Bull. Kyushu Inst. Tech.
(M. & N.S.) No. 20, 1973
p.TYPE TESTING FUNCTION SPACES AND A RELATED THEOREM DUE TO A. H. ZEMANIAN
By
Ky6ichi YosHiNAGA
(Received Oct. 20, 1972)
In a recent paper on the network theory [11], [12], A. H. Zemanian has introduced a concept of the Hilbert port as a natural extension of the electrical ntport. There, the most natural setting for such a concept is seen to be spaces of Banach space-valued distributions. And in order to study different types of such spaces within a single development, the notion of a p-type testing function space .Åë'r(E) is offered concerning Banach space-valued functions. Soon after, V.
K. Balakrishnan [1] has extended this notion to a more general case of vector- valued functions. The present article is concerned, first, with a classification of such y' (E) according to the type of the closed interval covers employed to define it. On the basis of such classification a theorem due to Zemanian (see below g3, Theorem Z) is then studied and extended from the angle of the tensor product of locally convex spaces. It is here noted that a theorem of this sort is used by Balakrishnan [1] to get an extension of the Paley-Wiener Theorem given by T.
K6mura [4].
Section 1 is devoted to the preliminary remarks. In the first plaee certain notations and terminologies used in this paper is explained. At the same time, regarding the diverse meanings involved, for instance, in M(E), particular mentions will be given. At the end of this section a brief summary of tensor products of locally convex spaces will be introduced in accordance with the general framework of L. Schwartz [7]. In Section 2 the p-type testing function space is defined and a classification of such spaces is given. In doing this one will observe that the nature of a p-type testing function space depends essenti- ally on the closed interval covers of the real line employed in defining it, and not so essentially on the weight functions but only on their growth properties.
Section 3 is concerned with a theorem due to Zemanian to the effect that there
is an algebraic linear isomorphism between Y(9)(E); F) and g2'(Yb(E; F)) in case
Eand F are both Banach spaces. Balakrishnan's extension of this theorem is
based on the additional assumption that a certain vector-valued distribution is
of finite order on any relatively compact open subset of real numbers. However, standing on the tensor product point of view, sueh a theorem may be reduced to a single identity e(E)=g7(i5BE for a given complete E. Thus all that we must do in practice is to find such E. The rest of this paper is devoted to obtain various extensions of this theorem by giving various familiar examples.
Except otherwise stated, the notations and terminologies of this paper are those of [2] (locally convex spaces) and those of [6], [7], [8] (distributions).
1. Preliminaries.
Let E and F be locally convex Hausdorff spaces over the complex field C and let us write Y(E; F) to denote the set of continuous linear applications of E into F. Yb(E; F) (resp. Y,(E; F), resp. Y.(E; F)) is used to indicate Y(E; F)
equipped with the topology of uniform convergence on each bounded subset (resp. convex balanced compact subset, resp. finite subset) of E. In case g;e' is a locally convex Hausdorff space of functions on the real line R, we often write .?e'(E) to mean the space MeE==YE(rw2; E)==Y,(E3; ee') where e of YE(or3; E) (resp. Y6(E3; .;eP)) designates the topology of uniform convergence on each equi- continuous subset of rw' (resp. Ei) [6, p. 34].
REMARK 1. In an early paper [5] ff(E) is used to denote the space of, roughly speaking, rw-type E-valued functions, while xesE is expressed as g;?(E).
For any quasi-complete E it is known that 8(E) defined thus above is the space of the infinitely differentiable E-valued functions on R with the topology of uniform convergence, for the derivative of each order including order O, on any compact subset of R [5]. For every closed subset K of R, let 8K be the set of functions q of 8 with the support supp q(K. 8K is a closed subspace of s and thus it is a nuclear Frechet space [3, Chap. II, p. 47, Th6orbme 9]. We write 8K(E) to denote the closed subspace of es(E) composed of all qcs(E) with supp op(K. If, in addition, K is bounded, 8K(E) (resp. esK) is frequently ex- pressed as 9K(E) (resp. g7K).
A sequence {K.; n ==O, 1, 2,•••} will be called a nesteel closed inteTval cover of R if each K. is a closed interval in R, K7.(K..i for every n and Nl/K.=:R. We n '=O
say that K. is of
type I if for each n, K. isabounded interval;
type II. if for almost all n, K. is a right half closed line;
type IIm if for almost all n, K. is a left half closed line;
type III if for almost all n, K. == R.
p-Type Testing Function Spaces and a Related Theorem due to A, H. Zemanian 9
Given nested closed interval cover {K.} of R, one obtains a sequence of spaces {esK.(E)} with the isomorphic injection: 8K.(E).8K..,(E). The induc- tive limit defined by this sequence is denoted as 9(E) (resp. 9.(E), resp. 9-(E)) if {K.} is of typeI(resp. type II. resp. type II-). The inductive limit correspond- ing to {K.} of type III is clearly seen to be the space s(E). As usual let us write 9=9(C), 9.==e.(C) and g-=g-(C) [8, p. 172].
REMARK 2. 0ne should observe that g(E)=ÅrF:2Z(E) in general, except that E is a normed space [5].
Assume as before E is quasi-complete and we here remark that y(E)= seEE is composed of all functions q(8(E) such that (1+t2)Mq(le)(t) is a bounded function on R for each pair of nonnegative integers k and m. The topology of Y(E) is seen to be given by the uniform convergence on R for each (1+t2)Mq(k)(t) [5].
In a previous paper [9] we have defined the space J`7' as the projective limit of the sequence {e-"vii+t2Y} with the continuous injection e-("'i)v'i+t2.g' ) q-Årq c
erv"vi
i+t2y and it is seen that Jo7- is the strong dual of a Silva space and so a
nuclear Frechet space. It is also known that ./07' is the space of functions qces such that
Hqlih,. =sup eM"i:.TT' 1 q(le)(t) I Åq oo
tER
for each pair of nonnegative integers k and m, equipped with the topology de- fined by the family {Hqllk,.} of semi-norms. Thus in a manner similar to the case of 90(E) one may infer that for quasi-complete Ethe space Jo7'(E)=Jf-eE is characterized as follows: It is the space composed of all functions qE8(E) such that each {eM"i+tiq(h)(t); tcR} is a bounded subset of E, supplied with the topology of uniform convergence on R for each eMviiTt,:'M
q(le)(t).Let va(E) be the space of functions q c 8(E) such that, for each nonnegative integer k, q(k)(t) is a bounded function on R. The topology of va(E) is given by the uniform convergence on R for each q(le)(t). It is known that va(E) coincides with va,(E) algebraically while the topology of the former is strictly finer than that of the latter (for EtÅrE=(O)) [5].
We finally refer to the tensor product of Iocally convex spaces [3], [7]. A family @ of subsets of a locally convex Hausdorff space E is called satuTateel if E=VA and if each subset of any scalar multiple of the convex balanced closed AE@
envelope of the union of any two elements of S belongs also to S. Let further
10 K. YosHINAGA
Z be a saturated family of subsets of another locally convex Hausdorff space F.
A bilinear application of ExF into a locally convex Hausdorff space G is @-%- hypocontinuous if both of its restrictions on A Å~F and on Ex B are continuous for each AE G5 and BE SI. If every element of @5 and of Sl is bounded, it is seen that this hypocontinuity is equivalent to the usual one [2, Chap. III, p. 39]. By a method similar to that employed in defining the projective tensor product [3], one may show that there exists exactly one locally convex topology on EXF, denoted as E@(E{)sF, in such a way that for any given locally convex Hausdorff space G, the continuous linear application of E@XsF into G may canonically be in one-to-one correspondence with the @-Z-hypocontinuous bilinear application of ExF into G. Making choice of particular @ and Z, E@CDsF gives the follow- ing topologies important in practice :
(T) G5 (or Sl) is composed of all subsets of E(or F). The corresponding topology on E(g)F is denoted as ECD.F, the topology of projective tensor product [3]. In this case "S-Z-hypocontinuous" is nothing but "continuous."
(B) @ (resp. Z) is composed of all bounded subsets of E (resp. F). The cor- responding topology is denoted as EXBF. In this case, instead of "@5-E$-hypo- continuous" one often makes use of "hypocontinuous with respect to bounded subsets".
(r) @ (resp. S) is composed of all subsets contained in convex balanced com- pact subsets of E (resp. F). The corresponding topology is written as EX7F•
(c) @ (resp. S) is composed of all bounded subsets of finite dimensions. The corresponding topology is denoted as EX,F, the topology of inductive tensor product [3]. In this case "S-Z-hypocontinuous" becomes "separately continu- ous."
There is another important topology on EXF defined in a manner entirely different from those given above.
(e) Grothendieck's s-topology EX,F is defined as the topology induced on EXF by the topology of EsF=Ye(E3 ; F) =Ye(F3 ; E).
It is not diMcult to see that the five topologies thus defined may be arranged as s f{gz:s{gB-ÅqrinÅqc according to coarser sg finer about topologies.
2. p-Type testing function spaces.
Let E be a locally convex Hausdorff space over C and let the topology of E is given by the family {p.; crEI} of semi-norms on E. For each pair (n, m) of nonnegative integers, let there be given a continuous real valued function S.,.(t) on R, called weight function, such that S.,.(t);}}le.+i,.(t)ÅrO for all t. Let {Kn}
and {I.} be two nested closed interval covers of R. For any pair (n,p) of non-
negative integers and for each cr c I, let us define a semi-norm pn,p,p.(q) fir
p-Type Testing Function Spaces and a Related Theorem due to A.H. Zemanian 11
qE8(E) as follows:
SUP Pa(6n,m(t)q'(le)(t))•
Pn,p,p.(q)=MaX OSk,m,qE;;P tEIq
For each n, let us define .f'.(E) as the set of functions q E 8(E) such that supp q is contained in K. and p.,p,p.(q) is finite for everyp and aE L It is not difficult to see that .fi.(E) is a locally convex Hausdorff space equipped with the topology defined by the family {p.,p,p.;p=O, 1, 2,•••, aE I} of semi-norms. On account of the condition 6..i,.(t)K6.,.(t), it is seen that p..i,p,p.(gp)Kp.,p,p.(q) and hence that J.(E)(i..i(E) with continuous injection. The space J-2f(E) is defined as the inductive limit of such increasing sequence {J.(E)} together with the continuous injections, i.e. ,.f-r(E)=O,.Åë-.(E) provided with the finest locally convex
n=O
topology such that each injection .i}.(E)--ÅrJil(E) is continuous. y(E) is called a p-type testing function space [11]. In case E= C, we write .Åë-r.(C) =Ji. and -1(C) = -Åë. It is noted that J.(E) is a Frechet space if E is a Frechet space.
The p-type testing function space .ii(E) defined thus above depends seem- ingly upon, besides the space E, the nested closed interval covers {K.}, {In} and the weight function 6.,.(t). In order to make a closer examination of the inter- ' relations among them,let us say that y- (E) is of type (i,i) provided that the closed interval covers {K.} and {I.} employed in defining it is of typeiand of
typei respectively where i,i=I, II., II., III. For any nonnegative n, nz, k, g and for any cr E I, let us define
Zn,m,k,q,p.(q')=SUPpa(6n,m(t)q)(le)(t)), q) E8(E), tE1'q
and we observe that
Pn,p,p.(q)== MaX Zn,m,le,p,p.(q)E{gCÅre.
of{;le,mgp
In case where 6.,.(t)=1 for all n, m, t, we write ZZ,,,p.(q), pS,p.(q) instead of Zn,m,k,q,p.(q), Pn,p,p.(q)• Therefore it holds that
ZZ, 4,p.(q) = SUP pa(Åë(k)(t)), tEIq
PS,p.(op)=MaX Z;,p,p.(q).
OSkSP
We now study j(E) separately in accordance with the classification of types
given above.
12 K. YosHiNAGA
PRoposmoN1. J2r(E) oftype (I,i),7'=I, II.,II-, III, is g-(E).
PRooF. Let n be given. Then for any qc8(E), suppq(K., it is not
diMcult to see that
an,mZZ,q,p.(gp)-ÅqZn,m,le,q,p.(q').Åqbn,mZl,q,p.(q'),
Where an,m==
,i}nKf.6n,m(t) and bn,m=,s,uKp.6n,m(t)År-an,mÅrO• Therefore putting an;p=
Min(an,o, an,i,•••,an,p) and bn,p=max(6n,o, b.,i,•••, b.,p), one obtains an;pPS,p.(q)-ÅqPn,p,p.(q)-Åqbn;pPS,p.(q)
for any qE8(E), supp q(K.. This proves ,J-if.(E)== g7K.(E) and so by taking the inductive limit it holds that ii(E)=g- (E) independently of {6.,.(t)}. This completes the proof.
PRoposiTioN2. Y(E) oftype (II.,I) or oftype (II., U-) is 9M.(E).
PRooF. Take any pair (n, g) of nonnegative integers. Observing, in any case, that every nonvoid I,AK. is a compact set, one obtains
(1) an,m,qZZ,4,p.(q')-ÅqZn,m,k,q,p.(q')Kbn,m,qZZ,q,p.(gP)
for every qEs(E) with suppq(K., where an,m,q= inf6.,m(t), and bn,m,4==
tElqAKn
t,S iU,.PKSn,m(t):}ilan,m,qÅrO• TherefOre,letting aA,p==min(a.,o,p, a.,i,p,•••,a.,p,p) and bA;p==MaX(bn,o,p, bn,i,p,•••, bn,p,p), it now follows that
(2) aA;pPS,p.(q)hÅqPn,p,p.(q)-Åqbh;pP;,p.(q)
for any qE8(E) with supp q(K.. Thus it is seen that, given any n enough Iarge to get that K. is a right half line, .J.(E) is the set of q E 8(E) with supp
op(K. provided with the subspace topology of 8(E), i.e. Y- .(E)==esM K.(E). Con- sequently y(E)= g.(E) is obtained as desired. This completes the proof.
Quite similarly we may state the following
PRoposmoN 3. -(E) of type (IL, I) oT of type (II-, U.) is e--(E).
PRooF. The proof is omitted.
PRoposmoN 4. Y(E) oftype (Ill, I) is 8(E).
PRooF. Take n sufliciently large so that K.==R is the case. Then for any
qcs(E) it holds that (1) is true by using the same a.,.,, and b.,.,, as above
with K.==R. It then follows that (2) is also true for any qe8(E). Therefore
p-Type Testing Function Spaces and a Related Theorem due to A.H. Zemanian 13
we get Y.(E)==es(E) and consequently .yr(E)==es(E). This completes the proof.
In order to study y(E) of further types, let us consider the growth property of a system of weight functions. We say that two systems of weight functions {e.,.(t)} and {6S,.(t)} has the same growth pTopeTty at t-År+oo (resp. t-•-oo), in symbols 6.,.(t)-v6E,.(t) (t.+cÅro) (resp. (t.-cÅro)), if for every pair (n, m) of nonnegative integers there exist positive numbers tn,., an,. and B.,. such that
6S,.(t)
(3) Ctfn,m-Åq- g.,.(t)-ÅqBn•m for each tl}irtn,m (resp. ts{:-tn,m)•
PRoposmoN 5. Given a system {6.,.(t)} of weight f2enctions, the space .Jf(E) of type (II., U.) coincides with the spaee Y(E) of type (U+, III) ana is con- tinuously eontaineel in g.(E). .Åë(E) remctins unehangeel if {S.,.(t)} is Teplaceel by another system of weight f2enctions of the same gTowth pToperty at t-År+ oo.
PRooF. The first part is clear. To prove the second, let {eA,.(t)} be another system of weight functions such that g.,.(t)"v6i,.(t) (t--År+oo). Therefore (3) is assumed for each tl2}it.,.. The semi-norms p and Z defined by means of eA,.(t) instead of 6.,.(t) are denoted as pA,p,p. and ZA,.,k,,,p. respectively. Then owing to the assumption (3) it holds that
(xA,mZn,m,le,q,p.(q')S:ZA,m,le,q,p.(q)-ÅqBA,mZn,m,k,q,p.(9P)
for every ifEif(E), suppq(K., where OÅqevA,m==min[,i,n.f.-f'\!:Xl , afn•m]
'
SIBA•m=max[,s,".p. f".':.MEIi , Bn,m]Åqoo•
It then follows that
aS;pPn,p,p.(9')nyÅqPA,p,p.(9')-ÅqBA;pPn,p,p.Cq)
for every qE8(E), supp q(K., with OÅqaA;p ==min(aA,o, crA,i, •••, crA,p)KBA;p==
max(BS,o, BS,i,•••, BA,p). This proves that {6.,.(t)} and {eE,.(t)} defines the same .i7.(E) and therefore the same Y(E). This completes the proof.
PRoposmoN 6. Given a system {S.,.(t)} of 2veight functions, the space y(E)
of type (ll-, ll-) coincides with the space .1(E) of type (llm, III) and is continu-
ozesly contained in sP-(E). Y(E) Temains unchanged if {S.,.(t)} is Teplaceel by anotheT system of weight f2Lnetions of the same gTowth puropeTty at t-År - oo.
PRooF. The proof is omitted.
PRoposmoN 7. Given a system {e.,.(t)} of weight functions, the space .yr-(E) of type (UI, ll.) (resp. (Ill, IIm)) coincicle$ with eaeh space ,.Åë.(E) foT sufiiciently
laTge n. Y(E) Temains 2Lnchangeel if {6.,.(t)} is Teplaeed by anotheT system of weight f2Lnetions of the same gTo2vth pToperty at t-År+ oo (Tesp. t-År- oo).
PRooF. Evident.
PRoposmoN 8. Given a system {Sn,m(t)} of weight funetions, the space .fnyr(E)
of type (Ill, III) eoincides with each spaee .yr.(E) foT su,fiieientey large n. y- (E) Temains zenchangeel if {6.,.(t)} is Teplaceel by anotheT system of 2veight f2enctions
of the same gTowth pTopueTty at t.Å} oo.
PRooF. Evident.
REMARK 3. Among the spaces .sf(E) of type (III, III) the followings are wellknown examples :
(i) ,90(E) and Y given by 6.,.(t) =(1+t2)m [5];
(ii) ./a7'(E) and ./aV given by 6.,.(t)==eM'iii•P2 [9], [10];
(iii) es(E) and as given by e.,.(t)==1[5].
3. Generalizations of a theorem due to Zemanian.
It is proved in A. H. Zemanian [11] that
THEoREM Z. Given any Banach spaces E and F, there is a one-to-one cor- respondence between the spaces Y(9(E); F) and g7t(Yb(E; F)) defined by T(qXe)== Åq S, qÅr e,
where op E 9, e c E, TE Y(9 (E); F) and SE 9t(Yb(E; F)).
V. K. Balakrishnan [1] has generalized this theorem for the case where E
and F are locally convex Hausdorff spaces. At the same time he has also taken
account of the circumstances where 9 is replaced by Y. In doing so the pro-
cedure followed by him is entirely standard and relies essentially upon the fact
that a certain vector-valued distribution appeared there is of finite order on any
relatively compaet open subset of R. We here take up the theorem once again
p-Type Testing Function Spaces and a Related Theorem due to A,H. Zemanian 15 .
from the viewpoint of the tensor product of locally convex spaces.
PRoposiTioN 9. Let E, Fand G be loeaLly convex HauseloTff spaces. Then by means of the relation
(4) T( gXe) == S( g)e,
wheTe gE C, e E E, eveTy TE Y(GC}{)BE; F) (Tesp. Y(GQ.E; F), resp. Y(G(g),E; F)) aejZnes 2eniqzeely an element SEY(G; Yb(E; F)) (resp. Y(G; Y,(E; F)), resp.
Y(G ; 9.(E; F))).
PRooF. We give here aproof only for the case TEY(GC8)BE;F), because
a proper change of it will be sufficient for the rest. Taking any g E G one defines a linear application S(g) of Einto F by the relation S(g)e== T(gXe) EF for any ecE. Since T isa continuous linear application of G(2ÅrBE into F, given any gEG and given any neighbourhood V of the origin of F, one may take a neigh- bourhood Uof the origin of E in sucha way that T(g(E{)U)(V is true. This proves S(g)(U)( V, which shows us that S(g) cY(E;F). By the definition of the topology of GXrsE together with the continuity of T, it is seen that the bilinear application GÅ~E) (g, e). T(gope) EF is hypocontinuous with respect to the bounded subsets. Thus given any neighbourhood V of the origin of F and given any bounded subset A of E, there exists a neighbourhood M of the origin of G such that T( "7XA)( V, i.e. S( n7) (A)( V, and consequently S c y(G;
yb(E;F)) is obtained. That S is determined uniquely by T is clear. This completes the proof.
CoRoLLARy. Given locally convex Ha2LseloTff spaces E, F and G, let TE Y(Gop,E; F) (Tesp. Y(GX.E;F)). Then theTe exists an SEY(G;Yb(E;F))
2Lniquely cletermineel by the Telation (4) with gEG, e E E.
PRooF. Aimong the topologies e, z, B that G(8)E should be supplied with, E is the coarsest and B is the finest, i.e. eKz:f{:B. Therefore our statement is an
easy consequence of the obvious relation Y(GXeE; F)(Y(GCD.E; F)(Y(GXBE;
F) together with Proposition 9. This completes the proof.
As a converse of Proposition 9, let us give
PRoposiTioN 10. Let E,F ana G be locally convex Hauselorff spaces and assume that E is bctrrelled. Then for any SE Y(G; Yb(E; F)) (Tesp. Y(G; Y,(E;
F)), Tesp. .s2f'(G; Y,(E;F))) one obtains TE Y(GCil)BE; F) (respu. Y(G8År,E; F), Tesp.
Y(G(g),E; F)) iLniquely elefinea by the Telation (4) with gEG, e E E.
PRooF. We only prove the case SeY(C;Yb(E;F)) and a slight change
thereof will do for the rest. Taking gEG and eEE, one defines a bilinear
application R of GÅ~E into F by the relation R(g, e)=S(g)e.
In order to see that R is hypocontinuous with respect to the bounded subsets, we first note that R is separately continuous. Let us then take any closed con- vex balanced neighbourhood V of F and let A and C be any given bounded subsets of E and G respectively. Owing to the continuity of the application S of G into Yb(E; F) it follows that R( "7Å~A)( V for a proper choice of a neigh- bourhood IU of the origin of C. Putting U={e; R(Cx{e})( V, ecE}, we may observe that U isa barrel of E and therefore a neighbourhood of the origin of E satisfying R(CxU)(V. Thus the hypocontinuity of R in question is proved and consequently by the definition of the tensor product and its 3- topology, one finds Tc Y(GXBE; F), uniquely determined in such a way that the relation
T(gXe)=:R(g, e)=S(g)e
holds true for any gEG and ec E. This completes the proof.
Proposition 10 together with Proposition 9 establishes the following theorem.
THEoREM 1. Let E, F ancl C be locally convex HazesaoTff spuaces anel szeppose furtheT E is baTTelled. Then theTe is a one-to-one coTTesponelence bet2veen the spaces Y(GXBE; F) (respu. Y(GX,E; F), resp. Y(G(g),E; F)) and Y(G; yb(E; F)) (Tesp. Y(G; Y,(E; F)), Tespu. Y(G; :2f',(E; F))) thTo2Lgh the Telation
T( gQe) = S( g)e,
wheTe TE :2f'(GXBE; F) (resp. Y(GX,E; F), Tesp. Y(G(El),E; F)), SE Y(C; :2f'b(E;
F)) (resp. Y(G; Y.(E; F)), resp. Y(G; Y,(E; F))), geC ana e c E.
PRoeF. Clear from Proposition 9 and Proposition 10.
In order to obtain Theorem Z, it is enough to take G=g in Theorem 1. To be precise, one may obtain the following
THEoREM 2. Let E be a Frechet space and let .if =g, g7., g., es, ,s;o oT ,/cz7'.
Then foT any loeally convex anel eomplete Hausdorff spaee F, theTe exists a one- to-one lineaT applieation of Y(Y(E); F) onto Y(.1; .S?b(E; F)) thTozegh the Tela- tion
T( if Xe) == S( g7)e, q E .1, e E E,
wheTe TE Y(Y(E) ; F) and S E Y(.yif ; Yb(E; F)).
p-Type Testing Function Spaces and a Related Theorem due to A.H. Zemanian 17
PRooF. Case .Åër == 9: It is already known that sD OX ,E =g(E) whenever E isaFrechet space [3, Chap. II, p. 84]. Since g and Eare both barrelled it is easy to see that 9([]JBE=g2g,E [7, p. 13], [2, Chap. III, p. 40, Proposition 6].
To prove the statement it is now enough to employ Theorem 1.
Case .Jr==g.: The essentials of the proof given below are the same as the
previous case. We first show that 9.(ii5,E=g- .(E) whenever E is a Frechet space. By definition the space g2- .(E) (resp. 9.) is the strict inductive limit of
8- K.(E) (resp. 8K.), where Kn=[n, +oo[ and esK.(E) (resp. 8K.) is the set of qc8(E) (resp.qces) with suppq(K.. Sinceeach 8K. isanuclearFrechet
space as a closed subspace of the nuclear Frechet space 8, it holds that esK.(E)=
8K.g.E [3, Chap. II, p. 80, Theoreme 13]. On the other hand one obtains esK.g.E=8K.Cl]5, E because of the fact that 8K. and E are both Frechet spaces.
Since the topology of the inductive limit of the inductive tensor products coincides with the topology of the inductive tensor product of the inductive limit [3, Chap. I, p. 76, Proposition 14], it holds that g.(E)== g7.g,E as desired.
We next see that 9+CISBE==9+(i]J,E as in the previous case and consequently Theorem 1 proves the statement for Case .1 =9..
Case y==9-: A slight modification of the preceding proof will do for this case.
Case Y=8: It is wellknown that 8(E)=8(E) [5]. And since s is nuclear in addition to that both 8 and E are Frechet spaces, one obtains es(E)= 8CiS.E==
ffCi5,E. Thus es- (E)==8(ll5BE is true and Theorem 1 proves the statement for this case.
Case .Åër==y: Quite similar to the preceding case.
Case J5==Y': It is known that JO" isanuclear Frechet space with the pro- perty of approximation by truncature and regularization [9]. Therefore it holds that J07'(E)=:Ja7-(li5eE==J67-([l5.E==J`7-gBE=J`O-g,E [3, Chap. II, p. 34, Th6oreme 6], [7, p. 13]. In order to see J`7"(E)= J67'(E), we first note that this equality holds
true in the sense of linear spaces. Then the continuous injection of JO'(E) onto
Jo7'
(E) is seen to be topological, i.e. Jd?"kE) = J`"'(E) as desired. Thus Theorem 1 proves the statement. This completes the proof.
REMARK 4. In order to get a further theorem of this sort, it is enough to
prove the identity of the type Jr(E)= YgxE, where E is a given locally convex
Hausdorff space and Y is any given function space. The topology Z of the tensor
product will be determined by the topology that the space Y(E; F) may be
supplied with. It will be seen as follows that such a identity is also necessary.
18 K. YdsHiNAGA
PRoposiTioN 11. Let Eana G be locally convex HazescloTff spaces and let E be a linear szebspace of C Tegarelless of the topology. Asszeme that, foT any Bauach space F, there is a o7ze-to-one lineaT apupeication of Y(E; F) onto Y(G; F) such that the Testrietion on E of the image of any u E Y(E; F) is ielenticctl with u.
Then E is a dense topotogical linear subspace of G. In conseq2eence it holas that Y(E; F)==Y(G; F) for any locally eonvex Hausaorff space F.
PRooF. We begin by taking F=C. By assumption there isaone-to-one
linear application L of Ei onto G' satisfying ÅqL(u), e,År==Åqu, eÅr for each
uEE',eeE. Take anyvcG' such that Åqv,eÅr =O for every eEE(G. Since v =L(u) for a proper ucE', one obtains Åqu, eÅr=ÅqL(u), eÅr =O for each eEE and therefore u=Ofrom which v=O. It follows thatEis dense in G. We next show that any convex balanced closed neighbourhood U of the origin of E is written as U=EAV with a suitable neighbourhood V of G. Let Eu be the normed space brought about by supplying the quotient E/IV of E by the subspace Nof x such that Zx E U for all scalar Z with the norm induced by the semi-norm
llxllu== I'
