�
EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 36, Num. 2 (2021), 147 – 155

doi:10.17398/2605-5686.36.2.147

Available online June 17, 2021

Rosenthal `∞-theorem revisited

L. Drewnowski ∗

Faculty of Mathematics and Computer Science

Adam Mickiewicz University

Uniw. Poznańskiego 4, 61–614 Poznań, Poland

drewlech@amu.edu.pl

Received April 27, 2021 Presented by M. González
Accepted May 20, 2021

Abstract: A remarkable Rosenthal `∞-theorem is extended to operators T : `∞(Γ,E) → F , where
Γ is an infinite set, E a locally bounded (for instance, normed or p-normed) space, and F any

topological vector space.

Key words: Vector-valued `∞-function space, locally bounded space, p-normed space, quasi-normed
space, isomorphism, Rosenthal `∞-theorem.

MSC (2020): 46A16.

1. Introduction and the standard Rosenthal `∞-theorem

I begin by quoting my antique result from 1975 which I dare to call Stan-
dard Rosenthal `∞-Theorem. It was first placed in [4] and, after a short time,
with a much refined proof, inspired very essentially by ideas from a work of J.
Kupka [11](1974), in [3] (the order of appearance happened to be just the re-
verse). It was quite a surprising common extension of the original `∞(Γ) and
c0(Γ) results of Haskell P. Rosenthal [18, Proposition 1.2; Theorem 3.4 and
Remark 1 after it] (announced in [17]) proved for operators acting from these
spaces to any Banach space F , and of similar in spirit results of N.J. Kalton
[8, Theorem 3.2;3.3, 4.3; Theorem 2.3] for Γ = N and any tvs F. Consult
[4, a comment after (R:N) on p. 209] for more precise information. Also see
N.J. Kalton [9]. In the `∞-case Rosenthal imposed a much stronger condition
on T than (R-0), namely, that “T|c0(Γ) is an isomorphism”, while in the c0-
case his condition was precisely (R-0). I now return to that old work of mine
with further extensions. The previous variants of Rosenthal’s theorem have
found significant applications to Banach spaces [18, 1, 14] and to more general
spaces ([6, Theorem 1.2], [7, p. 314 and ff.], [5]), and to vector measures ([12],

∗To my wife Krystyna and our daughters Monika and Karolina

ISSN: 0213-8743 (print), 2605-5686 (online)

© The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0)

https://doi.org/10.17398/2605-5686.36.2.147
mailto:drewlech@amu.edu.pl
https://publicaciones.unex.es/index.php/EM
https://creativecommons.org/licenses/by-nc/3.0/


148 l. drewnowski

[13], [2, Lemma 4.1.37, Lemma 4.1.39, Lemma 4.1.41], [15, Proposition 3]).
It is therefore natural to expect that the new, more general variants will
have an even wider range of applicability. A feature that all the Rosenthal-
type results encountered in this paper share is that they concern operators
T : X → F which send ‘primitive’ vectors in X that are away from zero to
vectors that are away from zero in F. In an unbelievable way ‘primitive’ can
be removed (though, in general, there is no good functional analytic road in
X from “primitive” vectors to those arbitrary ones), leading to the conclusion
that the restriction of T to a large subspace of X must be an isomorphism.
This means that these results are of a fundamentally basic nature.

Theorem 1.1. (Standard Rosenthal `∞-Theorem) Let F be a tvs
and X be a subspace of `∞(Γ) such that eγ ∈ X for all γ ∈ Γ. Also, let
T : X → F be an operator such that

(R-0) for some 0-neighborhood U in F and all γ ∈ Γ: T(eγ) /∈ U.

Then Γ has a subset A of the same cardinality m as Γ for which the restriction
T|X(A) is an isomorphism.

1.1. Notation, conventions, terminology, some basic facts. To
ensure proper understanding of was said above, and of the new material, I
clarify here a few points. The acronym tvs stands for (nonzero and Hausdorff)
topological vector space over the field of scalars K ∈ {R,C}, and operator
always means continuous linear operator.

For a tvs E and an infinite set Γ, `∞(Γ,E) is the space of bounded
functions x : Γ → E with the topology of uniform convergence on Γ. Thus for
any 0-neighborhood B in E the corresponding 0-neighborhood in `∞(Γ,E)
is B• := {x ∈ `∞(Γ,E) : x(Γ) ⊂ B}. Clearly, if E = (E,‖·‖) is an F-
normed space, then this topology is defined by the F-norm ‖·‖∞ given by
‖x‖∞ = supγ∈Γ ‖x(γ)‖. The notation Bc and B•c for the complement of B in
E, and of B• in `∞(Γ,E), resp., will also be used in the sequel. c0(Γ,E) is the
closed subspace of `∞(Γ,E) consisting of those x that have limit 0 along the
filter of cofinite subsets of Γ. If x,y ∈ `∞(Γ,E) and supp(x) ∩ supp(y) = ∅,
then x,y are said to be disjoint, and this is indicated by writing x ⊥ y. Notice
that whenever x,y ∈ B• and x ⊥ y, also x + y ∈ B•. This property will be
essential in what follows. If X is a set of functions x : Γ → E and A ⊂ Γ, then
X(A) := {x ∈ X : supp(x) ⊂ A}; 1A stands for the characteristic function
of A. If E = (K, | · |), then the notation for the spaces is simplified to `∞(Γ)
and c0(Γ). For γ ∈ Γ, eγ = 1{γ} are the standard unit vectors in `∞(Γ). In



rosenthal `∞-theorem revisited 149

the general case, their role will be taken over by the primitive vectors xeγ
(x ∈ E, γ ∈ Γ). No completeness or convexity assumptions will be imposed;
they are simply not needed in this paper. Finishing, we recall that by a well
known Aoki-Rolewicz theorem (see [16, Theorem 3.2.1]), the topology of any
locally bounded space E may be defined by some p-norm ‖·‖ (0 < p 6 1).
(‖ax‖ = |a|p‖x‖ for all x ∈ E,a ∈ K). See also [10, Chapter I.2] for a thorough
discussion of this and other types of normlike functionals (e.g., quasi-norms)
that are often used to define the topology of a locally bounded space, and
for the relevant terminology concerning spaces used in such contexts. The
author’s preference are p-norms. In principle, however, it is a matter of taste
and/or convenience (or, sometimes, necessity), whether one works with locally
bounded spaces or with, for instance, p-normed spaces. A more general type
of vector-valued `∞-spaces will appear in Theorem 2.4. Additional notation
of temporary character will be introduced when needed.

2. New extensions of Rosenthal `∞-theorem

As the main result (also in view of the versatility of its proof) of the paper
I consider the following.

Theorem 2.1. Let Γ be an infinite set, E a locally bounded tvs, F any
tvs, and X a subspace of `∞(Γ,E) such that for all (γ,x) ∈ Γ×E: xeγ ∈ X.
Also, let T : X → F be an operator and assume that

(R-1) for some bounded 0-neighborhood B in E there is a 0-neighborhood
U in F such that for all (γ,x) ∈ Γ ×Bc : T(xeγ) /∈ U.

Then there exists a subset A of Γ of cardinality m = |Γ| such that T|X(A) is
an isomorphism.

Proof. The reasoning is an adaptation, with appropriate modifications and
a very cautious treatment of the most ‘delicate’ points, of the proof of Theorem
1.1 given in [3].

Step 1: Denote Z := X ∩B• and notice the following:

(i) T(Z) is a bounded subset of F.

(ii) ∀(γ,x) ∈ Γ ×Bc : xeγ ∈ X(Γ) ∩B•c, hence T(xeγ) /∈ U.
(iii) If x,y ∈ Z and x ⊥ y, then x + y ∈ Z.
(iv) If x ∈ Z,γ ∈ Γ and xγ := x(γ), then x−xγeγ ∈ Z.



150 l. drewnowski

Step 2: Next, fix a base U of balanced neighborhoods of zero in F , choose
V ∈ U with V + V ⊂ U, then r ∈ N such that T(Z) ⊂ rV , and finally W ∈ U

for which W (r) := W+
r)
· · · +W ⊂ rV . The assertion of the theorem will be

reached once we show that:

(A1) There exists a set A ⊂ Γ such that |A| = |Γ| = m and Tx 6∈ W for
all x ∈ X(A) ∩ B•c. Beware: The validity of the statement concerning the
role of (A1) crucially depends on the local boundedness of X(A) implied by
the assumption that E has this property.

Before proceeding, observe that (A1) is equivalent to the following:

(A2) There exists a set ∆ ⊂ Γ with |∆| = m such that for any ∆′ ⊂ ∆ with
|∆′| = m one can find (α,s) ∈ ∆′×Bc with the property: ∀x ∈ X(∆r∆′)∩B•:
T(x + seα) /∈ W .

(A1)⇒(A2): Simply take ∆ = A.
(A2)⇒(A1): Since m · m = m, one can partition ∆ into m (disjoint) sets

Ci (i ∈ I, |I| = m), each of cardinality m. According to (A2), for each i ∈ I
one can find (γi,s) ∈ Ci×Bc such that T(x+seγi ) /∈ W for all x ∈ Z(∆rCi).
Then A := {γi : i ∈ I} is as required in (A1).

Now we are ready to show that (A1) or, equivalently, (A2) does indeed
hold. Suppose (A2) is false so that:

non(A2) For every set ∆ ⊂ Γ with |∆| = m there is ∆′ ⊂ ∆ with
|∆′| = m and such that ∀(α,s) ∈ ∆′ ×Bc: one can find x ∈ Z(∆ r ∆′)
with the property: T(x + seα) ∈ W .

Then, applying non(A2) r times, one arrives at a decreasing sequence of sets
Γ = ∆0 ⊃ ∆1 ⊃ . . . ⊃ ∆r such that, for every k ∈ [r] = {1, . . . ,r}, |∆k| = m,
and for each α ∈ ∆k and s ∈ Bc there exists x ∈ Z(∆k−1 r ∆k) for which
T(x+seα) = Tx+T(seα) ∈ W . Now, fix α ∈ ∆r and s ∈ Bc, and use them in
each of the r steps above. This will give for each k ∈ [r]: xk ∈ Z(∆k−1 r ∆k)
so that T(xk + seα) = Txk + T(seα) ∈ W . Summation over k ∈ [r] leads to
the following:

rT(seα) + T

( r∑
k=1

xk

)
∈ W (r) ⊂ rV.

Since the xk’s are pairwise disjoint, x :=
∑r

k=1 xk ∈ Z, hence Tx ∈ rV . It
follows that rT(seα) ∈ rV + rV ⊂ rU. Hence T(seα) ∈ U, contradicting
condition (R-1).



rosenthal `∞-theorem revisited 151

Remarks 2.2. (i) In condition (R-1) any other bounded neighborhood
B′ of zero in E can be used as well. In fact, if U is ‘good’ for B, then since
B ⊂ kB′ for some k ∈ N, k−1U is good for B′.

(ii) The most important examples of subspaces X are `∞(Γ) and c0(Γ) and,
of somewhat lesser importance, their further subspaces consisting of elements
x with either |supp(x)| 6 n or |supp(x)| < n, where n is any infinite cardinal
number 6 m. Note that for these X, if A ⊂ Γ and |A| = |Γ|, then X(A) is
isomorphic to X.

(iii) The theorem admits a seemingly stronger form, with condition (R-1)
requiring that for a bounded 0-neighborhood B in E there is a 0-neighborhood
U in E such that the set Γ′ := {γ ∈ Γ : ∀x ∈ Bc : T(xeγ) /∈ U} is infinite,
and the assertion saying that then there exists a subset A of Γ′ of the same
cardinality as Γ′ such that T|X(A) is an isomorphism. In reality, as easily
understood, both versions are of the same strength. A similar remark applies
to Theorem 2.3 below.

It is obvious that Theorem 2.1 can be equivalently restated for E being a
p-normed space (E,‖·‖) (0 < p 6 1), using the ball B = {x ∈ E : ‖x‖ 6 ε} for
some ε > 0 in condition (R-1). We won’t do that here. Yet two other variants
of Theorem 2.1, formulated in the framework of p-normed spaces are worth
explicit stating. In the first (and in its proof), close in spirit to the standard
Rosenthal Theorem we will use the following notation. B := {x ∈ E : ‖x‖ 6
1}, S := {x ∈ E : ‖x‖ = 1}, ‖·‖∞ for the associated p-norm in `∞(Γ,E),
B∞ for the closed unit ball, and S∞ for the unit sphere in `∞(Γ,E), resp.
Additionally, we let Ŝ∞ := {x ∈ S∞ : x(γ) ∈ S for some γ ∈ Γ}.

Theorem 2.3. Let E = (E,‖·‖) be a p-normed space (0 < p 6 1), F a
tvs, and Γ an infinite set. Further, let X be a subspace of the p-normed space
`∞(Γ,E) such that ∀(γ,x) ∈ Γ × E: xeγ ∈ X. Also, let T : X → F be an
operator such that

(R-2) For some 0-neighborhood U in F, T(seγ) /∈ U for all (γ,s) ∈ Γ ×S.

Then Γ has a subset A of the same cardinality m as Γ for which T|X(A) is an
isomorphism.

Proof. The result follows from Theorem 2.1. This will be achieved by show-
ing that the present condition (R-2) implies condition (R-1) in that theorem.
Fix ε > 0, and take any x ∈ E with λ = ‖x‖ > 1 + ε. Next, let y = x/λ1/p.
Then for any γ ∈ Γ: T(yeγ) /∈ U. Hence T(xeγ) /∈ λ1/pU ⊃ (1 + ε)1/pU =: U ′.



152 l. drewnowski

Thus (R-1) in Theorem 2.1 holds for the bounded 0-neighborhood (1 + ε)B∞
in E and the 0-neighborhood U ′ in F. To complete the story, let us yet see
that also Theorem 2.3 implies Theorem 2.1: Assume that in (R-1) of the latter
theorem we want to use 1

2
B in place of B (cf. Remark 2.2(1)). So take any

γ ∈ Γ and x ∈ E with λ = ‖x‖ > 1
2
. Set y = x/λ1/p. Then ‖y‖ = 1, so

that T(yeγ) /∈ U (U as in (R-2)). Hence T(xeγ) /∈ λ1/pU ⊃ 12p U =: U
′. Thus

T(xeγ) /∈ U ′ and (R-2) is satisfied with U ′ in place of U.

It may be of some interest to have an independent and direct proof of the
last result. Here it goes.

Another proof. Proceed precisely as in the proof of Theorem 2.1, replacing
B• and B•c by B∞ and Ŝ∞, keeping Steps 1 and 2 unchanged, and then
replacing conditions (A1), (A2) by the following two, resp.

(B1) There exists a set A ⊂ Γ such that |A| = |Γ| = m and Tx 6∈ W for
all x ∈ Z(A) ∩ Ŝ∞.

(B2) There exists a set ∆ ⊂ Γ with |∆| = m such that for any ∆′ ⊂ ∆
with ∆′| = m one can find (α,s) ∈ ∆′×S with the property: ∀x ∈ Z(∆ r ∆′):
T(x + seα) /∈ W , verifying that they are equivalent, and after that showing
that under the assumptions of the theorem (B1) does indeed hold. Having
done that select W ′ ∈ U with W ′ + W ′ ⊂ W , take the smallest m ∈ N
with m > 1/p, and next select 0 < ε < 1 so that for η := mpεp there holds
T((ηB∞)∩X) ⊂ W ′ (by the continuity of T), and then proceed to verify that

(B̂1) For the set A from (B1) one has: Tx 6∈ W ′ for all x ∈ Z(A) ∩S∞.
So, take any x ∈ Z(A)∩S∞, and then choose α ∈ A so that writing xα = x(α)
and λ = ‖xα‖, λ > 1−ε. Then let yα = xα/λ1/p, and define y := x−xαeα +
yαeα. Clearly, y ∈ Z(A) ∩ Ŝ∞ and hence Ty /∈ W . Since z := x − y =
(xα −yα)eα ∈ X(A),

‖z‖∞ = ‖xα −yα‖ = (1 −λ1/p)p 6 (1 −λm)p

= [(1 −λ)(1 + λ + · · · + λm−1)]p

< mpεp = η.

It follows that Tz ∈ W ′. Since y = x − z and Ty /∈ W while Tz ∈ W ′, we
conclude that Tx /∈ W ′. To finish, take any x ∈ X(A) with λ := ‖x‖∞ > 1.
Then y := λ−1/px ∈ S∞. By (B̂1), Ty = λ−1/pTx /∈ W ′. Hence Tx /∈
λ1/pW ′, and Tx /∈ W ′, too. From this the assertion follows.



rosenthal `∞-theorem revisited 153

The second promised variant, suggested by the referee, deals with “multi-
space” type `∞-spaces.

Theorem 2.4. Let Γ be an infinite set and, for a fixed p ∈ (0, 1], let
EΓ = (Eγ : γ ∈ Γ) be a family of p-normed spaces, each Eγ with its own
p-norm ‖·‖γ. Consider the vector subspace `∞(EΓ) of the product

∏
γ∈Γ Eγ

consisting of elements x = (xγ : γ ∈ Γ) (with xγ written often as x(γ)) such
that ‖x‖∞ := supγ ‖xγ‖γ < ∞, and equip it with the p-norm ‖·‖∞ defined by
this equality. Next, let X be a subspace of `∞(EΓ) such that xeγ ∈ X for all
γ ∈ Γ and all x ∈ Eγ. Further, let F be any tvs, T : X → F be an operator,
and assume that

(R-3) for some ε > 0 there is a 0-neighborhood U in F such that T(xeγ) /∈ U
for all γ ∈ Γ and all x ∈ Eγ with ‖x‖γ > ε.

Then there exists a subset A of Γ of cardinality m = |Γ| such that T|X(A) is
an isomorphism.

Proof. Again, we argue as in the proof of Theorem 2.1, with necessary
changes. To avoid any ambiguity, all details are given. We first fix the addi-
tional notation to be used in the present proof. Thus, we denote B(Eγ) =:
{x ∈ Eγ : ‖x‖γ 6 ε}, Bc(Eγ) =: {x ∈ Eγ : ‖x‖γ‖ > ε}, B• =: {x ∈ `∞(EΓ) :
‖x‖∞ 6 ε}, B•c =: {x ∈ `∞(EΓ) : ‖x‖∞ > ε} . Next, keep Steps 1 and 2
unchanged, and replace conditions (A1), (A2) by the following two, resp.

(C1) There exists a set A ⊂ Γ such that |A| = |Γ| = m and Tx 6∈ W for
all x ∈ X(A) ∩B•c.

(C2) There exists a set ∆ ⊂ Γ with |∆| = m such that for any ∆′ ⊂ ∆
with |∆′| = m one can find (α,s) ∈ ∆′ × Bc(Eα) with the property: ∀x ∈
X(∆ r ∆′) ∩B•: T(x + seα) /∈ W .

Then we show that (C1) ⇐⇒ (C2).
(C1)⇒(C2): Simply take ∆ = A.
(C2)⇒(C1): Since m·m = m, one can partition ∆ into m (disjoint) sets Ci

(i ∈ I, |I| = m), each of cardinality m. According to (C2), for each i ∈ I one
can find (γi,s) ∈ Ci×Bc(Eγi ) such that T(x+seγi ) /∈ W for all x ∈ Z(∆rCi).
Then A := {γi : i ∈ I} is as required in (C1).

Then we go on to show that (C1) or, equivalently, (C2) does indeed hold.
Suppose (C2) is false so that:



154 l. drewnowski

non(C2) For every set ∆ ⊂ Γ with |∆| = m there is ∆′ ⊂ ∆ with |∆′| =
m and such that ∀(α,s) ∈ ∆′ × Bc(Eα): one can find x ∈ Z(∆ r ∆′)
with the property: T(x + seα) ∈ W .

Then, applying non(C2) r times, one arrives at a decreasing sequence of sets
Γ = ∆0 ⊃ ∆1 ⊃ . . . ⊃ ∆r such that, for every k ∈ [r] = {1, . . . ,r}, |∆k| = m,
and for each α ∈ ∆k and s ∈ Bc(Eα) there exists x ∈ Z(∆k−1 r ∆k) for
which T(x + seα) = Tx + T(seα) ∈ W . Now, fix α ∈ ∆r and s ∈ Bc(Eα),
and use them in each of the r steps above. This will give for each k ∈ [r]:
xk ∈ Z(∆k−1 r ∆k) so that T(xk + seα) = Txk + T(seα) ∈ W . Summation
over k ∈ [r] leads to the following:

rT(seα) + T

( r∑
k=1

xk

)
∈ W (r) ⊂ rV.

Since the xk’s are pairwise disjoint, x :=
∑r

k=1 xk ∈ Z, hence Tx ∈ rV . It
follows that rT(seα) ∈ rV + rV ⊂ rU. Hence T(seα) ∈ U, contradicting
condition (R-3).

Acknowledgements

I am grateful to Iwo Labuda, Zbigniew Lipecki, and Marek Nawrocki
for their interest in this work and numerous helpful suggestions. I
also thank the editors of Extracta Mathematicae, especially Jesús M.F.
Castillo, and the referee, for their criticism and engagement in improving
the look of this publication.

References

[1] S.A. Argyros, J.M.F. Castillo, A.S. Granero, M. Jimenez-
Sevilla, J.P. Moreno, Complementation and embeddings of c0(I) in
Banach spaces, Proc. London Math. Soc. 85 (2002), 742 – 768.

[2] C. Constantinescu, “ Spaces of Measures ”, De Gruyter Studies in Mathe-
matics 4, Walter de Gruyter & Co., Berlin, 1984.

[3] L. Drewnowski, Un théorème sur les opérateurs de `∞(Γ), C.R. Acad. Sci.
Paris Sér. A 281 (1975), 967 – 969.

[4] L. Drewnowski, An extension of a theorem of Rosenthal on operators acting
from `∞(Γ), Studia Math. 57 (1976), 209 – 215.

[5] L. Drewnowski, Generalized limited sets with applications to spaces of type
`∞(X) and c0(X), European J. Math. (to appear).

[6] L. Drewnowski, I. Labuda, Copies of c0 and `∞ in topological Riesz
spaces, Trans. Amer. Math. Soc. 350 (1998), 3555 – 3570.



rosenthal `∞-theorem revisited 155

[7] L. Drewnowski, I. Labuda, Topological vector spaces of Bochner measur-
able functions, Illinois J. Math. 46 (2002), 287 – 318.

[8] N.J. Kalton, Exhaustive operators and vector measures, Proc. Edinburgh
Math. Soc. (2) 19 (1974/75), 291 – 300.

[9] N.J. Kalton, Spaces of compact operators, Math. Ann. 208 (1974), 267 – 278.

[10] N.J. Kalton, N.T. Peck, J.W. Roberts, “ An F-space sampler ”, Lon-
don Mathematical Society Lecture Note Series, 89, Cambridge University
Press, Cambridge, 1984.

[11] J. Kupka, A short proof and a generalization of a measure theoretic disjoin-
tization lemma, Proc. Amer. Math. Soc. 45 (1974), 70 – 72.

[12] I. Labuda, Sur les mesures exhaustives et certaines classes d’espaces vectoriels
topologiques considérés par W. Orlicz et L. Schwartz, C. R. Acad. Sci. Paris
Sér. A 280 (1975), 997 – 999.

[13] I. Labuda, Exhaustive measures in arbitrary topological vector spaces, Studia
Math. 58 (1976), 239 – 248.

[14] A. Avilés, F. Cabello Sánchez, J.M.F. Castillo, M. González,
Y. Moreno, “ Separably Injective Banach Spaces ”, Lecture Notes in Math-
ematics, 2132, Springer, 2016.

[15] Z. Lipecki, The variation of an additive function on a Boolean algebra, Publ.
Math. Debrecen 63 (2003), 445 – 459.

[16] S. Rolewicz, “ Metric Linear Spaces ”, Second edition, PWNùPolish Scientific
Publishers, Warsaw; D. Reidel Publishing Co., Dordrecht, 1984.

[17] H.P. Rosenthal, On complemented and quasi-complemented subspaces of
quotients of C(S) for stonian S, Proc. Nat. Acad. Sci. U.S.A. 60 (1968),
1165 – 1169.

[18] H.P. Rosenthal, On relatively disjoint families of measures, with some ap-
plications to Banach space theory, Studia Math. 37 (1970), 13 – 36.
(Correction, ibid., 311 – 313.)


	Introduction and the standard Rosenthal -theorem
	Notation, conventions, terminology, some basic facts.

	New extensions of Rosenthal -theorem