© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
Rev. Acad. Canar. Cienc., XII (Núms. 1-2), 107-124 (2000) - (Publicado en julio de 2001)
Sorne properties in the dual of a Fréchet space1
by
Manuel Valdivia
Abstract
In the dual of a Fréchet space we study Schauder basic sequences as well as certain
bounded subsets.
1 Preliminaries
Al! linear spaces we shall be dealing with in this paper are assumed to be defined over the
field JK of real or complex numbers. The topologies to be defined in these linear spaces will be
separated. lN stands for the set of positive integers.
If (E, F) is a dual pair of linear spaces, by (-, ·) we denote its associated bilinear
functional. For a subset A of E, Aº and A.L will be the polar and the orthogonal set, respectively,
of A in F . The weak topology in E will be represented by a(E, F). If B is a closed bounded
absolutely convex subset for the topology a(E, F), then Es will denote the normed space
defined by the linear hull of B normed by its Minkowski functional. We say that a sequence
(xn) of E converges to x in the sense of Mackey when there is a subset A of E which is closed
bounded and absolutely convex for the topology a(E, F) such that Xn, x E A, n E lN, and (xn)
converges to x in EA· A series ·L;:"=1 Yn in E converges to y in the sense of Mackey whenever
the sequence (E~=l Yn)::.'=i converges to y in the sense of Mackey.
If E is a locally convex space, then E' will denote its topological dual. The duality
(E, E') is the usual one, that is, if x E E, u E E', then (x, u) = u(x). If T is the topology of
E and A is a subset of E, then A [r] will stand for the topological space induced in A by the
topology T. We say that E is locally separable if given an arbitrary bounded subset M of E,
11991 Mathematics Subject Classification: 46 A 04, 46 A 35.
Key words and phrases: Fréchet spaces, basis.
The author has been partially supported by Programa Sectorial, DGES pro. 96-0758.
107
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
there is a bounded subset B of E, closed and absolutely convex, such that M C B and EB is
separable.
We say that a sequence (xn) in a locally convex space E is a Schauder basis, or simply
a basis, if there is a sequence (un) in E' such that
and every element x of E may be expressed as the sum of a series of the form E::"=t anxn,
an E IK, n E IN. It is then immediate that (x, un) = an, n E IN. We call Un, n E IN, the
associated functionals of the basis (xn) ·
A sequence (xn) is basic in the locally convex space E if it is a Schauder basis of the
subspace F of E given by the closed linear hull of {xn: n E IN}. Making use of Hahn-Banach's
Theorem one obtains that a sequence (xn) is basic if and only if there is a sequence (un) in E'
such that
and every element x of F can be expressed as the sum of a series of the form E::"=t anXn, an E IK,
n E IN. It is then immediate that
00
X = L (x,un) Xn .
n=l
Proposition 1 Let (xn) be a sequence in a locally convex space E. Let r be a positive integer.
If the sequence (xn) is basic and x 1 , x2, ... , Xr are linearly independent, then a finite number o/
terms may be omitted in the sequence (xn) so that one obtains a subsequence (zn) o/ (xn) with
the following properties:
1. (zn) is basic.
2. Zn = Xn, n = 1,2, ... ,r.
Proof Let F be the closed linear span of { Xr+i, Xr+2, . . . } . We find a sequen ce ( vn)::"=r+l in E'
so that
(xm, Vn) = O, m # n, (xm, Vm) = 1, m, n = r + 1, r + 2, ... ,
and, if x E F, then
00
X L (x, Vn) Xn·
n=r+l
108
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
Let G be the linear hull of {x1, X2, ... , Xr }. If F n G = fO}, we put Zn := Xn, Wn .- Vn,
n = r + 1, r + 2, ... Suppose now that F n G #{O}. We take y E F n G, y# O. Then
00
Y = L (y, Vn)Xn
n=r+l
and so there is a positive integer no > r such that (y, Vn0 ) # O. If we put L to denote the closed
linear span of
it follows then that L n G e F n G and y fÍ: L n G. Proceeding in this way we may omit a
finite number of terms of (xn);:;,r+i so that it turns into a subsequence (zn)::"=r+l of (xn)::"=r+l
in such a way that if M is the closed linear hull of { Zr+i, Zr+2, ... } then M n G = {O}. A finite
number of terms of (vn)::"=r+l may also be omitted so that it becomes a subsequence (wn)::"=r+i
of ( Vn)::"=r+l with
(zm, wn) =O, m # n, (zm, Wm) = 1, m, n = r + 1, r + 2, ...
In any case considered above we have obtained a subsequence (zn)::"=r+l of (xn)::"=r+l• by omitting
a finite number of terms in this last sequence, so that, if H is the closed linear hull of
{z,+l, Zr+2, ... }, then HnG ={O}. We have also obtained a sequence (wn)::"=r+l in E' such that
(zm, Wn) = O, m /= n, (zm1.wm) = 1, m, n = r + 1, r + 2, ...
We write z1 := x1, j = 1, 2, ... , r. We choose j E JN. If 1 :S j :S r, we find an element u1 in E'
such that it vanishes in H and in {xi, x2 , •.. , Xr} \ { x1} and takes value 1 in x1. If j > r, we take
u1 in E' such that it coincides with w1 in H and vanishes in G. Then
(zm, Un) =O, m /= n, (zm, um) = 1, m, n E JN,
and, if x E G + H, it follows that
X = Y. + z, y E G, z E H.
Hence,
r 00
y = L (y,u1) z;, z = L (z,u;) z;,
j=l j=r+l
from where we obtain that
00
x = L (x,u;) z1.
j=l
109
q.e.d.
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
Proposition 2 Let (xn) be a basic sequence in a locally convex space E. Let F be the subspace
of E' orthogonal to {x 1 ,x~ .... ,xn, ... }. Then, there is a basis (un) in E' [a(E',E)]/F whose
associated functionals coincide with Xn, n E JN.
Proof Let ( vn) be a sequence in E' such that
(xm, Vn) =O, m # n, (xm, Vm) = 1, m, n E JN.
Let G be the closed linear hull of {X¡, x2 , •. . , Xn, .. } Let <p be the canonical mapping from E'
onto E'/ F. We put
Un:= cp(vn), n E JN.
We see next that (un) is a Schauder basis in E' [a(E', E)]/ F . We identify in the usual way G
with the topological dual of E' [a(E', E)]/ F. Then
We take an arbitrary element u in E' [a(E', E)]/ F . We choose v E E' with cp(v) =u. Let x be
in G. Then
00
X L (x,vn) Xn
n=l
and so
r
(x, u) = (x, v) = lim(L (x, Vn)Xn, v)
r n=l
r r
= li~(x, L(Xn,v)vn) = li~(x, L(Xn,u)un),
n=l n=l
cunsequently
in E' [a(E',E)]/F.
q.e.d.
Corollary 1 Let E be a Prér.het space. Let (un) be a basir. sequence in E'[a(E',E)]. Let F be
the subspace of E orthogonal to {U¡, u2 , ..• ,Un, ... } . Then, there is 11 basis (xn) in the Préchet
space E/ F whose associated linear functionals coincide with Un, n E: 1N.
Proof AftP.r the last proposition, there is a basis (xn) in E [a(E,E')]/F with associllted
functionals being un, n E 1N. Since E/ F is a Fréchet space, every weak basis of this space is a
basis, [3, p. 296], and so (xn) is a basis in E/F.
110
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
q.e.d.
If X is a Banach space, X*, and also (X)*, is the Banach space conjugate of X. We
write X**, and also (X)**, for the second conjugate of X. We identify X with a subspace
of X** by means of the canonical injection. B(X) denotes the closed unit hall of X. Unless
stated, 11 · 11 will be the norm of X, and also the norm of X*. A Banach space X is said to be
quasi-reflexive when the dimension of X**/ X is finite. Clearly, every reflexive Banach space is
quasi-reflexive. The first example of a quasi-reflexive non-reflexive Banach space is given in [2].
If E is a Fréchet space, f3(E', E) is the strong topology on E'. By E" we mean the
topological dual of E' [f3(E', E)]. The strong topology on E" will be represented by f3(E", E').
Clearly, in the case that E is a Banach spaceX, E" [f3(E", E')] is X** with the norm topology.
We identify the Fréchet space E with a subspace of E" [f3(E", E')] via the canonical injection. If
A is a subset of E , A is the closure of A in E" [u(E", E')] and A* is the subset of E" [u(E", E')]
formed by those elements x which are adherent points of sequences of A. If Y is the linear hull
of E U A, we say that A is of finite order whenever the dimension of Y/ E is finite. When this
dimension is infinite we say that A is of infinite order.
We say that a basis (xn) in a Fréchet space E, with associated functionals (un), is
locally shrinking if, for any u in E', the series I:~=l (xn, u)un converges to u in the sense of
Mackey.
Let E be a Fréchet space. Let A be a bounded subset of E' [u(E', E)]. We say that
A is locally of finite order when there is a compact absolutely convex subset B of E' [u(E', E)],
B :J A, such that A is of finite order in the Banach space E~.
If E is a Fréchet space and Mis a compact absolutely convex subset of E' [u(E', E)],
by E(M) we denote the linear space formed by the linear functionals on EM whose restrictions to
M [u(E', E)] are continuous; we assume E(M) endowed with the norm defined by the Minkowski
functional of the polar set of M in E(M)· Then, E(M) is a Banach space whose conjugate
identifies with EM.
2 Sequences in the dual of a Fréchet space
For a sequence (un) in a linear space, we say that (vn) is a block-convex sequence of (un) if
there are positive integers
1 n 1 < n2 < . . . < ni < ...
111
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
and, for each j E JN, there are
a;,r ~O, r:::::: n;, n;+i. ... , n;+l - 1
such that
n;+1-l n;+i-1 L a;,r :::::: 1, v; L a;,rUr.
v=n; r=n;
lt is clear that every subsequence of (Un) is a block-convex sequence of (un)· Besides, every
subsequence of (vn) is a block-convex sequence of (un)·
In the next Lemma 1, Theorem 1 and Theorem 2, E is a Fréchet space and
Ai e A2 e ... e Am e ...
is a fundamental system of compact absolutely convex subsets of E'[u(E', E)]. Tm is a locally
convex topology in EAm which is coarser than the norm topology, finer than the one induced
in EAm by T m+l • m E 1N and finer that the induced topology by u(E', E) in EAm.
Lemma 1 Let (un) be a sequence in A1 such that u(E', E)-converges to the origin. IJ, for each
m E 1N, (un) does not Tm-converge to the origán, then .there is a block-convex sequence (vn) of
(un) which does not have the originas a Tm-adherent point.
Proof. For each s E 1N, we find in E'A. a r,-continuous seminorm /, anda subsequence (v,.;)
of (u;) such that
inf {f,(v,.;) : j E JN} > O, f,(u) $; 1/2, u E A,.
We write p1 ::::::: Ji. If s > 1 and Hr.; is the hyperplane of E orthogonal to {vr.;}, j E 1N,
r :::::: 1, 2, ... , s - 1, we then have that
U{ Hr.; : j E 1N,r:::::: 1,2, ... ,s-1}
is a subset of the first category-in E, hence there is an element
x,EE\u{Hr.;: jE1N,r=l,2, ... ,s - 1}
such that
1 1(x.,u)1 $; 2' u EA,.
We now write
p,(u) := f,(u) + 1 (x., u) 1, u E E'A.·
112
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
lt then follows that, for each s E lN, p. is a r.-continuous seminorm in E~. such that
inf {p,(v,,;) : j E lN} > O, p,(u) ~ 1, u E A,,
and, if s > 1,
p,(vr,;) ::/=O, j E lN, r = 1,2, ... ,s-1.
If there exista q E lN, a subsequence (tq,;) of (vq.;) anda sequence of positive integers
m1 < 1712 < ... < mn < ...
such that
inf {p,(tq.;) : j E lN} > O, s E { m1, m2, ... , mn, ... }
it suffices to put v; := tq.;,J E lN, to reach the conclusion. If it is not so, we proceed inductively
beginning with a sequence ( v1.;) and with an arbitrary sequence of positive integers
di < d2 < . .. < dn < ...
We find an element
such that
inf { p.1•1 (v1.;) : j E lN} = O.
Now, since s1,1 > 1, it follows that p.1•1 (v1.;) ::/=O, j E lN and so there is a subsequence (vt,;) of
( V1.;) such that
lim p,1,1 {vt.;) = O.
J
Let us assume that,. for a positive integer r, we have already found the subsequence (ví.;) of
(v1.;) and the integer s1,r > 1 so that
We then find an element
S1,r+l E {di, d2, ... , dn, ... , }, 81,r+l > S1,ri
such that
inf {P.1,r+i(vr.;). j E lN} = O.
113
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
Since s1,r+l > 1, we have that Psi,r+i(ví) ¡.O, j E lN, and so we find a subsequence (víj1) of
ví) such that
This ends the induction process. We now define u1,n := ví',n• n E lN. Then
lim p.1 •• (u1,j) = O, q E lN.
J
Again proceeding by complete induction, we assume that for a positive integer n we have found
the sequence of positive integers
Sn,l < Sn,2 < .. · < SnJ < ...
Proceeding as we <lid before, we start with the sequence (v,n,nJ) instead of (v1,i) and with (snJ)
instead of (di) · We then obtain a subsequence (sn+i ,j) of (sn,j) such that
lim Psn+l,q ( Un+lJ) = O, q E lN.
J
This concludes the induction process. We now put r 1 := 1, Tm+i := Sm,mi m E lN. If we now
fue m, i E lN, it follows that Sm+i-l,m+i-1 is an element of the form sm,di d E lN. Then
On the other hand, (umJ) is a subsequence of (vrmJ) and consequently,
We now write Qm := Prm• m E lN. Then, Qm is a Trm-continuous seminorm on EArm such that
Qm(u):::::; 1, U E Arm> inf{qm(UmJ): j E lN} = hm >O,
lim Qm+i(UmJ) = O, i E lN.
J
We may choose the sequence (umJ), for each m > 1, in such a way that UmJ goes after
um- l,j in the order of the sequence (un), j E lN. Clearly, the linear span of { UmJ : j E
lN} has infinite dimension and therefore we can take the vectors UmJ• j E lN, to be linearly
independent. Besides, for each m > 1, we can choose um,j so that it is linearlv ;ndependent
from u1J, u2J, .. ., Um- lJ• j E lN. So, we may assume that, for each j E lN, the vectors u;J,
i E lN, are linearly independent.
114
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
We now take numbers
1 > k1 > ... > k,,, > ... > o
such that
O < k,,, < h,,., km+l < r(m+2l k~, m E lN.
We find a positive integer n1 such that
and, assuming that for a positive integer m we have already obtained nm E 1N, we find an
integer n...+i > nm such that
( ) k;,+2
Qm+2 u;,; < 4(m + 2)' j = 1, 2, ... , m - 1, i ~ nm+l·
Given the positive integers p $ s, the vectors u1,np ' u2,np> ... , Up,np are linearly independent and
nence there is a vector zp,• in E such that
(zp,., U1,np) = 1, (.ip,., Uj,np) =O, j = 2, 3, .. ,p.
We write
•
g,(u) := q,(u) + L 1 (zp,., u) 1, u E E~ •• .
p= l
Then, g, is a Tr, -continuous seminorm on E~ ••. We clearly have that
is a subsequence of (un)· We define
m
fJ;m := k; (L k;)-1, j = 1, 2, ... , m, m E JN.
i=l
The sequence ( vm) such that
m
Vm := L fJ;mUj,nm>
j=l
is a block-convex sequence of (un) · We fix a positive integer s > l. Then, for m > s,
a-1 m
g,(vm) ~ q,(vm) ~ q,(fJsmUs,nm) - L q,(fJ;mUj,nm) - L q,(fJ;mUj,n.,)
j=l s+l
m -1 m m
~ f3amka - (L k;t1 L k;q,(u;,nm) - (L k;)-I L k;q,(u;,nm)
i=l j=l i= l j =s+l
115
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
>_ f3 sm k s - ~4f 3s m k s - (L~.., k•) -1 ~L.., r(i+ilk32 -l
i=l j=l
:'.:'.: f3smks - 41f 3smks - 41f 3smks > ;1/ •2 (~ )-1 L.., k; ,
i=l
and, ifm:::; s,
m
g,(vm) 2:1(zm,.,vm)1=1L(Zm,.,f3jmUj,nm)1
j=l
= f31m = kl(f k;)-l > ~k2 d=k·)-l
i=l 2 • i=l ' .
Then, for any m E IN,
g,(vm) > ~k;(f k;)- 1,
i=l
from where we deduce that (vm) <loes not have the originas a Tr,-adherent point in EÁ,, .
q.e.d.
Note. In (6], two particular results of the previous lemma are proved: When Tm coincides with
the topology of the norm of EÁm and when r is the weak topology of EAm • m E IN.
For the proof of Theorem 1 we shall need the following result to be found in (7]: a)
Let E be a separable Fréchet space. Let
be a fundamental system of zero neighbourhoods in E . Let 11 · llm denote the norm of Eh,
m E IN. Let (un) be a sequence in UJ. that a(E', E)-converges to the origin and such that
in/{11 Un llm: n E IN} > O, m E IN.
Then, there is a subsequence (wn) of (un) such that if F is the subspace of E orthogonal to
{wn : n E IN}, there is a basis (xn) of E/F whose associated functionals coincide with Wn,
n E IN.
Theorem 1 Let (un) be a sequence in A1 such that it a(E', E)-converges to the origin and it
does not converge to the origin for the topology Tm, m E IN. If E is separable, then there is a
block-convex sequence ( wn) of (un) satisfying the following conditions:
1. (wn) is basic in E' [a(E', E)].
2. (wn) does not have the origin in EAm as a Tm adherent
point, m E IN .
116
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
Proof We apply Lemma 1 to obtain a block-convex sequence (vn) of (un) with the property
there mentioned. If 1 · lm is the norm of EÁm, we then have that
inf{I Vn lm: n E lN} > O, m E lN.
Applying now resulta), we obtain a subsequence (wn) of (vn) such that it satisfies the properties
there mentioned. Clearly, (wn) is a block-convex sequence of (un) that accomplishes l. and 2.
q.e.d.
For the proof of Theorem 2 we shall need the following result, [4]: b) Let X be a
Banach space with X* separable. Let (un) be a sequence in X* that converges to the origin for
the weak* topology but it does not converge to the origin for the norm topology. Then there
is a subsequence ( wn) of (un) such that, if Y is the subspace of X orthogonal to { Wn : n E JN},
then X/Y has a shrinking basis whose associated functionals coincide with Wn, n E JN.
Theorem 2 Let (un) be a sequence in A1 that a(E', E)-converges to the origin and it does not
converge to the origin for the topology Tm, m E JN. If E' [a(E', E)] is locally separable, there is
a block-convex sequence (vn) of (un) such that it satisfies the following conditions:
1. (vn) is basic in E' [a(E', E)].
2. (vn) does not have the origin in EÁm as a Tm-adherent
point, m E lN.
3. If F is the subspace of E orthogonal to { Vn : n E JN},
then the basis (xn) of E/ F whose functionals
coincide with Vn, n E lN, is locally shrinking.
Proof We may assume with no loss of generality that EÁm is separable, m E lN. Clearly,
E' [/3(E', E)] is separable and so E is also separable. We apply Theorem 1 to obtain a blockconvex
sequence (wn) of (un) with the properties l. and 2. there stated. The Banach space
E(Ai) has EÁ, as its conjugate, which is separable. On the other hand, (wn) converges to
the origin in EÁ, [a(EÁ,, E(AiJ)] and it is not norm-convergent in EÁ, . Applying result b),
we obtain a subsequence (w1n) of (wn) such that if F1 is the subspace of E(Ai) orthogonal to
{w1n: n E JN}, then E(Ai)/F1 has a shrinking basis whose associated functionals coincide with
w1n, n E JN. Proceeding inductively, let us assume that we have found, for a given positive
integer r, the subsequence (wrn) of (wn) such that if F, is the subspace of E(A,) orthogonal
to {wrn : n E JN}, then E(A,)/F, has a shrinking basis whose associated functionals are Wrn,
117
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
n E IN. The Banach space E(A,+1) has EA,+1 as its conjugate which is separable. On the
other hand, (wrn) converges to the origin in EA,+1 [a(EA,+i•E(A,+i))] and it <loes not converge
in norm in EA,+1 · We apply again result b) and thus obtain a subsequence(w(r+lJn) of (wrn)
such that if Fr+l is the subspace of E(A,+il orhtogonal to { W(r+l)n : n E IN}, then E(A,+1)/ Fr+l
has a shrinking basis whose associated functionals coincide with W(r+l)n, n E IN. Having in
mind Proposition 1, the sequence (wcr+1Jn) may be chosen in such a way that W(r+l)n = w,n,
n = 1, 2, ... , r. This concludes the complete induction process. We now define Vn := Wnn, n E IN.
Then, (vn) is a subsequence of (wn) and so it is a basic sequence in E' [a(E', E)]. Clearly, (vn)
<loes not have the origin in EA= as a Tm-adherent point, m E IN. Let G be the closed linear
hull of {vn : n E IN} in E' [a(E' , E)J and !et F be the subspace of E orthogonal to G. Let
(xn) be the Schauder basis in E/ F whose associated functionals are Vn, n E IN. Let t.p be the
canonical mapping from E onto E/F. We take Yn E E such that t.p(Yn) = Xn, n E IN. Let u be
an arbitrary element of G. Then, in E' [a(E', E)J,
00 00
U = L(Xn,u) Un = L(Yn,u) Un.
n=l n=l
The sequence (í::~=l (yn, u)un)~=l is bounded in E' [a(E', E)J and therefore there is r E IN such
that
m
L(Yn,u)un EA,, mEIN.
n=l
n=l
In E(A,) / F, there is a shrinking basis whose associated functionals are ( wrn). On the other
hand, (vn) is a subsequence of (w,n) and consequently, if G, is the subspace of E(A,) orthogonal
to { VN : n E IN}, we have that E(A,)/G, has a shrinking basis (x~) whose associated functionals
are Vn, n E IN. Besides, in EA, ·we have that
00
u L (x~, u) Vn·
n=l
Thus, for each m E IN,
00
(Ym, u) = (Ym, L (x~, u)vn) (x:-,., u).
n=l
Hence, the series I:::'=i (xn, u)vn converges to u in EA, .
q.e.d.
118
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
3 Bounded sets in the dual of a Fréchet space
Let F and H be two subspaces of the linear space E. Let cp and 'lj; be the canonical mappings
from H onto H / H n F and from E onto E/ F, respectively. If x belongs to E/ H n F, we choose
y in H such that cp(y) = x and we define Tx := 'l/;(y). It is clear that Tx does not depend on
the element y ch osen and T : H / H n F -t E/ F is a one-to-one linear mapping. We shall say
that T is the canonical mapping from H / H n F into E/ F . Later on we shall need the following
result, [8]: c) If E is a locally convex space, F is closed in E and H n F is dense in F, tben
the canonical mapping T : H / H n F -t E/ F is a topological isomorpbism from H / H n F
onto the subspace T(H/H n F) of E/F.
Lemma 2 Let A be a bounded subset o/ a Fréchet space E . Let H be a closed subspace o/ E
and let cp be the canonical mapping from E onto E/ H . If A• is contained in E+ H, then cp( A)
is weakly relatively compact in the Fréchet space E/ H.
Proof We show that cp(A) is weakly relatively compact making use of Eberlein's Lemma.
Let 'lj; be the canonical mapping from E/ H onto E" ¡H. Let T be the canonical mapping from
E/ H into E" ¡H. After result c), we have that T is an isomorphism from E/ H , with the weak
topology, onto the subspace T(E / F) of E" [u(E", E')]/H. We now take an arbitrary sequence
(xn) in cp(A). Let (Yn) be a sequence in A such that cp(yn) = Xn, n E JN. Let y0 be a point in
A* which is u(E", E')-adherent to (Yn)· It follows that y0 =u+ v, u E H, v E E . Then, the
sequence ('l/;(Yn - v)) in E" [u(E",E')JIH has the originas an adherent point, from where we
deduce that the sequence (T- 1 ('1/;(Yn - v) )) in E [a(E, E')]/H has the originas adherent point
and so, in E/ H, cp(v) is a weakly adherent point of the sequence (xn) ·
q.e.d.
Theorem 3 LetA be a bounded subset o/ a Fréchet space E. Let F be the linear hull o/ E u A.
Jf the dimension o/ E/ F is infinite, then there is a countable subset B o/ A such that, if G is
the linear hull o/ E U B, then the dimension o/ G /E is also infinite.
Proof Let us assume that the property is not true. Let L be the closed linear hull of E U A•.
It is immediate that the dimension of L/ E is finite. Let P be a finite-dimensional subspace of
L such that E+ P =L. We find a separable closed subspace H of E such that H :::>P. Then,
A• e E + H. Let now cp and 'lj; be the canonical mappings from E onto E/ H and from E"
119
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
onto E" /fi, respectively. We take in E a fundamental system of closed absolutely convex zero
neighbourhoods
Let el> be the canonical mapping from E' onto E'/ H .L. We ha ve that L # F and hence we may
find an element z in A not in L. Let {zi : i E J, 2} be a net in A that a(E", E')-converges to
z. The net {cpzi: i E J, 2 } is weakly Cauchy in E/H. After the former lemma, cp(A) is weakly
relatively compact, and so {cpz;·: i E J, 2} converges to a certain point t. We choose w in E so
that cpw = t. Then, it is clear that z - w belongs to H. We put M to denote the subspace of
E" [¡J(E", E')] given by the closed linear hull of (H n L) U {z - w} . For every m E JN, <I>(U,';.)
is equicontinuous on M and since this Fréchet space is separable, it follows that
<I>(U:;,) [u(E' I H.L, M)]
is metrizable and separable. In this topological space, we take a countable dense subset Am·
Let Bm be a countable subset of u:;. such that <I>(Bm) = Am. Then
separates points in M. We now find a sequence (im) of elements of I such that
In E/ H, the set
1
1 (z; - z, Uj) 1 < -, j = 1, 2, ... , n, i 2 Ín .
n
Pn := { .,Pz; : i E J}
(1)
has the point t in its weak closure, hence, applying Kaplanski's Theorem, [5, p. 312], there is
a countable subset Dn of Pn that contains t in its weak closure. We have then that the set
is countable and weakly relatively compact, hence metrizable for the weak topology. Let Vn,
n E JN, be a fundamental system ofneighbourhoods oft for the weak topology. For each n E JN,
we find in {zi : i 2 in} an element Yn such that IPYn E Vn n Dn· Then (cpyn) converges weakly
to t. We shall see next that (Yn) converges to z in the topology a(E", E') and, consequently, z
will be in A*, which is a contradiction. So, !et us take an element y which is a(E", E')-adherent
to (Yn) · Since (cp(yn - w)) converges weakly to the origin in E/H, it follows that
y-w E HnL e M
120
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
and, since z - w E M , we have that y - z E M. From (1) we conclude that
and so
1 .
1 (Yn - z, Uj) 1 < -, J = 1, 2, ... , n,
n
(y - Z, Uj) = Ü, j = 1, 2, ....
The set { ui, u2 , ... ,un, ... } separates points in G, therefore y = z.
q.e.d.
Theorem 4 Let E be a Fréchet space such that every countable bounded subset o/ E' [a(E', E)]
is o/ locally finite order. Then, there is a fundamental system o/ zero neighbourhoods in E,
such that the following conditions hold:
1. Eh is quasi-reflexive, m E IN.
2. E' [.B(E', E)] is the inductive limit o/ the sequence
o/ Banach spaces Eú::,, m E IN.
Proof. We take a compact absolutely convex subset A of E' [a(E', E)] . Let us suppose that
A is not of locally finite order. We take in E' [a(E', E)] a fundamental system
A e A1 e A2 e ... e Am e ...
of compact absolutely convex subsets. Since A is of infinite order, we apply Theorem 3 to obtain
a countable subset Mm of A of infinite order in EAm. Then, the countable set M := U::';'=1 Mm
is not of locally finite order in E' [a(E', E)], which is a contradiction. Therefore, there is a
compact absolutely convex subset B of E' [a(E', E)] such that A is contained in B and A is of
finite order in E's . We identify E's with the Banach space conjugate of E(B)- For each r E IN,
we put
Wr := 2r A + rr B.
Let 1 · Ir be the gauge of Wr . Since W, is a(E's, EcB¡)-compact, we have that 1 · Ir is a dual
norm equivalent to the norm of E's . So, the maoping
U --+I U In U E E~,
121
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
is CJ(E'n, Ecsi)-Iower semicontinuous. For each u in E', we put
00
1 u 12:= :L 1 u 1; .
r==l
We take u0 E E'n and a E IR with a <I u0 12 . We find s E lN such that
• L 1uo12 > a.
r::::::l
The mapping
s
u---+ L 1u1;, u E E'n,
r=l
is CJ(E'n, Ec8 i)-Iower semicontinuous and so there is a neighbourhood U of u0 for the topology
C!(E'n, Ec8 >) such that
s L 1 u ¡; > a, u E U.
r=l
Consequently,
1 u 12 > a, u E U,
and so the mapping
is CJ(E'n , Ecsi)-Iower semicontinuous and thus
is CJ(E'n, Ec8 >)-closed. Since D is contained in the CJ(E'n, Ec8 >)-compact set 2A + 2-1 B, it follows
that D is a compact absolutely convex subset of E' [CJ(E', E)]. If u E A, then 2ru E Wr and
thus 1 u Ir ::; 2r. Consequently,
00 1 u 12 ::; :L r2r < i,
r=l
and A e D. We see next that E'o is quasi-reflexive. In (E'n)**, !et F and G be the linear hulls
of E'n U A and E'n U D, respective[ y. For each r E lN, D is contained in 2r A + 2-r E and so
l5 e 2r A+ rrB.
We deduce from here that
and, consequently,
11 V - V¡ 11 = 11 V2 11 ::; rr,
122
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
so the distance from v to G is zero. Hence F = G. If J is the canonical injection from E'v
into EÉJ, the second conjugate J** of J is one-to-one, [1], and it turns the closed unit hall of
(E'v)** into D. We deduce from here that E'v has finite codimension in (E'v)**, that is, E'v is
quasi-reflexive.
In E' [<7(E', E)], we take now a fundamental system of compact absolutely convex
subsets
C1 e C2 e ... e Cm e ...
We find, for each m E IN, a compact absolutely convex subset Dm such that Cm e Dm and
E'vm is quasi-reflexive. We now find integers
1 = n 1 < n2 < .. . < n, < ...
such that Dn, C Dn>+" s E IN. If U, is the polar set of Dn, in E, then
is a fundamental system of zero neighbourhoods in E with Eú~ quasi-reflexive, m E IN.
It is now easy to show that E" [/3{E", E'))/ E is separable. Hence, E is the direct
topological sum of two subspaces E1 and E2 such that E1 is refiexive and E~ [/3(E~, EDJ is
separable, [9). Consequently, E is distinguished and so E' [f3(E', E)] is ultrabornological. It
follows now that E' [/3{E',E)) is the inductive limit of the sequence of Banach spaces Eh,
m E IN.
q.e.d.
References
[1) DAVIS, W.J., FIGIEL, T., JOHNSON, W.B., PELCZYNSKI, A.: Factoring weakly com-pact
operators, J. Funct. Anal. 17, 256-280 {1955).
(2) JAMES, R. C.: Bases and reflexivity of Banach spaces, Ann. Math. 52, 518-527 {1950).
[3) JARCHOW, H.: Locally Convex Spaces. Stuttgart: Teubner 1981.
[4) JOHNSON, W.B., ROSENTHAL, H.P.: On w*-basic sequences and their applications to
the study of Banach spaces, Studia Math. 43, 77-92 {1972).
[5) KÓTHE, G.: Topological Vector Spaces l. Berlin-Heidelberg-New York: Springer 1969.
[6) VALDIVIA, M.: Basic sequences in the dual of a Fréchet space, Math. Nachr. {To appear)
123
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
[7] VALDIVIA, M.: A Characterization of Totally Reflexive Fréchet Spaces, Math. Z. 200,
327-346 (1989).
[8) VALDIVIA, M.: Br-complete spaces which are not E-complete, Math. Z. 185, 253-259
(1984).
[9) VALDIVIA, M.: Descomposiciones de espacios de Fréchet en ciertas sumas topológicas
directas, (Artículo en homenaje al Profesor Pablo Bobillo Guerrero). Universidad de
Granada, 1992.
Author's address:
M. Valdivia
Departamento de Análisis Matemático
Universidad de Valencia
Dr. Moliner, 50
46100 Burjasot (Valencia)
Spain
124