GariVeraAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 7, No. 2, 2006

pp. 165-170

Extension of Compact Operators from
DF-spaces to C(K) spaces

Fernando Garibay Bonales and Rigoberto Vera Mendoza

Abstract. It is proved that every compact operator from a DF-
space, closed subspace of another DF-space, into the space C(K) of
continuous functions on a compact Hausdorff space K can be extended
to a compact operator of the total DF-space.

2000 AMS Classification: Primary 46A04, 46A20; Secondary 46B25.

Keywords: Topological vector spaces, DF-spaces and C(K) the spaces.

1. Introduction

Let E and X be topological vector spaces with E a closed subspace of X.
We are interested in finding out when a continuous operator T : E → C(K)

has an extension T̃ : X → C(K), where C(K) is the space of continuous real
functions on a compact Hausdorff space K and C(K) has the norm of the
supremum. When this is the case we will say that (E,X) has the extension
property. Several advances have been made in this direction, a basic resume
and bibliography for this problem can be found in [5]. In this work we will
focus in the case when the operator T is a compact operator. In [4], p.23 , it
is proved that (E,X) has the extension property when E and X are Banach
spaces and T : E → C(K) is a compact operator. In this paper we extend this
result to the case when E and X are DF-spaces (to be defined below), for this,
we use basic tools from topological vector spaces.

2. Notation and Basic Results in DF-spaces.

We will use basic duality theory of topological vector spaces. For concepts
in topological vector spaces see [3] or [2]. All the topological vector spaces in
this work are Hausdorff and locally convex.

Let (X,t) be a topological vector space and E < X be a closed vector
subspace. Let X′ = (X,t)′, E′ = (E,t)′ be the topological duals of X and E
respectively.



166 F. Garibay and R. Vera

A topological vector space (X, t) possesses a a fundamental sequence of
bounded sets if there exists a sequence B1 ⊂ B2 ⊂ · · · of bounded sets in
(X, t), such that every bounded set B is contained in some Bk.

We take the following definition from [3], p. 396.

Definition 2.1. A locally convex topological vector space (X,t) is said to be a
DF-space if

(1) it has a fundamental sequence of bounded sets, and
(2) every strongly bounded subset M of X′ which is the union of countably

many equicontinuous sets is also equicontinuous

A quasi-barrelled locally convex topological vector space with a fundamental
sequence of bounded set is always a DF-space. Thus every normed space is a
DF-space. Later we will mention topological vector spaces which are DF-spaces
but they are not normed spaces.

First, we state some theorems to be used in the proof of the main result.

If K is a compact Hausdorff topological space, we define, for each k ∈ K

the injective evaluation map k̂ : C(K) → R , k̂(f) = f(k) which is linear and

continuous, that is k̂ ∈ C(K)′. Let K̂ = {k̂ |k ∈ K} ⊂ C(K)′ and cch(K̂) the

balanced, closed and convex hull of K̂ (which is bounded).

Theorem 2.2. With the notation above we have

(1) K̂ is σ(C(K)′,C(K))-compact and K is homeomorphic to

(K̂,σ(C(K)′,C(K)) . Here σ(C(K)′,C(K)) denotes the weak-* topol-
ogy on C(K)′.

(2) If T : E → C(K) is a compact operator then A = T ′(cch(K̂))
β

is
β(E′,E)-compact. Here β(E′,E) is the strong topology on E′, this
topology is generated by the polars sets of all bounded sets of (E, t).

Proof. See [1], p. 490. �

Theorem 2.3. If (X, t) is a DF-space then (X′, β(X′,X) ) is a Frechet
space.

Proof. See [3], p. 397 �

Theorem 2.4. Let M be paracompact, Z a Banach space, N ⊂ Z convex
and closed, and ϕ : M → F(N) lower semicontinuous (l.s.c.) Then ϕ has a
selection.

Proof. See [6] �

In the above theorem, F(N) = {S ⊂ N : S 6= ∅, S closed in N and convex};
ϕ : M → F(N) is l.s.c. if {m ∈ M : ϕ(m)∩V 6= ∅} is open in M for every open
V in N, and f : M → N is a selection for ϕ if f is continuous and f(m) ∈ ϕ(m)
for every m ∈ M.

Theorem above remains true if Z is only a complete, metrizable, locally
convex topological vector space (see [7]).



Extension of Compact Operators 167

3. Main Results

Lemma 3.1. Let A ⊂ E′. If there is a continuous map

f : (A,σ(E′,E)) → (X′,τ′), σ(X′,X) ≤ τ′ ≤ β(X′,X)

such that

(1) f(a)|E = a and
(2) f(A) is an equicontinuous subset of X′.

Then every linear and continuous map T : E → C(K) has a linear and con-

tinuous extension T̃ : X → C(K).

Proof. Let us define T̃ : X → C(K) in the following way: for each x ∈ X ,

T̃(x) : K → R is given by T̃(x)(k) = f(T ′(k̂))(x). Here, k̂ is the injective

evaluation map defined before Theorem 2.2. It is easy to check that T̃ is
linear and extends T .

First, let us show that T̃(x) ∈ C(K) for each x ∈ X. For this let O ⊂

R be an open set. We have that T̃(x)−1(O) = T ′−1(f−1(x−1(O) ) ). Since
x : X′[σ(X′,X)] → R , f and T ′ are all continuous maps with the weak∗

topology, T̃(x)−1(O) is open in K. This proves that T̃(x) ∈ C(K).

Let us check that T̃ is continuous. Let {xλ}Λ
t
→ 0 in X, we need to show

that {T̃(xλ)}
||·||C(K)
−→ 0 .

For this, let ǫ > 0. By hypothesis f(A) is a equicontinuous subset of X′, so
that, ǫf(A)◦ ⊂ X is a t-neighborhood of 0. Here f(A)◦ denotes de polar set
of f(A). Hence, there is λ0 ∈ Λ such that xλ ∈ ǫf(A)

◦ for all λ ≥ λ0. From

part 2 of Theorem 2.2 we have T ′(K̂) ⊂ A, hence

|T̃(xλ)(k̂)| = |f(T
′(k̂))(xλ)| ≤ ǫ for all λ ≥ λ0

This implies that

||T̃(xλ)||C(K) = sup{ |f(T
′(k̂))(xλ)|/k ∈ K} ≤ ǫ for all λ ≥ λ0

This proves that {T̃(xλ)}
||·||C(K)
−→ 0 . �

Let i : E → X be the inclusion map and i′ : X′ → E′ the dual map of i,
that is, if y ∈ X′, i′(y) = y|E .

Let P(X′) = {Y | Y 6= ∅, Y ⊂ X′} and define ψ : E′ → P(X′) by ψ(e′) =
{ extensions of e′ to X}. Notice that y ∈ ψ(i′(y)) for all y ∈ X′ and ψ(e′) ∈
F(X′).

With this notation, we have

Proposition 3.2. Let (E, t) and (X, t) be DF-spaces, with E < X a closed
subspace. If O ⊂ X′ is a β(X′, X)-open set then the set UO = {z ∈ E

′ |ψ(z) ∩
O 6= ∅} is an open set in (E′, β(E′, E) ).

Proof. Notice that UO = {z ∈ E
′ | ψ(z) ∩ O 6= ∅} = i′(O). By Theorem 2.3

(X′,β(X′, X) ) and (E′,β(E′, E) ) are Frechet spaces. By the Banach-Schauder
theorem (see [3], p. 166), the map i′ : (X′, β(X′, X) ) → (E′, β(E′, E) ) is an
open map. Since i′(O) is open in E′, UO is also open. �



168 F. Garibay and R. Vera

Corollary 3.3. Let (E, t) and (X, t) be DF-spaces, with E < X a closed

subspace. Let A = T ′(cch(K̂))
β

be as in part 2 of Theorem (2.2) Then ϕ :
(A,β(E′,E)) → P(X′) given by ϕ = ψ|A is a lower semicontinuous function,
X′ provided with the strong topology β(X′, X).

Proof. It follows from

{z ∈ A | ϕ(z) ∩ O 6= ∅} = {z ∈ E′ | ψ(z) ∩ O 6= ∅} ∩ A

and Proposition 3.2. �

With the notation in Corollary 3.3, we have

Proposition 3.4. If (X,t) is a DF-space then ϕ : (A,β(E′,E)) → P(X′)
admits a selection, that is, there is a continuous function f : (A,β(E′,E)) →
(X′,β(X′,X)) such that f(a) ∈ ϕ(a).

Proof. From Theorem 2.3, (X,t) DF-space implies (X′,β(X′,X)) Frechet.
From Theorem 2.2, part 2, A is β(E′,E)-compact, hence A is a paracompact
set. By Corollary 3.3, ϕ is a lower semi continuous function, therefore, by
Theorem 2.4, ϕ admits a selection. �

Theorem 3.5. If (X,t) and the closed subspace E are DF-spaces then every

compact operator T : E → C(K) has a compact extension T̃ : X → C(K).

Proof. Let A be as in Proposition 3.4 and f : (A,β(E′,E)) → (X′,β(X′,X))
a selection function. Since A is β(E′,E)-compact and f is continuous, f(A) is

compact, hence f(A) is an equicontinuous set. Let T̃ be the linear extension
of T given in Lemma 3.1.

Let us prove that T̃ is a compact operator. For this, we need to show that
there is a t-neighborhood V such that T̃(V ) is a relatively compact set.

Since f(A) ⊂ X′ is an equicontinuous set and X is a DF space, [2] (p.
260 and p. 214) tells us that there is V ⊂ X a balanced, closed and convex
t-zero-neighborhood such that f(A) ⊂ V ◦ and the topologies β(X′, X) and
ρV ◦ coincide on f(A). Here ρV ◦ is the Minkowski functional of V

◦. In this
case ρV ◦ is a norm and (X

′
V ◦, ρV ◦ ) is a Banach space.

By using the Arzela-Ascoli Theorem, we will show that T̃(V ) ⊂ C(K) is
relatively compact.

First, T̃ (V ) is pointwise bounded because, for each x ∈ V and k ∈ K ,

|T̃(x)(k)| = |f(T ′(k̂))(x)| ≤ 1 since f(A) ⊂ V ◦.

Now let us prove that T̃(V ) is equicontinuous in C(K).
Choose and fix k0 ∈ K and ǫ > 0. Since the chain of functions

K−̂→K̂
T

′

−→ (A,β(X′,X))
f

−→ (f(A),β(X′,X))

is continuous, given a β-neighborhood W of f(T ′(k̂0)) on f(A) , there exists

O ⊂ K neighborhood of k0 such that k ∈ O ⇒ f(T
′(k̂)) ∈ W . Since

ρV ◦|f (A) = β(X
′, X)|f (A), we can say that

k ∈ O ⇒ ρV ◦
(

f(T ′(k̂)) − f(T ′(k̂0))
)

< ǫ



Extension of Compact Operators 169

For each x ∈ X , x : (X′V ◦, ρV ◦ ) → R is linear and continuous, moreover,
|x′(x)| ≤ ||x||ρV ◦ ρV ◦ (x

′) for all x′ ∈ X′ , where

||x||ρV ◦ = sup{|x
′(x)| | x′ ∈ V ◦}

If x ∈ V , ||x||ρV ◦ ≤ 1. Therefore, for every k ∈ O and every x ∈ V
∣

∣

∣
(f(T ′(k̂)) − f(T ′(k̂0))(x)

∣

∣

∣
≤ ||x||ρV ◦ ρV ◦

(

f(T ′(k̂)) − f(T ′(k̂0))
)

≤ (1)(ǫ)

This proves that T̃(V ) is equicontinuous in C(K) and, by the Arzela-Ascoli

Theorem, T̃(V ) is relatively compact which means that T̃ is a compact oper-
ator. �

In [3] (p. 402) it is shown that the topological inductive limit of a sequence
of DF-spaces is a DF-space. In particular, if (En) is a sequence of Banach
spaces such that En is a proper subspace of En+1, its inductive limit is DF-
space. This inductive limit is not metrizable (see [8] p. 291). For this kind of
spaces, Theorem 3.5 can be applied, i.e., given a fixed n, a compact operator
T : En → C(K) can be extended to a compact operator of the inductive limit.

Acknowledgements. The research of the authors was supported by the
Coordinación de la Investigación Cient́ıfica de la UMSNH.

References

[1] N. Dunford and J. Schwartz, Linear Operators, vol. I., Wiley Interscience, New York,
1957.

[2] H. Jarchow, Locally Convex Spaces, B. G. Teubener Stuttgart, 1981.
[3] G. Kothe, Topological Vector Spaces, vol. I., Springer Verlag, New York, 1969.
[4] W. B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces.

Preprint. ( 2001).
[5] W. B. Johnson and M. Zippin, Extension of operators from weak*-closed subspaces of

l1 to C(K), Studia Mathematica 117 (1) (1995), 43–45.
[6] E. Michael, Continuous Selections I, Annals of Mathematics 63 (2) (1956), 361–382.
[7] E. Michael, Some Problems, Open Problems in Topology, Elsevier, Amsterdam, 1990.
[8] L. Narici and E. Beckenstein, Topological Vector Spaces, Marcel Dekker, Inc., New York,

1985.

Received November 2004

Accepted June 2005



170 F. Garibay and R. Vera

F. Garibay Bonales (fgaribay@zeus.umich.mx)
Facultad de Ingenieŕıa Qúımica, Universidad Michoacana de San Nicolás de
Hidalgo, Morelia, Michoacán 58060, México.

R. Vera Mendoza (rvera@zeus.umich.mx)
Facultad de F́ısico-Matemáticas, Universidad Michoacana de San Nicolás de
Hidalgo, Morelia, Michoacán 58060, México.