If(Z)

COMPATIBLE SEMINORMS IN A VECTOR LATTICE* BY CASPER GOFFMAN

UNIVERSITY OF OKLAHOMA Communicated by S. Bochner, May 16, 1956 Let X be a vector space. A locally convex topology r in X is given by a system of seminorms pal A, for which XJ is a Hausdorff space. We suppose that there is an order relation w in X for which X<, is a vector lattice. We suppose, further, that the order dual, X,', which consists of all bounded linear forms on Xs, is a total subspace of the algebraic dual X' of X, i.e., for every x $ 0 there is a bounded f such that f(x) $ 0. In particular, this implies that X, is Archimedean. A seminorm p is compatible with w if x| > yI implies p(x) > p(y). A locally convex topology r is compatible with co if its seminorms may be chosen so as to be compatible with W. For general information see N. Bourbaki, Elements de mathcmatique.' Under the above assumptions, we observe that for every w there is a finest r compatible with w. In particular, we show that there is at most one Banach topology compatible with w, and, if it exists, it is the finest locally convex topology compatible with w. Thus, for various function spaces, the norm in use is seen to yield the only possible Banach space compatible with the natural order relation of the space. We first note that if p is a seminorm compatible with w, then the continuity seminorm set of p is contained in X,'. For, if f(x) is a linear form on X such that, for some M, f(x) < Mp(x), for all x e X, then, for every y > 0, zJ . y implies p(z) < p(y), so that If(z)l ' Mp(Z) < Mp(y), which means that f(x) is bounded. We thus have the fact that if r is compatible with w, then X' c X<'. We next note that there is a finest convex topology r compatible with W, namely, the one given by all the compatible seminorms. We point out that for this topology XT= X,'. For, if f(x) e X,' is positive, the function p(x) = f(| xI) is a seminorm compatible with w for which f(x) is in the continuity seminorm set of p(x).. But every bounded linear form on X. is the difference of two positive linear forms. Hence, for the finest topology r compatible with w, we have XT' v X,', so that X,' = X<'. For a total subspace Y c X', we call the finest locally convex space X, for which XT' = Y a Mackey space, and summarize the above remarks in THEOREM 1. If T is compatible with X, then XT' c X,,'. If r is the finest convex topology compatible with w, then XT' = X<,'. If r is compatible with w, XT is a Mackey space, and XT' = X(,', then r is thefinest convex topology compatible with W. In applying this theorem to Banach spaces, we first prove LEMMA 1. If X, is a vector lattice and T: lIx|I is a compatible norm for which X, is a Banach space, then, if {IXn is a Cauchy sequence of increasing positive elements, x = lim Xn implies X _ Xn for every n. n X co Proof: Suppose that (Xn- x) + > 0 for some n. Then (x, - x) + > (x, - x) + for every p > n. But 536 Downloaded by guest on September 26, 2021 VOL. 42, 1956 MATHEMATICS: C. GOFFMAN 537- IxP- xl > (xP -X) so that

iixP - xii| ll(Xn - X)+|l- Since II(xn - x+|I > 0, it follows that IxpI does not converge to x. THEOREM 2. If X., is a vector lattice and T: IIxII is a compatible norm for which X, is a Banach space, then it is the finest convex topology compatible with X,. Thus a vector lattice may be normed as a Banach lattice in at most one way. Proof: Since XT is a Mackey space and XT' c X.', we need only show that every positivef(x) on X., belongs to XT'. Suppose, then, that f(x) is not continuous but is positive. Then, for every n, there is an x for which lxIi _ 1 and If(x) _ n. Since, by compatibility, xiII = IIx!! and since (fix! ) =fAx +xjX f(x+ -x- = If(x)i, it follows that there is an x >O for which IIx!! 1 and If(x) _ n. Now let {Xn}[be a sequence of positive elements such that, for every n, iix-I _ 1 and If(xn)i _ 29". Let

n = X1 + 1 X2 + Xn 2 22 1+2 Then {nyI is an increasing sequence of positive elements. It is a Cauchy sequence since

1 Yn+pI ~- yil =2= x+lXn+1 ++2n.+p+ Xn+Px 2+< for every n, p. Hence, if y = lim yn, then, by the lemma, y _ Yn, so that

f(y) > f(yN) > - f(x.) _ 2n for every n, which is impossible. For all the spaces of real functions and sequences of analysis the order relation is the one for which the positive elements are those for which f(x) > 0 everywhere, or almost everywhere, as the case may be, or x_ > 0 for all n. As examples of Theorem 2, we mention only the two most important cases. COROLLARY 1. For the vector lattice of continuous functions on a compact set S the only compatible Banach space topology is given by If!! = max If(x)I. COROLLARY 2. For the vector lattice Lp of pth-power integrable functions, p > 1, on a measure space S, the only compatible Banach space topology is given by

if!= [f If(x) s PdM]l/v Downloaded by guest on September 26, 2021 538 MATHEMATICS: HARISH-CHANDRA PROC. N. A. S.

There are important vector lattices whose associated finest compatible convex topology is not a Banach space topology, and we hope to be able to discuss the general situation in the future. * Supported by National Science Foundation Grant NSF G-2267. 1 N. Bourbaki, PA1ments de mathbmatique; Livre V: Espaces vectoriels topologiques, chaps. ii, iii, iv, Paris, 1953; Livre VI: Integration, chap. ii, Paris, 1952.

A FORMULA FOR SEMISIMPLE LIE GROUPS BY HARISH-CHANDRA INSTITUTE FOR ADVANCED STUDY Communicated by Marston Morse, June 26, 1956 In order to save space we shall follow strictly the notation of an earlier paper.' Define to, po as usual,2 and suppose that to = bo nf to ± bo n po. Then we say that to is fundamental if to n to is maximal Abelian in to. Any two fundamental Cartan subalgebras of go are conjugate under Go. THEOREM 1. Let b, be a connected component of [o'. Then there exists a real number c such that lim Ff(H; 6(7r)) = cf(O) (H e H-O bi) for every f e(g0). Moreover, c = 0 if bo is not fundamental. THEOREM 2. Suppose that to is fundamental and bl,, .. . , bW are all the distinct connected components of fbo'. Let cj be the real number of Theorem 1 corresponding to tj(1 < j < r). Thencl +c2+.... ±+Cr O. The following result plays an essential role in the proofs of the above theorems. Let dH denote the Euclidean measure on to and dk the Haar measure on the compact analytic subgroup Ko of Go corresponding the to. LEMMA 1. Assume that to c to, and for any g e (e((bo) put g^(X) = f exp (iB(X, kH))7r(H)2 A, g(sH) dk dH (X e Go) KoXbo SeW Then the integral F.(H) = 7r(H) JUGS 9(x*H)dx* is convergent for H C bo', and there exists a constant c $ 0 such that eic(s) F;(sH') = c f b. E c(s) exp (iB(H', sH))rr(H)g(H) dH SeW seW for all H' e bo' and g e e(fo). Let G be any connected Lie group whose Lie algebra is g0, and let A be the Cartan subgroup of G corresponding to bo. LEMMA 2. Let E be the centralizer in G of an element ao e A. Denote the natural mapping of G on U = G/Z by x -x t (x e G). Then we can find a neighborhood B of a0 in A with the following property. Given any compact set w in G, there exists a compact set Q in G satisfying the condition that if xax' e X for some a e B and x e G, then eU.C Downloaded by guest on September 26, 2021