -^^ = ^,-+-^o.-*- ••'^YT PBOPOSITIOH 1.1.6 (BACHMAS [j] , p. 25, Theorems 1,1 and 1.2). If K is a complete field with respect to a non-archimedian valuation and if {CLyi] is a sequence - 10 - of elements of K then (1.1.29) il Ct-y^ convei^^es if and only if XO^v\^ (X^'^', 11=1 '^ (1.1.30) If X-\OL^| converges, then 21 Ct^v converges. (1.1.29) is a mariced departure from the classical theory of series (i.e. pertaining to archimedian valuation) where -Yw \eL.Y^\ -^ is only a neceesaiy condition for 00 the conve rgence of XI (X «A • The conclusions of the following easily proved lemma are handy tools in the study of series in non-archimedian fields. LEMMA 1.1.1 (LENSKOI [l3l , p. 37). If K is a complete non-archimedian valued field and if {61^] is a sequence of elements ot K then (1.1.31) l£cc^| ^ ^^'t.fjl^-l^ (1.1.32) If la^^(>lCLv^l for all Y\4'Y^o then ^1=1 - 11 - 1,2. Power series over ralued fields Throughout this section K will denote a oonsplete non- archimedian valued field. Let (1.2.1) a^-t- a,Ccc-:x^,^+a2.C3c-3::^)^ Ha^Cac-aCj,)^ - - be a power series with coefficients from K aja<5 oc.oCo^K. We denote such a series by o< or o< Cac) • ^ (1,1.29) the power series (1.2.1) converges at a point oc of K ii^ and only it\cly^\\:xl~oco\^-^ 0,'n-^oo . The set of all oc in K for which the series (1.2.1) converges is called the domain of convergence of the power series, and, as in the case of the real and complex fields with the usual absolute value, one establishes that 51. Cly, C^-^o ) converges for j cc-aCj>l<'i;^ and divex^ges for Ix-oipl ?> "H , where (1.2,2) -R"^::. -^yr^^^ The case (oc-oipl^^'R is easier to analyse here because of the non-archimedian property of the valuation namely for \X-'a:ol==T^ ^^* series converges if and diverges if ixnrr\ I CLyi I "f^ ^ 0 . The series o( -n ' given by (1,2,1) is said to be an entire function over K if it converges for all oc in K » equivalently if - 12 - (1.2.3) ja^ I '^__^o, nn-> (X7, as is clear from (1,2,2). Following Lenskol Jl3l we call the power series (1.2,1) a regular fimotion. Associated to each fixed regular function o( and a point oc^^ in K , we have a function M(Y) defined on [pjR) with real values defined as follows : (1.2.4) MCY) =r -^uf. la^lY*^ To avoid ambiguity we sometimes write MCC<,DCO,Y) instead of K(Y)» indicating explicitly the regular function to which M(Y)i8 associated. lM(Y)is well-defined since for TG [.0,R) , '^'YwI a«^\ Y'^= 0 . aiven TG [o, R ) , tenns of (1.2.1) for which MCY") =\CX^\Y^will l3« called central terms. If, for a given T , there exists one and only one central term, then T is called an ordinary point. If, for Y , there exist atleast two central terms^ then Y is called a critical point. If o(Cx) = 0 then eveiy v is critical. In the sequel, we omit this trivial case and assume that o(Cxy^o * We denote hy Yl ,CT) "t*^® smallest suffix of the central teims for V If d =^ 0, then for any T and if O/Q = 0 then for any'r>0. • An additional reference, kindly pointed out by Prof. Dr. A.F.Monna wherein most of the results of §,1.2 can be found is Guntaer [103 . - 13 - there exists a central tezm with the laxgest suffix. Otherwise Ictnlr"^ cannot tend to 0 as it should forTp'O, We denote this suffix hy M -(7) • At an ordinary point, we obviously have 'TioiCy)- No^Cv) • At a critical point we have NO(CT) > 'V\e< CY) • It is easy to see that if Y, < Y^ , THi ^ TY) a- then \ a^j Y,"^' ^ I OLynJ ^^^ implies that \(^'rr.,\yr' ^ ICl^Jr^ Therefore (1.2.5) MpcCr,) < r\^ CT^). This means that No( (T,) ^ Mo((Yj_) and Tl^^Cr,) ^Tl^CT^^,) . Let now { Yj/'j he monotonically increasing and { Tu ] be monotonically decreasing sequences with the limit Y In view of (1.2.5). n^ C T^ ) t f^o(CY^) are monotonically non-increasing and Y1O^(Y|^") , '^' o(^^K^ are monotonically non-decreasing sequences of non-negative integers, bounded by •YIO^(T) and N^Cr) respectively. Thus, fork->ko,'^o(CY,;^), Yi^(T^), N^(r;^y, ^/o(CT^) are constants. Consequently thei^ exist interv^als (Y, y^^ in which Y^o(( v) , N^ (y) are constants. Let their values in these intervals be 'Vl , Ni and 10^ ,N' respectively. Then the terms whose suffixes are 1^\ Ti^^t U , N" are central at the point Y . * ?or Y= 0, we consider (it is clear that Yj^ > Y only) only inequality (1.2.7) to have been obtained. - 14 - Therefore '^o((^) ^ Ti'' < M" < N;.CY) But, by (1.2.5), M" < ^O(CY) and W^C Cr) :< r\' We thus have (1.2.6) T^otC-Y) = -YA''- N''^ l\io<(r)^ (1.2.7) .T\O The inequalities (1.2.6) and (1.2.7) ehow that for V there exists an interval (T", Y' ) (or, if Y=0, an interval Lo , Y' ) ) in which, every point, besides peihaps T itself, is ordinary. Therefore the set of all critical points in any segment [.o, nrj is finite. For otherwise, there would exist a point ^ to which the sequence of critical points different from y% , would cq(verge, which is impossible. In view of the above, we shall write the critical points (in case they exist) in the form of an increasing sequence To = 0 < T, -C T^ ^ - - - ^ Yj^ -C - • (Vj , is the first critical point z^O, 0 may or may not be a critical point). Suppose there are infinitely many critical points and Consider the interval(YJ^ ,^te-»-i ) * ^°^ ^K ^ ^ ^ ^k+\ we have - 15 - We get Ho((.y^•)=•r\^Cr^^^y=.Y^^^ say. Then in CY^ ^Tj,^,) This means that in ^"^k,'^fe+.i) , M(Y^ is continuous and is monotonically inozeasing. Since CLY\ IT, — 1 (X^ \T the critical points are also points of continuity. It TOmains to study the behaviour of HCr) in the inter vals COj r ) and C^^ , "R) (in the first case it is assumed that is not a critical point and in the second, it is assumed that there exists a last critical point ). For thi8» we may assume that CLQ =^ 0 . It is easy to see that in C 0,y^) and InCfc ,-R) Obviously Y->0-V Ce- HCr) is continuous at 0 from right. If there exists a last critical point y^ , then M(r) is defined for Y='R. In this case H(v)is continuous (from left) at "R . If the set of critical points is countable, but not finite, then icofw MCY) = 0O and for Y^'R , MCk) has no meaning (We can 'Y-^R-O put KC'R)=oo). - 16 - Thus we have shown that MCt) Is continuous on Co,'R^ and monotonlcally increases. As for classical complex valued functions we have here too the maximum modulus principle embodied in THEOREM 1.2.1 (LEHSKOI, §.3] , Theorem 3.10 p. 85). If o( is a regular function over a complete non-locally compact non-archimedian valued field K and v is in the value group of K, then (1.2.8) MCY)=^ A-iAJp lo^CDc)!. PTOOP. If Y is an ordinary point then Mfv) :=:lo Since, by assumption K is complete and non-locally compact, in view of Theorem 1.1.1, either (i) the residue class field is not finite or (ii) the valuation is non- discrete. We first consider case (i) when the residue class field of K is infinite. Let Tl, ^ Yla.^ - • • ^'^-m be the suffixes of the central tenns for V , T, CL £ K and I col = V , I Tl = MCv) . We denote by Ac the residue class to which the element ou^ ou T (L=l,2.,>Tn ) belongs and choose VQ^^ 0 in the residue class field such that - 17 - Let us take e e XQ • '^^^^ la^ Cae^'H- - - . -\- av^ cae)'^'^! = 1^1 = M(Y) It follows that, for 5c = ^-^-as such that I 5c-DCol = r \c< Coc:)l ^ H(Y) . Since in case (i) I 0( C ^) I < M (Y) C I CC- CCOI = V) we get the required result, (1.2.9) M(Y)=. ^^f lo(Coc)l. (In fact, in this case, the supiemum is attained). TOT the remaining part of the proof, we notice that, in case (ii), when the valuation is non-discrete, (1.2.10) M (X, ,-Y) c^ MCX:^,Y) if I cc,- 2C:^( ^ Y • In fact, if I ^c,- oc^.! ^ V then \ oc- DC,| = Y if and only if \ zc- Xj,) = Y . It follows that i.e. we get (1.2.10). Since IMCr^is continuous, (1.2.10 ) is true even when l^Cj-iXa.! =v. We now observe that for any non-archimedian valued field, it is possible to constzuot a complete non-archimedian extension with an infinite residue class field (for example its algebraic closure). In the extension field (1.2.10) holds and since the definition of - 18 - MC'V) <5oe8 not depend on it, (1.2.10) remains true for a regular function orer K t whether we consider cKC-^c) over K or over any complete extension of K • Let now K he non-locally compact having a finite residue class field. This means that the valuation is dense. We choose ordinary points Y' in the value group of K so near to Y that lAere & is an arbitrarily given positive number. The above inequality gives us Analogously for 5c such that | 5c- DCQ\-=^Y \x-Dc\'^Y But \ X- 5c 1 ^ Y implies that | x- x^ \ - Y and M (o(, XO,Y) = M (o<, X , Y) . Thus ^^-^ |o(C:^)l >y \M(O(,3CO,Y;. But the strict inequality does not hold and hence the equality. COBOLLAFY 1.2.1 (LEUSKOI jlj) , p. 87, Theorem 3.11). With the assumptions as in Theorem 1.2.1, A^ i o( C X ) 1 =• ^-^ I o< C ^ ) I . We next show that we have in the case of entire functions defined on an algebraically closed non-archimedian field an - 19 - analogue of the classical PIcard Theorem. To start with, oo let us consider infinite products | | (X-y^ > CL €K» where K is any complete non-archimedian field. We have, with convergence of the infinite product defined in the usual %ray, via. PTOPOSITION 1.2.1 (LEHSKOI [13] p.38, Theorem 1.27). If 00 PTOOP. If ixTYV (X„ = 1 then there exists n^ such that lav^l^l , y^>r\^. Putting A=M I <^'nl , and choosing N satisfying, for a given £ > 0 , I a^-l| < &/A , -vn > Ni-rio^ we have Consequently, by Proposition 1.1.5 TT Ct^ converges to a non-zero limit (The limit is non-eero because |Ct^| = 1^ Conversely, if CU] - CL , (L ^ 0 , then there exists tip with \'f\~^ d ^ — I Ct I forTi>Y)o. - 20 - Por £ > 0 , if we choose N such that it follows that |a| la^-l |<. £ |a>l for Y) > N + Ylo . In other words I a^^- ll < £ for such an Yl , which means that Zzyru Oi^-[ With the aid of the concept of infinite product we can prore : PBOPOSITION 1.2.2 (LENSKOI [l5] p.96, Theorem 3.19). Let K he further algebraically closed non-archimedian valued field. If { 61Y, ] is a sequence of elements of K such that \(^i\ < 10.^1 ^'••^la-»,l<- • i %^lat»l=<»; a,= a^= • =£i«-o; then there exists an entire function oCCx.) for which ^yv 'Y\ CX3 are zeros and in fact, ocCoc)= oc ° "j]~ r \ — 2^ \ Conversely we can show as follows that an entire function possesses a countable number of zeros (a finite number of zeros if it is a polynomial). In the remaining part of this section we assume that K is an algebraically closed non-archimedian valued field and CO o^C^) = Jl (X^ 3c' is an entire function over K . If o(Coc; Tiro is not identically a constant then it has critical points. In fact, without loss of generality we can assume that CX^ ^ 0 , Consider the set S of all Y for which ( a^, | > I a^| y^ for all n :^ I . It is easy to see that S = [o , Y^j whew y*' ( ^ 0, OQ) is the first critical point. - 21 - If o<,(x) is not a polynomial, then the set of critical points of o(C3c) is countable and JC^'W YK =. ex? . Por this, we have only to note that there is no last critical point for o(C^") . In fact, if there is such a point Y^ then we would have (1.2.11) \o<^'x\^\y^^ y l^-VYiK^ for all sufficiently large Y and YTl =^ Y\^ , Since o^(x") is not a polynomial, it is possible to find an "YTl such that TTi >y\^ and a,^ ^ 0 . But then ( 1*2.11) cannot hold for all sufficiently large Y , Suppose, now, CU =^0 in K is a zeiro of o(C3c") • Then \(X.\ is a critical point. I'or, otherwise, M(Y)= I^IY^ for only one V^ , where 10^1= "^ . But then, for eveiy YA^'^t \Oi^\y'^ y K^l^*^- Consequently lo(Ca)l = [|L^avxCri= IO.^IY'^C by (1.1.32) ). Therefore, o(Ca.") =^ o - a contradiction. But ©((x") has a countable number of critical points as already proved. Thus, an entire function which is not a polynomial, possesses countable number of zerosi Arranging them in the order of increasing valuations clearly we have ^T^ I (^ u 1 = ^^ • Defining, for XG-K an a-point of a function c>(Cx") as a point xeK such that o((oc'>=CX/, we obtain the Picard Theorem below by considering the zeros of the function o((sO—0^, - 22 - THEOREM 1.2,2 (LEHSKOI [ij] , p.97, Theorem 3.21). It o(Coc) is an entlxe function defined on the non-arohlmedian valued field K ^ich is algebraically closed, then for any Ct £ K» there is a countable number of a-pointe {CIYV\ of c^Cx.) with Xc-YW/ I avi I = 00. We now give an analogue of the complex line integral for a non-arohimedian complete algebraically closed field K . For all integers TI in K s^oh that IY)\ c: | we set ^^(x) = x^-1 = cx-1^'^ cx- %T) • -• (X- O. dlearly, for all i,r\, | ^^."^^l -1 . We fix a, Y € K and consider for all Y\ as above the set (X4-Y^^^\ (X+V^ w- + V s^ . Such a sequence of points in \\ is called a discrete circle with centre CL and radius |T|r p . DEFINITION 1.2.1 (ADAMS [l] , pp. 297-298). Let -^^(jc-:) be a K -valued function defined for all points on the above discrete circle. Then we define, if it exists, =. ^-^^-Jr^ f(a+v^,r)- The above integral is called the Schnirelman integral. THEOREM 1.2.3 (ADAMS [l] , p. 299, Theorem 4). Let oiCx.)- ao + «:i,3c + - - --hdyyX^. - be a power series over K converging for all x in K such that Ix I ^ R C R> O) . - 23 - If 0., T £ K such that | ex. I ^ T^ and lr(-^R thtn ^ <'(Cx)c;(x exists and Is equal to o{ CO/) • COK>LLA!?T 1.2.2. If o(C3c)n XC^y^X^ is an entiiw function over V< then o^CoOotx = C3< Ceo) exists for all Ct, r G K . THEOREM 1.2,4 (ADAMS [\\, p. 504, Corollaiy to Theorem 13). Let o<(oc) be a power series convergent for JCG K such that loc\-«i-R CR?>o). Let Ye K he such that \^\^^ and 6i{x) ^ (x-a,) 'cx-Qv) -^ • Cx-a^^ he a polynomial such that {(L^i ^ \y\ for l^t^Yi- Let further ac:>c) - ?^^^ . Then, if acK such that laUR I ^Cx) CX-OL-) dx- X- Vesictue of ^x) a,v the summation for oc ranging oyer all poles of <^ such that |oc-aj|< \YI and residue of ^CPC) at a pole CO being defined in the usual way as the coefficient of —'—- in cc-co the expansion of g (c>c) in powers of (x-co) . 1.3. Linear topological spaces over a Non->archimedian valued field DEFINITION 1.3.1. By a non-arohimedian normed linear space over a non-archimedian field K '^^ mean a linear - 24 ^ space E over K together with a non negative real valued function II |{ satisfying the conditions (1.3.1) llacil 2 0 if and only if oc = 0 ; (1.3.2) llAXH = Ulll^cll i (1.3.3) llx+^H4 max CllxihWy-U). for all DC, If GE , A e K. The function \\ H. ExE--^^ induces a metric on E defined by clc^c, t^)= \| x-^^H, DC,I^ € E. BEFIKITIOH 1.3.2. A non-archimedian noimed linear spaceE is called a non-archimedian Banach space if E is complete with respect to the metric cLcx, ^^. For non-archiraedian spaces we have the notion of convexity of a subset given by DEFINITION 1.3.3 (MONNA (l6] , p.532). Let E be a linear space over a non-archimedian valued field K . A subset C of E is said to possess the property CO if :^^ d , ^GC imply that Aoc+K*^ £ (i for all IAi,lK^ ^ 1- ^ subset C of E is called K -convex if C possesses the property C<^) or can be written as C. = DC^-\- d where C possesses the property (^C ) and -^o is a fixed vector in E . DEFIHITION 1.3.4 (MORHA [l8] , p,353). Let E be a linear topological space over a non-archimedian valued field K . A subset A of E is said to be bounded if for evexy neighbouihood U of 0 there exists A ^ 0 , A G K such that A ^ A U . - 25 - The following theorem is an analogue of Kolmogoioy's criterion for no inability of topological linear spaces. THEOREM 1.3.1 (MOMA [18]^ p. 360). A necessaiy and sufficient condition for the topology of a linear space ^ over a non-archimedian valued field to he defined by a non- aichimedian nons is that the79 exists a bounded K -convex neighbouihood of 0. What follows is an exact analogue of the criterion for continuity of a linear operator from one noxmed linear space to another. THBOHEM 1.3.2 (MOMA [l5] , pp. 1134-1135, Theorems 9 and 10), Let K be a non-archimedian non-trivial valued field and B and p be non-archimedian normed linear spaces over K • Then a linear operator T: E -> F i" continuous if and only if there exists a number M>0 such that for all 2^ in E . llrocK < M llocll. DEFINITION 1.3.4. A nest of (closed) spheres in a non- archimedian valued field K is a set of (closed) spheres totally ordered by inclusion. The following notion seems to be a natural restriction on K in certain situations. DEFINITION 1.3.5. A non-archimedian field K is ^^i^ ^^ be spherically complete if eve17 nest of spheres in the field has a common point. . 26 . DEFINITION 1.3.6. A non-archlmeaian field K is said to hare Hahn-Banach property if, for any non-archimedian noxmed linear space E over K every linear functional defined on a subspace of E has an extension possessing the same nozn defined on the whole of E . THEOREM 1.3.3 (INGLETON [ij ). A non-arohimedian valued field has the Hahn-Banaoh property if and only if it is spherically complete. DEFINITION 1.3.7. Let E he a linear space over a non- archimedian valued field K . By a non-archimedian deminoxm on B we mean a real valued function ^c^c) on E satisfying (1.3.4) t^CAJc) = I X\ \>Cx^; (1.3.5) |pCx-t-«^) < YYiapc (f5C:)c), \^c^)) , pcx)^o, \pi^)^V for all oc , t^ e E, A e K. DEFINITION 1.3.8 (MONNA [ITI pp. 394-395). Lefp be a set of non-archimedian seminozms on a linear space E over a non-archimedian valued field K . Given a>Owe consider the sets (1.3.6) U. Co) = {^cjaceE^ ^.CX)-=c£,Ul,2,--Tl,K^'rj. By talcing the sets in (1.3.6) as the fundamental system of neighbourhoods of 0 » E becomes a linear topological space over K . E with this topology is called a locally K -convex space. - 27 - THEOREM 1.3.4 (MOMNA [l?!, p. 400). Let E and F be locally K -convex spaces. Suppose that the topology of E. is defined by a unique semi-norm \pcx) and that of F by a unique semi-nozm C^C^l . Suppose further that F Is spherically complete. Let V be a linear subspace of E and T a linear operator of V into F • Then there exists a linear operator TQ of E into F extending T and such that the noun of TQ is equal to the noim of T. DEFINITION 1.3.9 ( VAN TIEL C24] , p. 255, Definition 2.6). Let A and B be two subsets of a locally K -convex space F . Ve say that A absorbs & if there exists a real number Cu>o such that B c XA for IXl> Ou CAGK) A subset of F is said to be absorbing if it absorbs eve 17 singleton subset of F . DEFINITION 1.3.10 (VAN TIEL [24l , p. 268, Definition 3.4). A K -convex, absoxbing and closed subset of a locally R -convex space E is called a K -barrelled set. DEFINITION 1.3.11 (VAN TIEL [24] , p. 268, Delfinition 3.5). A locally K -convex space F is called K -barrelled if eveiy K -barrelled subset of E is a neighbourhood of 0 • THBOHEM 1.3.5 (VAN TIEL i24l , p. 268 Theoreme 3.15). Every locally K -convex space which is of second category (i.e. which is a Baire space) is K -barrelled. - 28 - Let E and f- be locally K -convex spaces and ^CCE^F) the vector space of continuous linear maps of E in p Let ^JV be the fundamental system of neighbourhoods of o ifl F. DEFINITION 1.3.12 (VAN TIEL 124] , p. 274, Definition 4.1) A subset H of ^C£.,F) is said to be equicontinuous if, for eveiy V G "^ there exists a neighbourhood (J of O in E such that H C U") c^ V. We denote ^CE-, K") ^7 E^ THEOREM 1.3.6 (VAN TIEL [24] , p. 275, Theore'me 4.3 ) If E is a K -barrelled space and if H is a weakly bounded subset (i.e. bounded vdth respect to the weak topology) of £.' then H is equicontinuous. With natural topologies arising out of valuations we will have occasion to use the general notion of 0 -dimensionality defined as follows t DEFINITION 1.3.13 A 0 -dimensional topological space is a topological space admitting a neighbouzhood basis at each point consisting of sets which are both open and closed. THEOREM 1.3.7 (TAYLOR [25] p.76, Theorem 2.41-C). Let K be a non-empty complete metric space. Then X is of second categoiy as a subset of itself. - 29 - THB0I5EM 1.3.8 (TAYLOP [25] p. 77, Theortm 2.41-D) Let X b« a metric space and S a set of second category In x • Let 5l= /-f V be a collection of real valued continuous functions defined on X such that for each ace S there exists a real number mnCai) with the property fOr-) ^ yY\Cx)toT •••zy -5 € 5i * ^hen there exists a real constant N and an open sphere 0 In x such that fCoc") ^ N if oceOand f G^f. NCE-j. In tenas of the metric this means PCd^, OL'TA) -> 0 as n,'rv\ -> oo , independently of each other. However, it is clear from the following proposition that the independence of VL , rrt in the above definition is of no significance in a non- archimedian valued field. - 9 -