isibang/ms/2013/10
May 2nd, 2013
http://www.isibang.ac.in/e statmath/eprints
Transitivity of various notions of proximinality in
Banach spaces
C. R. Jayanarayanan and Tanmoy Paul
Indian Statistical Institute, Bangalore Centre
8th Mile Mysore Road, Bangalore, 560059 India
TRANSITIVITY OF VARIOUS NOTIONS OF
PROXIMINALITY IN BANACH SPACES
C. R. JAYANARAYANAN AND TANMOY PAUL
Abstract. We derive transitivity of various degrees of proximinality
in Banach spaces. When the transitivity does not carry forward to
the bigger space we investigate these properties under some additional
assumptions of the intermediate space. For instance, we show that if
Z ⊆ Y ⊆ X where Z is a finite co-dimensional subspace of X which is
strongly proximinal in Y and Y is an M-ideal in X then Z is strongly
proximinal in X. As a consequence of these properties we derive that
in a L1 -predual space strongly subdifferentiable points and quasi polyhedral points are same. We also discuss the transitivity of intersection
properties of balls in the context of proximinality. Various examples and
counter examples are given.
1. Introduction
We work with real scalar. Let X be a real Banach space and Y be
a closed subspace of X. For x ∈ X, let d(x, Y ) = inf y∈Y ∥x − y∥ and
PY (x) = {y ∈ Y : ∥x − y∥ = d(x, Y )}. We call the subspace Y is proximinal
if for every point x ∈ X \ Y , PY (x) ̸= ∅.
One of the interesting problems in approximation theory is about the transitivity of various degrees of proximinality(viz. proximinality, strong proximinality (Definition 1.2), 1 21 -ball property, semi M-ideal (Definition 1.4)
etc.). Precisely, let (P ) be one of these proximinality properties and let Y
and Z be subspaces of X with Z ⊆ Y ⊆ X such that Z has property-(P )
in Y then is it necessary that it has property-(P ) in X? It is well known
that the answer is not affirmative if the property-(P ) is proximinality, strong
proximinality or 1 21 -ball property though semi M-ideal property is transitive. In this paper we discuss mainly about the above mentioned transitivity
problem and their natural consequences.
2000 Mathematics Subject Classification. Primary 46B20, 46E15.
Key words and phrases. proximinality, strong proximinality, ideals, semi M -ideals, M ideals.
1
2
JAYANARAYANAN AND PAUL
Our first result in the next section states that proximinality pass through
the M (or L)-summands in a Banach space. Let us recall the following notion
in a Banach space.
Definition 1.1. A subspace Y of a Banach space X is an M-ideal in X if
there is a linear projection P on X ∗ with ker(P ) = Y ⊥ and for all x∗ ∈ X ∗ ,
∥x∗ ∥ = ∥P x∗ ∥ + ∥x∗ − P x∗ ∥.
It is well known that an M-ideal is proximinal. In fact, it has several
stronger proximinality properties which we will discuss below. [12] is a well
known reference for this concept.
We recall from [11] the following stronger version of proximinality.
Definition 1.2. A subspace Y of X is strongly proximinal if it is proximinal
and for a given x ∈ X \ Y and ε > 0 there exists a δ > 0 such that
PY (x, δ) ⊆ PY (x) + εBX , where PY (x, δ) = {y ∈ Y : ∥x − y∥ ≤ d(x, Y ) + δ}.
It is also well known that an M-ideal is strongly proximinal. In Proposition 2.7 we characterize the finite co-dimensional strongly proximinal subspaces of Banach spaces. Our main result in this section is if Z is a finite codimensional subspace of X and Y is an M-ideal in X such that Z ⊆ Y ⊆ X
and Z is strongly proximinal in Y then Z is strongly proximinal in X (Theorem 2.8). A natural correspondence between strongly proximinal hyperplane
and the ’subdifferentiability’ of the dual norm is given in [7, Proposition 2.1].
Definition 1.3. [11]
(a) The norm on X is strongly subdifferentiable (SSD in short) at
x ∈ X if the one sided limit
∥x + ty∥ − ∥x∥
d+ (x)(y) := lim
t
t→0+
exists uniformly for y ∈ SX . In this case we call x is a SSD point of
X.
(b) We call x ∈ X is a quasi polyhedral (QP in short) point of X if
there exists δ > 0 such that
JX ∗ (z) ⊆ JX ∗ (x) if ∥z − x∥ < δ, ∥z∥ = ∥x∥,
where JX ∗ (x) = {f ∈ BX ∗ : f (x) = 1}.
It is known that a QP point is also an SSD point but the converse is not
true in general [11].
TRANSITIVITY OF VARIOUS NOTIONS OF PROXIMINALITY IN BANACH SPACES 3
In this section we prove that for a positive measure µ, any SSD-point of
L1 (µ) is also a QP-point. In Example 4.4 it is observed that for a subspace
Y ⊆ X the Hahn-Banach extension of SSD point of Y ∗ need not be an SSD
point of X ∗ but it is SSD if Y is an M-ideal in X (Corollary 2.14). For a finite
co-dimensional strongly proximinal space Y ⊆ X it is known that Y ⊥ ⊆
{SSD points of X ∗ } but the converse is still open[11]. In Proposition 2.16
we observe that the converse is true in the class of ’L1 -predual’ space.
Recall that a Banach space X is said to be an L1 -predual if X ∗ is isometric
to L1 (µ) for some measure µ.
In Section 3 we discuss intersection properties of balls in Banach spaces.
We restrict ourselves only on (strong)1 12 -ball property and semi M-ideals.
Definition 1.4. [21]
(a) A subspace Y of a Banach space X is said to have the (strong)n
-ball property if, whenever {B[ai , ri ]}ni=1 are closed balls in X with
∩n B[ai , ri ] ̸= ∅ and Y ∩ B[ai , ri ] ̸= ∅, then Y ∩ (∩i B[ai , ri + ε]) ̸= ∅
for every (ε = 0)ε > 0.
(b) A subspace Y of a Banach space X is said to have the (strong)1 12
-ball property if (a) is true for n = 2 and one of the centres of the
balls is in Y .
(c) A subspace Y is a semi M-ideal in X if there is a (nonlinear)
projection P from X ∗ onto Y ⊥ such that for all x∗ , y ∗ ∈ X ∗ , λ ∈ R
P (λx∗ + P y ∗ ) = λP x∗ + P y ∗
∥x∗ ∥ = ∥P x∗ ∥ + ∥x∗ − P x∗ ∥.
Such a projection is called semi L-projection and in this case Y ⊥ is a
semi L-summand. It is known that a subspace having 1 12 -ball property is
strongly proximinal[6, Proposition 3.3]. From [18, Example 6] it follows
that this property also fails to be transitive. Due to the fact that in a
finite dimensional space 1 12 -ball property implies strong 1 12 -ball property,
[18, Example 6] also ensures strong 1 12 -ball property fails to be transitive.
It is known that semi M-ideals are precisely the subspaces having 2-ball
property [15, Theorem 6.10].
The main result in this section states that both of these properties pass
through the M-ideals of a Banach space (Theorem 3.4). Theorem 3.4 is a
particular case of [18, Theorem 5] but our arguments are completely different
4
JAYANARAYANAN AND PAUL
and should be of interest. We conclude this section by arguing that [14,
Corollary 2.5] is incorrect but necessarily true for L1 -predual spaces.
Several examples are given in the Section 4. In particular we prove that
the property strong proximinality is not transitive (Example 4.3) and also
we show that if Y ⊆ X then Hahn-Banach extension of SSD points of y ∗
not necessarily SSD in X ∗ (Example 4.4).
For a Banach space X, BX , SX and B[x, r] denote the closed unit ball, the
closed unit sphere and closed ball with centre at x and radius r respectively.
N A(X) = {f ∈ SX ∗ : f attains its norm on X}.
X will always denotes a Banach space and by a subspace we always mean
closed subspace.
[12] is a standard reference for any unexplained terminology.
2. Transitivity Property of Proximinality and Strong
Proximinality
It is observed in [13] that proximinality is not transitive in Banach space.
In [4, Remark 3.25] the author observed that there exists a subspace V
such that V ⊆ c0 is proximinal but not in ℓ∞ . So proximinality does not
pass through even M-ideals.
Our next result shows that the transitivity holds under more stronger
assumption than M-ideal.
We call φ : R≥0 × R≥0 → R≥0 is a monotone map if φ(α1 , β) ≥ φ(α2 , β)
whenever α1 ≥ α2 .
Proposition 2.1. Let X = Y ⊕ Z and φ : R≥0 × R≥0 → R≥0 be a monotone
map such that for x ∈ X ∥x∥ = φ(∥y∥, ∥z∥) if x = y + z where y ∈ Y, z ∈ Z
and also φ(αn , β) → φ(α, β) implies αn → α. If W ⊆ Y is a proximinal
(strongly proximinal) subspace then it is proximinal (strongly proximinal) in
X.
Proof. If W is proximinal in Y , then the proximinality of W in X follows
from the fact that PW (y) ⊆ PW (x). Note that the convergence assumption
on φ is not used in this case.
Next let W be strongly proximinal in X.
Clearly d(x, W ) =
φ(d(y, W ), ∥z∥). Now let (wn ) be a sequence in W such that ∥x − wn ∥ →
d(x, W ). Then, by the assumption on φ, ∥y − wn ∥ → d(y, W ) and
hence, by the strong proximinality of W in Y , d(wn , PW (y)) → 0. Since
TRANSITIVITY OF VARIOUS NOTIONS OF PROXIMINALITY IN BANACH SPACES 5
PW (y) ⊂ PW (x), d(wn , PW (x)) → 0 and hence the strong proximinality of
W in X follows.
As an immediate consequence of Proposition 2.1 we have that if Y is an
L-summand of X then any proximinal (strongly proximinal) subspace W of
Y is proximinal (strongly proximinal) in X. When Y is an M-summand in
X, Proposition 2.1 also ensures that W is proximinal in X if it is so in Y .
Proposition 2.2. Let X be a Banach space and Y be an M -summand in
X. Then a subspace W ⊂ Y is stongly proximinal in X if W is strongly
proximinal in Y .
Proof. Following the same notations used in Proposition 2.1 we have that
d(x, W ) = max{d(y, W ), ∥z∥} and PW (y) ⊆ PW (x). Let ε > 0.
Suppose ∥z∥ > d(y, W ).
Then PW (x) = B[y, ∥z∥] ∩ W , and PW (x, δ) = B[y, ∥z∥ + δ] ∩ W . Hence
strong proximinality of W in X follows by choosing δ = ε.
Suppose ∥z∥ ≤ d(y, W ).
Then PW (x) = PW (y) and PW (x, δ) = PW (y, δ). And in this case the
strong proximinality of W in X follows from the same for W in Y .
We now introduce some notations from [11] in order to state a characterization of finite co-dimensional strongly proximinal subspace in Banach
space.
Let {f1 , f2 , . . . , fn } ⊆ N A(X).
Define JX (f1 ) = {x ∈ BX : f1 (x) = 1}, JX ∗∗ (f1 ) = {x∗∗ ∈ BX ∗∗ :
x∗∗ (f1 ) = 1} and let M1 = M1∗ = 1.
Define M2 = sup{f2 (x) : x ∈ JX (f1 )} and let JX (f1 , f2 ) = {x ∈ JX (f1 ) :
f2 (x) = M2 }.
Similarly M2∗ = sup{x∗∗ (f2 ) : x∗∗ ∈ JX ∗∗ (f1 )} and also JX ∗∗ (f1 , f2 ) =
{x∗∗ ∈ JX ∗∗ (f1 ) : x∗∗ (f2 ) = M2∗ }.
Inductively get Mi , Mi∗ and JX (f1 , . . . , fi ), JX ∗∗ (f1 , . . . , fi ) for 1 ≤ i ≤ n.
For ε > 0 define JX (f1 , ε) = {x ∈ BX : f1 (x) > 1 − ε}. If JX (f1 , . . . , fi , ε)
is defined then JX (f1 , . . . , fi+1 , ε) = {x ∈ JX (f1 , . . . , fi , ε) : fi+1 (x) >
Mi+1 − ε}.
Throughout this section we use the following characterization of finite
co-dimensional strongly proximinal subspace
6
JAYANARAYANAN AND PAUL
Theorem 2.3. [11] Let Y be a finite co-dimensional proximinal subspace of
X. Then Y is strongly proximinal if and only if for any basis {f1 , . . . , fn }
of Y ⊥
lim [sup{d(x, JX (f1 , . . . , fi )) : x ∈ JX (f1 , . . . , fi , ε)}] = 0
ε→0
for 1 ≤ i ≤ n.
In other words a necessary and sufficient condition for strong proximinality of Y is: given ε > 0 there exists δε > 0 such that d(x, JX (f1 , . . . , fi )) < ε,
whenever x ∈ JX (f1 , . . . , fi , δε ), for 1 ≤ i ≤ n.
We now exhibit some relations between the notations defined above.
Proposition 2.4. Let X be a Banach space and let Y be a finite codimensional strongly proximinal subspace of X and {f1 , f2 , . . . , fn } ⊆ SY ⊥
be a basis of Y ⊥ . Let Mi , Mi∗ , JX (f1 , . . . , fi ) and JX ∗∗ (f1 , . . . , fi ) are defined
as above. Then for 1 ≤ i ≤ n
(a) Mi = Mi∗ .
w∗
(b) JX ∗∗ (f1 , . . . , fi ) = JX (f1 , . . . , fi ) .
Proof. (a). Clearly we have M2∗ ≥ M2 . Since JX ∗∗ (f1 , f2 ) is a w∗ -compact
set, f2 attains its supremum at some x∗∗
0 ∈ JX ∗∗ (f1 , f2 ). Choose a net (xα ) ⊆
∗∗
∗
BX such that xα → x0 in w sense. Since f2 (xα ) → M2∗ , d(xα , JX (f1 )) → 0.
∗
Choose (yα ) ⊆ JX (f1 ) such that ∥xα −yα ∥ → 0. Then yα → x∗∗
0 in w sense.
∗
Since f2 (yα ) → x∗∗
0 (f2 ), M2 ≥ M2 . And so on.
w∗
(b). Since f1 is an SSD point we have JX (f1 ) = JX ∗∗ (f1 ). Next we have
w∗
JX (f1 , f2 ) ⊆ JX ∗∗ (f1 , f2 ). Choose ϕ ∈ JX ∗∗ (f1 , f2 ) and let (xα ) ⊆ BX
such that xα → ϕ in w∗ -sense. Since f1 (xα ) → ϕ(f1 ), d(xα , JX (f1 )) → 0.
Choose (yα ) ⊆ JX (f1 ) such that ∥xα − yα ∥ → 0. Hence yα → ϕ in w∗ -sense.
Since we have f2 (yα ) → ϕ(f2 ) = M2 , d(yα , JX (f1 , f2 )) → 0. Hence there
exists (zα ) ⊆ JX (f1 , f2 ) such that ∥yα −zα ∥ → 0, which in turn gives zα → ϕ
w∗
in w∗ -sense. i.e. JX (f1 , f2 ) = JX ∗∗ (f1 , f2 ). The rest follows in a similar
arguments.
Proposition 2.4(b) is a generalization of [11, Remark 1.2(2)].
Our next result generalizes a known fact related to the strongly proximinal
hyperplane in Banach space, see [11, Remark 1.2].
Lemma 2.5. Let Y be a finite co-dimensional strongly proximinal subspace
of X. Let {f1 , . . . , fn } be a basis of Y ⊥ , where ∥fi ∥ = 1, 1 ≤ i ≤ n. Then
for x ∈ BX , d(x, JX (f1 , . . . , fi )) = d(x, JX ∗∗ (f1 , . . . , fi )).
TRANSITIVITY OF VARIOUS NOTIONS OF PROXIMINALITY IN BANACH SPACES 7
Proof. If n = 1 then the conclusion follows from [11, Remark 1.2(1)].
Since no new ideas are required for n > 2 we only prove the case for n = 2.
Hence we have to show that for x ∈ BX , d(x, JX (f1 , f2 )) =
d(x, JX ∗∗ (f1 , f2 )).
Let d = d(x, JX ∗∗ (f1 , f2 )). Since JX ∗∗ (f1 , f2 ) is w∗ -compact, it is proximinal. Choose ϕ ∈ JX ∗∗ (f1 , f2 ) such that ∥x − ϕ∥ = d.
Since Y is strongly proximinal, given ε > 0 there exists δε > 0 such that
d(x, JX (f1 , f2 )) < ε, whenever x ∈ JX (f1 , f2 , δε ). Choose ε > 0 arbitrarily.
Let E = lin{x, ϕ} ⊆ X ∗∗ and F = lin{f1 , f2 } ⊆ X ∗ . By principle of
local reflexivity there exists T ∈ B(E, X) such that T x = x, (1 − ε′ ) ≤
∥T (z ∗∗ )∥ ≤ (1 + ε′ ) if z ∗∗ ∈ SE and fi (T (z ∗∗ )) = z ∗∗ (fi ), i = 1, 2. Where
ε
ε′ < min{δε/22 , 2(d+1)
}.
Let x1 =
Tϕ
∥T ϕ∥ .
Now
Tϕ
∥
∥T ϕ∥
= ∥T (x − ϕ)∥ + |1 − ∥T ϕ∥|
∥x − x1 ∥ ≤ ∥x − T ϕ∥ + ∥T ϕ −
≤ (1 + ε′ )d + ε′
= d + ε′ (1 + d) < d +
ε
2
And for i = 1, 2
)
Tϕ
fi
∥T ϕ∥
ϕ(fi )
∥T ϕ∥
Mi
∥T ϕ∥
Mi
1 + ε′
Mi ε′
Mi −
1 + ε′
Mi − δε/22
(
fi (x1 ) =
=
=
≥
=
>
Which shows that x1 ∈ JX (f1 , f2 , δε/22 ) and hence d(x1 , JX ∗∗ (f1 , f2 )) ≤
d(x1 , JX (f1 , f2 )) < ε/22 . Get ϕ1 ∈ JX ∗∗ (f1 , f2 ) such that ∥x1 − ϕ1 ∥ < ε/22 .
Again by P.L.R. there exists x2 ∈ BX such that ∥x1 − x2 ∥ < ε/22 and
fi (x2 ) > Mi − δε/23 .
8
JAYANARAYANAN AND PAUL
Proceeding thus inductively we construct a sequence (xn ) ⊆ BX such that
∥xn − xn−1 ∥ < ε/2n and fi (xn ) > Mi − δε/2n+1 , i = 1, 2 and for all n.
Clearly (xn ) is cauchy and hence there exists z ∈ BX such that z =
limn xn . Now fi (z) = Mi for i = 1, 2 and hence z ∈ JX (f1 , f2 ). Also for
all n we have ∥x − xn ∥ ≤ d + ε/2 + . . . + ε/2n . Taking limit n → ∞ we
have ∥x − z∥ ≤ d + ε. Since ε > 0 is arbitrary and z ∈ JX (f1 , f2 ) we have
d(x, JX (f1 , f2 )) ≤ d = d(x, JX ∗∗ (f1 , f2 )), hence the result follows.
As an immediate consequence of [8, Theorem 1.2], [11, Lemma 1.1] and
[7, Proposition 2.1] we have that
Proposition 2.6. A hyperplane Y in X is strongly proximinal if and only
if Y ⊥⊥ is strongly proximinal in X ∗∗ .
Our next result generalizes Proposition 2.6 for finite co-dimensional case.
Proposition 2.7. If Y is a finite co-dimensional proximinal subspace of a
Banach space X, then Y is strongly proximinal in X if and only if Y ⊥⊥ is
strongly proximinal in X ∗∗ .
Proof. We first show that the condition is necessary.
Choose a basis of Y ⊥⊥⊥ in X ∗∗∗ . Clearly this gives a basis of Y ⊥ in X ∗ .
Let the basis be {f1 , . . . , fn }.
Now for a given ε > 0 choose δ(ε) > 0 by the arguments defined next to
Theorem 2.3.
Now if x∗∗ ∈ JX ∗∗ (f1 , . . . , fi , δ) then x∗∗ (fj ) > Mj − δ for 1 ≤ j ≤ i. Let
(xα ) ⊆ BX such that xα → x∗∗ in w∗ -sense, without loss of generality assume
fj (xα ) > Mj −δ for all α for 1 ≤ j ≤ i. Hence there exists zα ∈ JX (f1 , . . . , fi )
such that ∥xα − zα ∥ < ε. Passing through a subnet of (zα ), if necessary, we
may assume zα → ϕ in w∗ -sense, for some ϕ ∈ JX ∗∗ (f1 , . . . , fi ).
Hence we have (xα − zα ) → (x∗∗ − ϕ) in w∗ -sense. Thus ∥x∗∗ − ϕ∥ ≤
limα ∥xα − zα ∥ ≤ ε. And finally we have d(x∗∗ , JX ∗∗ (f1 , . . . , fi )) ≤ ∥x∗∗ −
ϕ∥ < ε, justifies the necessity.
To prove sufficiency observe that JX (f1 , . . . , fi , δ) ⊆ JX ∗∗ (f1 , . . . , fi , δ).
Now for a given ε > 0 choose δ(ε) > 0 again using the arguments next to
Theorem 2.3. Then for x ∈ JX (f1 , . . . , fi , δ) we have d(x, JX (f1 , . . . , fi )) =
d(x, JX ∗∗ (f1 , . . . , fi )) < ε and this completes the proof.
We are now ready to prove the main theorem of this section.
TRANSITIVITY OF VARIOUS NOTIONS OF PROXIMINALITY IN BANACH SPACES 9
Theorem 2.8. Let X be a Banach space and Z be a finite co-dimensional
proximinal subspace of X. Let Y be an M-ideal in X and Z ⊆ Y ⊆ X. Then
Z is strongly proximinal in X if Z is strongly proximinal in Y .
Proof. ¿From necessity part of Proposition 2.7 it follows that Z ⊥⊥ is strongly
proximinal in Y ⊥⊥ . Since Y ⊥⊥ is an M-summand in X ∗∗ , Z ⊥⊥ is strongly
proximinal in X ∗∗ .
Now the theorem follows from the sufficiency part of Proposition 2.7. In Example 4.3 it is observed that the transitivity of strong proximinality
not necessarily true in general. We do not know whether a further weakening
on the assumption of Y in Theorem 2.8 is possible or not.
Question 2.9. Let Y be a semi M-ideal in X and Z be a strongly proximinal
subspace of Y such that Z is of finite co-dimension in X. Is Z also strongly
proximinal in X ?
Example 4.2 asserts that the condition of semi M-ideal in the above question can not be weakened to ideals.
Remark 2.10. We do not know whether the finite co-dimensionality assumption on Y in Theorem 2.8 is necessary or not. The answer is not
known even if the strong proximinality in Theorem 2.8 is replaced by proximinality.
For an SSD point f of X ∗ , there always exists an Hahn-Banach extension
of f to X ∗∗ which is an SSD point of X ∗∗∗ but we do not know what happen
if F is any Hahn-Banach extension of f to X ∗∗ . Coming to a more general
set up it is natural to ask the following. First recall the following definition.
Definition 2.11. [19] A subspace Y of a Banach space X is said to be an
ideal if Y ⊥ is kernel of a norm one projection on X ∗ .
Every Banach space, under the canonical embedding, is an ideal in its
bidual.
Question 2.12. Let Y be an ideal in X and y ∗ be an SSD point of Y ∗ . If
x∗ is an Hahn-Banach extension of y ∗ then is x∗ an SSD point of X ∗ ?
The answer is affirmative under a slight modification on the assumption
on Y in the above question. Our next Theorem is a particular case of [8,
Proposition 2.1] but for the sake of completeness we briefly state the proof.
10
JAYANARAYANAN AND PAUL
Theorem 2.13. Let Y be a semi L-summand in a Banach space X and let
y ∈ Y be an SSD point of Y . Then it is also an SSD point of X.
Proof. Let P : X → Y be a semi L-projection, then
d+ (y)(x) = d+ (y)(P x) + ∥x − P x∥
Hence the conclusion follows from
(
)
x
∥−1
∥y + t∥P x∥ ∥PP x∥
∥y + tx∥ − 1
P
x
− d+ (y)(x) = ∥P x∥
− d+ (y)(
) .
t
∥P x∥t
∥P x∥
It is now immediate that
Corollary 2.14. If Y is an M-ideal in X and f ∈ SY ∗ is an SSD point of
Y ∗ then the unique Hahn-Banach extension of f to X is also an SSD point
of X ∗ .
In contrast to Example 4.4, Corollary 2.14 gives a sufficient condition on
Y such that the Hahn-Banach extension of SSD points of Y ∗ is SSD in X ∗ .
Combining [5, Theorem 2.1] with Proposition 2.7 we derive a characterization for finite co-dimensional strongly proximinal subspaces in L1 -preduals.
Our next result gives a class of Banach spaces where SSD-point and QPpoint coincide.
Proposition 2.15. If X is an L1 -predual then an SSD-point of X ∗ is also
a QP-point of X ∗ .
Proof. Let f ∈ L1 (µ) be an SSD-point. Since L1 (µ) is an L-summand in
its bidual, f is an SSD point of X ∗∗∗ = C(K)∗ (up to an isometry), for
some compact Hausdaurff space K. By [5, Theorem 2.1], f is a QP-point of
C(K)∗ hence a QP-point of X ∗ = L1 (µ).
And hence we have
Proposition 2.16. If X is an L1 -predual and Y ⊆ X be a finite codimensional proximinal subspace of X. The following are equivalent
(a) Y is strongly proximinal.
(b) Y ⊥ ⊆ {x∗ ∈ X ∗ : x∗ is an SSD-point of X ∗ } = {x∗ ∈ X ∗ :
x∗ is a QP-point of X ∗ }.
TRANSITIVITY OF VARIOUS NOTIONS OF PROXIMINALITY IN BANACH SPACES11
Proof. Since (a) ⇒ (b) follows from [11] it only remains to prove that (b) ⇒
(a). It is enough to show that Y ⊥⊥ is strongly proximinal. Now Y ⊥ ⊆
{SSD points of X ∗ } implies Y ⊥⊥⊥ ⊆ {SSD points of X ∗∗∗ }. The rest of the
arguments follows from [5, Corollary 2.3]
Remark 2.17. In fact, from the proof of Proposition 2.15 it follows that
any SSD-point of L1 (µ) is also a QP-point.
3. Transitivity Property of some ball intersection properties
In this section we discuss few ball intersection property and their transitivity which are closely related to proximinality in Banach space. Though
we discuss only about (strong) 1 12 ball and semi M-ideals, our next result
follows in a general setup without any further clarifications.
Lemma 3.1. Let Y be an M -summand in a Banach space X and Z be a
subspace of Y . Let n ∈ N.
(a) If Z has (strong) n-ball property in Y then Z has (strong) n-ball
property in X.
(b) If Z has (strong) 1 12 -ball property in Y then Z has (strong) 1 21 -ball
property in X also.
Proof. (a) Let ε > 0 and let {B[xi , ri ]}1≤i≤n be a family of n balls in X such
that
n
∩
B[xi , ri ] ∩ Z ̸= ∅ for all i = 1, . . . , n and
B[xi , ri ] ̸= ∅.
∩n
i=1
Let x ∈ i=1 B[xi , ri ] and P : X → Y be a M-projection.
∩
It is clear that P x ∈ ni=1 B[P xi , ri ] and B[P xi , ri ] ∩ Z ̸= ∅. By n∩
ball property of Z in Y get z ∈ Z (∩ni=1 B[P xi , ri + ε]), then ∥z − xi ∥ ≤
max{∥z − P xi ∥, ∥xi − P xi ∥} ≤ ri + ε, 1 ≤ i ≤ n.
Now strong n-ball property of Z in X follows by taking ε = 0 above.
The proof of (b) is similar.
Lemma 3.2. Let Y be a subspace of a Banach space X. Then
(a) Y is a semi M -ideal in X if and only if Y ⊥⊥ is a semi M -ideal
in X ∗∗ .
(b) Y has 1 12 -ball property in X if and only if Y ⊥⊥ has 1 12 -ball property in X ∗∗ .
12
JAYANARAYANAN AND PAUL
Proof. (a). Suppose Y is a semi M -ideal in X. i.e. Y ⊥ is a semi L-summand
in X ∗ . Then by [15, Theorem 6.14], Y ⊥⊥ is a semi M -ideal in X ∗∗ .
Conversely suppose that Y ⊥⊥ is a semi M -ideal in X ∗∗ .
Let ϵ > 0 and let {B[xi , ri ]}i=1,2 be two balls in X such that
B[xi , ri ] ∩ Y ̸= ∅ for i = 1, 2 and B[x1 , r1 ] ∩ B[x2 , r2 ] ̸= ∅.
Let x ∈ B[x1 , r1 ] ∩ B[x2 , r2 ] and yi ∈ B[xi , ri ] ∩ Y . Since Y ⊥⊥ is a semi
M -ideal in X ∗∗ and is a weak∗ -closed subspace of X ∗∗ , Y ⊥⊥ has strong 2-ball
property in X ∗∗ . Hence there exists a x∗∗ ∈ Y ⊥⊥ such that ∥x∗∗ − xi ∥ ≤ ri .
Let E = span{x1 , x2 , y1 , y2 , x, x∗∗ } and r = max{r1 , r2 }.
Then by an extended version of Principle of Local Reflexivity(see [3, Theorem 3.2]) there exists an operator Tϵ : E → X such that
(1) Tϵ (z) = z if z ∈ E ∩ X.
(2) Tϵ (E ∩ Y ⊥⊥ ) ⊂ Y .
ϵ
(3) ∥Tϵ ∥ ≤ 1 + .
r
Now take z = Tε (x∗∗ ). Then z ∈ Y and ∥z − xi ∥ ≤ ri + ε for i = 1, 2.
Hence Y is a semi M -ideal in X.
(b). Follows immediately from [21, Theorem 3].
Corollary 3.3. Let Y be a semi M -ideal in a Banach space X. Then Y is
a semi M -ideal in X ∗∗ if and only if Y is an M -embedded space.
Proof. Suppose Y is a semi M -ideal in X ∗∗ . Then Y is a semi M -ideal in
Y ⊥⊥ = Y ∗∗ and hence, by [16, Corollary 3.4], Y is an M -ideal in Y ∗∗ .
Conversely suppose that Y is an M -embedded space. Since Y is a semi
M -ideal in X, by Lemma 3.2, Y ⊥⊥ is a semi M -ideal in X ∗∗ . Then, by [18,
Theorem 5], Y is a semi M -ideal in X ∗∗ .
Theorem 3.4. Let Y be an M -ideal in X and Z be a subspace of Y .
(a) If Z is a semi M -ideal in Y then Z is a semi M -ideal in X.
(b) If Z has 1 12 -ball property in Y then Z has 1 12 -ball property in X.
Proof. (a). Since Z ⊂ Y ⊂ X, Z ⊥⊥ ⊂ Y ⊥⊥ ⊂ X ∗∗ . Then by Lemma 3.2,
Z ⊥⊥ is a semi M -ideal in Y ⊥⊥ and by Lemma 3.1, Z ⊥⊥ is a semi M -ideal
in X ∗∗ . Then by Lemma 3.2, Z is a semi M -ideal in X.
(b). Since Z ⊂ Y ⊂ X, Z ⊥⊥ ⊂ Y ⊥⊥ ⊂ X ∗∗ . Then by Lemma 3.2, Z ⊥⊥
has 1 21 -ball property in Y ⊥⊥ and by Lemma 3.1, Z ⊥⊥ has 1 21 -ball property
in X ∗∗ . Then by Lemma 3.2, Z has 1 21 -ball property in X.
TRANSITIVITY OF VARIOUS NOTIONS OF PROXIMINALITY IN BANACH SPACES13
Remark 3.5.
(a) We do not know the corresponding analogue of
Theorem 3.4(b) in the context of strong 1 12 -ball property.
(b) Recently another stronger version of proximinality also attracted
people’s attention viz. ball proximinality, [1, 14, 20] are some recent
study on this property. In [14] the authors proved that a Banach space
has strong 1 12 -ball property if and only if it is ball proximinal and has
1 21 -ball property. The Example 13 in [21] proves the existence of an
M -ideal which don’t have the strong 1 12 -ball property. Since all M ideals have the 1 12 -ball property, by [14, Theorem 2.11], this M -ideal
cannot be ball proximinal. This example reveals that Corollary 2.5
in [14] is incorrect.
Now we will give a class of Banach spaces in which M -ideals are ball
proximinal.
Definition 3.6. Let X be a Banach space and n ∈ N. Then X has n.2.I.P.
if any pairwise intersecting family of n balls in X actually intersects.
It is well known that a real Banach space is L1 -predual if and only if it
has 4.2.I.P.[17].
Theorem 3.7. If X has 3.2.I.P. then every M -ideal in X satisfies strong
3-ball property. In particular, an M -ideal in an L1 -predual space has strong
3-ball property.
Proof. Let Y be an M -ideal in X and let {B(xi , ri )}3i=1 be a family of 3
closed balls satisfying
∩
B(xi , ri ) ∩ Y ̸= ∅ for all i = 1, 2, 3. and 3i=1 B(xi , ri ) ̸= ∅.
Let ϵ > 0. Since Y is an M -ideal in X there exists a y0 ∈ Y such that
∩
y0 ∈ 3i=1 B(xi , ri + ϵ).
Now {B(xi , ri )}3i=1 ∪ {B(y0 , ϵ)} is a mutually intersecting family of closed
balls in X.
For every i, {B(xj , rj )}j̸=i ∪ {B(y0 , ϵ)} is a mutually intersecting family
of 3 balls in X. Since X has 3.2.I.P the intersection of these three balls is
nonempty. Since Y is an M -ideal, for every i, 1 ≤ i ≤ 3, and every δ > 0,
there is a point yi = yi (δ) satisfying
∥yi − xj ∥ ≤ rj + δ, j ̸= i and ∥yi − y0 ∥ ≤ ϵ + δ for all i.
We now follow the technique used in [17, Lemma 4.1] in rest of the proof.
14
JAYANARAYANAN AND PAUL
∑
Let y = 13 3i=1 yi , then ∥y − y0 ∥ ≤ ϵ + δ and ∥y − xj ∥ ≤ rj + δ + 32 ϵ.
Now for δ ≤ ϵ/6 we have
∥y − y0 ∥ ≤ 2ϵ and ∥y − xj ∥ ≤ rj + 5ϵ/6, j = 1, · · · , n
¿From above equation it follows that there exists a sequence zm in Y (with
z0 = y0 ) satisfying ∥zm+1 −zm ∥ ≤ 2( 56 )m ϵ and ∥zm+1 −xj ∥ ≤ rj +( 65 )m ϵ, j =
1, 2, 3.
∩
Let z = limm→∞ zm , then z ∈ 3j=1 B(xj , rj ) ∩ Y and this concludes the
proof of the theorem.
Corollary 3.8. If X has 3.2.I.P. then every M -ideal in X is ball proximinal.
In particular M -ideals in an L1 -predual space is ball proximinal.
Proof. This follows immediately from 3.7 and [14, Theorem 2.4].
4. Some Examples
Our first example shows that the strong proximinality assumption on a
subspace is not sufficient to guarantee that any proximinal subspace of it is
also proximinal in the bigger space.
Example 4.1. There exist two subspaces Z and Y of finite co-dimension
in X such that Z is proximinal in Y and Y is strongly proximinal in X but
Z is not proximinal in X.
Proof. Let X be C[0, 1]. Choose k ∈ [0, 1] \ {0, 1, 12 , 13 , . . .}. Now define
µ, ν ∈ C[0, 1]∗ as
µ=
∞
∑
1
δ1
2n n
n=1
and
1
ν = (δ0 − δk ).
2
Then ∥µ∥ = ∥ν∥ = 1. Now take Z = ker(µ) ∩ ker(ν) and Y = ker(ν). Since
supp(ν) is finite, by [5, Theorem 2.1], ker(ν) is strongly proximinal in C(K).
Since 1 ∈ ker(ν) and µ(1) = 1, µ|ker(ν) is a norm-attaining functional on
ker(ν). Hence ker(µ) ∩ ker(ν) = ker(µ|ker(ν) ) is a proximinal subspace of
ker(ν). But ν is not absolutely continuous with respect to µ on supp(µ).
Hence, by [10], ker(µ) ∩ ker(ν) is not proximinal in C[0, 1].
Our next example is a variant of Example 4.1.
Example 4.2. There exist two subspaces Z and Y of finite co-dimension
in X such that Z is strongly proximinal in Y and Y is an ideal in X but Z
is not proximinal in X.
TRANSITIVITY OF VARIOUS NOTIONS OF PROXIMINALITY IN BANACH SPACES15
Proof. Let X, µ, ν and k be as in the proof of Example 4.1. Take Z =
ker(µ) ∩ ker(ν) and Y = ker(µ). Choose a continuous function g : K →
[−1, 1] such that g( n1 ) = g(0) = 1 for n ≥ 2 and g(1) = g(k) = −1. Then
g ∈ ker(µ) and ν(g) = 1. Since ν|ker(µ) attains its norm over ker(µ),
∞
∑
1
ker(µ) ∩ ker(ν) = ker(ν|ker(µ) is proximinal in ker(µ). Let λ = −
δ1.
2n n
n=2
Then ker(µ) = ker(λ − δ1 ) and ∥λ∥ ≤ 1 and hence, by [2], ker(µ) is an
L1 -predual space. Then, by [19, Proposition 1], ker(µ) is an ideal in C[0, 1].
But ν is not absolutely continuous with respect to µ on supp(µ). Hence, by
[10], ker(µ) ∩ ker(ν) is not proximinal in C[0, 1].
We now show that strong proximinality is not transitive
Example 4.3. There exist two subspaces Z and Y of finite co-dimension
in ℓ1 such that Z is strongly proximinal in Y and Y is strongly proximinal
in ℓ1 but Z is not strongly proximinal in ℓ1 .
Proof. Let f = (0, 1, 1, . . .) and g = (1, − 12 , − 13 , . . .). Then f and g are
QP-points of ℓ∞ . Let Z = ker(f ) ∩ ker(g) and Y = ker(f ). Since f is a
QP-point of ℓ∞ , Y is strongly proximinal in ℓ1 . Also since g attains its norm
on Y and g is a QP-point of ℓ∞ , by the proof of [6, Proposition 4.2], g|Y is
a QP-point of Y ∗ . Hence Z = ker(g|Y ) is strongly proximinal in Y . Since,
[8, Theorem 1.2] and [9, Theorem 5], f + g ∈ Z ⊥ is not an SSD-point of ℓ∞ ,
Z is not strongly proximinal in ℓ1 .
Example 4.3 also leads to the fact that if Y is a subspace of X then a
Hahn-Banach extension of SSD point of Y ∗ not necessarily SSD in X ∗ .
Example 4.4. There exists a subspace Y of ℓ1 and an SSD-point of Y ∗
such that one of its Hahn-Banach extension is not an SSD-point of ℓ∞ .
Proof. Let f , g, Z and Y be as in Example 4.3. Since g|Y is an SSD-point
of Y ∗ and f + g is an Hahn-Banach extension of g|Y , the conclusion follows
from Example 4.3.
Our next example shows semi M-ideal may not pass through L-summands.
Example 4.5. There exists an L-embedded space X and a semi M -ideal Y
in X such that Y is not a semi M -ideal in X ∗∗ .
16
JAYANARAYANAN AND PAUL
Proof. Take X = ℓ1 . Then X is an L-summand in its bidual. For the
constant sequence 1 ∈ ℓ∞ , Y = ker(1) is a semi M -ideal in ℓ1 (see [12,
Remark 2.3]). But ker(1) is not a semi M -ideal in (ℓ∞ )∗ . For, if ker(1)
is a semi M -ideal in (ℓ∞ )∗ , then it is a semi M -ideal in ker(1)⊥⊥ . Then,
by [16], ker(1) is an M -ideal in ker(1)⊥⊥ . Since, by [12, Corollary 3.3.C
and Theorem 3.4], a non-reflexive M -embedded space contains a subspace
isomorphic to c0 , ker(1) is reflexive . But this is a contradiction(since ℓ1
cannot have an infinite dimensional reflexive space). Hence ker(1) is not a
semi M -ideal in (ℓ∞ )∗ .
Example 4.5 also leads to the following example.
Example 4.6. There exists a Banach space X and subspaces, say J and Y ,
of X such that Y is a semi M -ideal in J, J is an ideal X but Y is not a
semi M -ideal in X.
Proof. Since all Banach spaces are ideal in its bidual, Example 4.5 gives
required example.
Acknowledgements. The authors would like to thank Professor
T.S.S.R.K. Rao for many helpful discussions and valuable suggestions. A
major part of this work was done when the second author was visiting Indian
Statistical Institute Bangalore center as an NBHM post doctorate fellow and
he would like to thank the NBHM for their financial support.
References
[1] P. Bandyopadhyay, Bor-Luh Lin, T. S. S. R. K. Rao, Ball proximinality in Banach
spaces, Banach Spaces and Their Applications in Analysis (Oxford / USA, 2006),
B. Randrianantoanina et al (ed.), Proceedings in Mathematics, de Gruyter, Berlin,
(2007) 251–264
[2] J. B Bednar and H.E Lacey, Concerning Banach spaces whose duals are abstract
L-spaces, Pacific J. Math. 41 (1972), 13–24.
[3] E. Behrends, On the principle of local reflexivity, Studia Math. 100 (1991), no. 2,
109–128.
[4] J. Blatter, Grothendieck spaces in approximation theory, Mem. Amer. Math. Soc. 120
(1972), Amer. Math. Soc.,Providence, RI.
[5] S. Dutta and D. Narayana, Strongly proximinal subspaces of finite co-dimension in
C(K), Colloq. Math. 109 (2007), no. 1, 119–128.
[6] S. Dutta and Darapaneni Narayana, Strongly proximinal subspaces in Banach spaces,
Contemp. Math. 435, Amer. Math. Soc., Providence, RI, 2007, 143152.
TRANSITIVITY OF VARIOUS NOTIONS OF PROXIMINALITY IN BANACH SPACES17
[7] S. Dutta and P. Shunmugaraj, Strong proximinality of closed convex sets, J. Approx.
Theory 163 (2011), no. 4, 547-553.
[8] C. Franchetti and R. Payá, Banach spaces with strongly subdifferentiable norm, Boll.
Un. Mat. Ital. B (7)7 (1993), no. 1, 45–70.
[9] C. Franchetti, Lipschitz maps and the geometry of the unit ball in normed spaces,
Arch. Math. (Basel) 46 (1986), no. 1, 76–84.
[10] A. L Garkavi, The problem of Helly and best approximation in the space of continuous
functions, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 641–656.
[11] G. Godefroy and V. Indumathi, Strong proximinality and polyhedral spaces, Rev. Mat.
Complut. 14 (2001), no. 1, 105–125.
[12] P. Harmand, D. Werner and W. Werner, M -ideals in Banach spaces and Banach
algebras, Lecture Notes in Math., 1547, Springer-Verlag, Berlin, 1993.
[13] V. Indumathi, Proximinal subspaces of finite co-dimension in general normed linear
spaces, Proc. London Math. Soc. 45 (1982), 3, 435-455.
[14] Indumathi, V and Lalithambigai, S, Ball proximinal spaces, J. Convex Anal. 18
(2011), no. 2, 353–366.
[15] A. Lima Intersection properties of balls and subspaces in Banach spaces, Trans. Amer.
Math. Soc. 227 (1977), 1–62.
[16] A. Lima Uniqueness of Hahn-Banach extensions and liftings of linear dependences,
Math. Scand. 53 (1983), no. 1, 97–113. Theory 93 (1998), no. 2, 331-343.
[17] J. Lindenstrauss Extension of compact operators, Mem. Amer. Math. Soc. 48 1964.
[18] R. Payá and D. Yost, The two-ball property: transitivity and examples, Mathematika
35 (1988), no. 2, 190–197.
[19] T. S. S. R. K Rao, On ideals in Banach spaces, Rocky Mountain J. Math. 31 (2001),
no. 2, 595–609.
[20] F. B. Saidi, On the proximinality of the unit ball of proximinal subspaces in Banach
spaces: a counter example, Proc. Amer. Math. Soc. 133 2005, 2697-2703.
[21] D. Yost The n-ball properties in real and complex Banach spaces, Math. Scand. 50
(1982), no. 1, 100–110.
(C. R. Jayanarayanan) Stat–Math Unit, Indian Statistical Institute, Bangalore center, 8th Mile Mysore Road, R. V. College Post, Bangalore 560059,
India, E-mail : [email protected]
(Tanmoy Paul) Department of Mathematics, Indian Institute of Technology Hyderabad, ODF Estate, Yeddumailaram 502205, India, E-mail :
[email protected]