On Semi-Uniform Kadec-Klee Banach Spaces

Abstract

Inspired by the concept of U-spaces introduced by Lau, (1978), we introduced the class of semi-uniform Kadec-Klee spaces, which is a uniform version of semi-Kadec-Klee spaces studied by Vlasov, (1972). This class of spaces is a wider subclass of spaces with weak normal structure and hence generalizes many known results in the literature. We give a characterization for a certain direct sum of Banach spaces to be semi-uniform Kadec-Klee and use this result to construct a semi-uniform Kadec-Klee space which is not uniform Kadec-Klee. At the end of the paper, we give a remark concerning the uniformly alternative convexity or smoothness introduced by Kadets et al., (1997).

1. Introduction

Let 𝑋 be a real Banach space with the unit sphere 𝑆_𝑋={𝑥∈𝑋∶‖𝑥‖=1} and the closed unit ball 𝐵_𝑋={𝑥∈𝑋∶‖𝑥‖≤1}. In this paper, the strong and weak convergences of a sequence {𝑥_𝑛} in 𝑋 to an element 𝑥∈𝑋 are denoted by 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥, respectively. We also let sep{𝑥_𝑛}=inf{‖𝑥_𝑛−𝑥_𝑚‖∶𝑛<𝑚}. (1.1)

Definition 1.1 (see [1]). We say that a Banach space 𝑋 is a Kadec-Klee space if {𝑥_𝑛}⊂𝐵_𝑋(KK)∶𝑥_𝑛⇀𝑥sep{𝑥_𝑛}>0}⇒‖𝑥‖<1. (1.2)

A uniform version of the KK property is given in the following definition.

Definition 1.2 (see [2]). We say that a Banach space 𝑋 is uniform Kadec-Klee if for every 𝜀>0 there exists a 𝛿>0 such that {𝑥_𝑛}⊂𝐵_𝑋(UKK)∶𝑥_𝑛⇀𝑥sep{𝑥_𝑛}≥𝜀}⇒‖𝑥‖≤1−𝛿. (1.3)

Two properties above are weaker than the following one.

Definition 1.3 (see [3]). We say that a Banach space 𝑋 is uniformly convex if for every 𝜀>0 there exists a 𝛿>0 such that (UC)∶𝑥,𝑦∈𝐵_𝑋1‖𝑥−𝑦‖≥𝜀}⇒2‖𝑥+𝑦‖≤1−𝛿. (1.4)

Let us summarize a relationship between these properties in the following implication diagram: UC⇒UKK⇒KK. (1.5)

In the literature, there are some generalizations of UC and KK.

Definition 1.4 (see [4]). We say that a Banach space 𝑋 is a 𝑈-space if for every 𝜀>0 there exists a 𝛿>0 such that (𝑈-space)∶𝑥,𝑦∈𝐵_𝑋⟨𝑥−𝑦,𝑓⟩≥𝜀forsome𝑓∈∇_𝑥1}⇒2‖𝑥+𝑦‖≤1−𝛿. (1.6)

Here ∇_𝑥={𝑓∈𝑆_𝑋^∗∶⟨𝑥,𝑓⟩=‖𝑥‖}.

Definition 1.5 (see [5]). We say that a Banach space 𝑋 is semi-Kadec-Klee if (semi-KK)∶{𝑥_𝑛}⊂𝑆_𝑋𝑥_𝑛⇀𝑥∈𝑆_𝑋}⇒⟨𝑥,𝑓_𝑛⟩→1∀{𝑓_𝑛}⊂𝑆_𝑋^∗satisfying𝑓_𝑛∈∇_𝑥𝑛∀𝑛.. (1.7)

Some interesting results concerning semi-KK property are studied by Megginson [6].

Remark 1.6. It is clear that 𝑈-space⇒semi-KK. (1.8)

Remark 1.7. A Banach space 𝑋 is semi-KK if and only if {𝑥_𝑛}⊂𝐵_𝑋𝑥_𝑛⇀𝑥inf{⟨𝑥_𝑛−𝑥,𝑓_𝑛⟩∶𝑛∈ℕ}>0forsome{𝑓_𝑛}⊂𝑆_𝑋^∗satisfying𝑓_𝑛∈∇_𝑥𝑛,∀𝑛}⇒‖𝑥‖<1. (1.9)

We now introduce a property lying between 𝑈-space and semi-KK.

Definition 1.8. We say that a Banach space 𝑋 is semi-uniform Kadec-Klee if for every 𝜀>0 there exists a 𝛿>0 such that (semi-UKK)∶{𝑥_𝑛}⊂𝑆_𝑋𝑥_𝑛⇀𝑥⟨𝑥_𝑛−𝑥,𝑓_𝑛⟩≥𝜀,forsome{𝑓_𝑛}⊂𝑆_𝑋^∗satisfying𝑓_𝑛∈∇_𝑥𝑛∀𝑛}⇒‖𝑥‖≤1−𝛿. (1.10)

In this paper, we prove that semi-UKK property is a nice generalization of 𝑈-space and semi-KK property. Moreover, every semi-UKK space has weak normal structure. We also give a characterization of the direct sum of finitely many Banach spaces which is semi-KK and semi-UKK. We use such a characterization to construct a Banach space which is semi-UKK but not UKK. Finally we give a remark concerning the uniformly alternative convexity or smoothness introduced by Kadets et al. [7].

2. Results

2.1. Some Implications

For a sequence {𝑥_𝑛}⊂𝑆_𝑋 and {𝑓_𝑛}⊂𝑆_𝑋^∗ satisfying 𝑓_𝑛∈∇_𝑥𝑛 for all 𝑛, we let sep_{𝑓𝑛}{𝑥_𝑛}=inf{⟨𝑥_𝑛−𝑥_𝑚,𝑓_𝑛⟩∶𝑛<𝑚}. (2.1) It is clear that sep_{𝑓𝑛}{𝑥_𝑛}≤sep{𝑥_𝑛}.

Theorem 2.1. A Banach space 𝑋 is semi-UKK if and only if for every 𝜀>0 there exists a 𝛿>0 such that {𝑥_𝑛}⊂𝑆_𝑋𝑥_𝑛⇀𝑥sep_{𝑓𝑛}{𝑥_𝑛}≥𝜀,forsome{𝑓_𝑛}⊂𝑆_𝑋^∗satisfying𝑓_𝑛∈∇_𝑥𝑛,∀𝑛}⇒‖𝑥‖≤1−𝛿. (2.2)

The following theorem shows that our new property is well placed.

Theorem 2.2. The following implication diagram holds: UC⇒UKK⇒KK⇓⇓⇓𝑈-space⇒semi-UKK⇒semi-KK. (2.3)

Remark 2.3. The implication U -space⇒semi-UKK strengthens the result of Vlasov. In fact, it was proved by Vlasov ([5, Theorem 7]) that every uniformly smooth Banach space is semi-KK and by Lau ([4, Corollary 2.5]) that every uniformly smooth Banach space is a 𝑈-space.

2.2. Sufficient Conditions for Weak Normal Structure

Recall that a Banach space 𝑋 has weak normal structure (normal structure, resp.) if for every weakly compact (bounded and closed, resp.) convex subset 𝐶 of 𝑋 containing more than one point there exists a point 𝑥₀∈𝐶 such that sup{‖𝑥₀−𝑧‖∶𝑧∈𝐶}<𝑑𝑖𝑎𝑚𝐶 (see [8]). It is clear that normal structure and weak normal structure coincide whenever the space is reflexive. It was Kirk [9] who proved that if a Banach space 𝑋 has weak normal structure, then every nonexpansive self-mapping defined on a weakly compact convex subset of 𝑋 has a fixed point. In this subsection, we present a new and wider class of Banach spaces with weak normal structure.

Lemma 2.4 (Bollobás [10]). Let 𝑋 be a Banach space, and let 0<𝜀<1. Given 𝑧∈𝐵_𝑋 and ℎ∈𝑆_𝑋^∗ with 1−⟨𝑧,ℎ⟩<𝜀²/4, then there exist 𝑦∈𝑆_𝑋 and 𝑔∈∇_𝑦 such that ‖𝑦−𝑧‖<𝜀 and ‖𝑔−ℎ‖<𝜀.

Theorem 2.5. If a Banach space 𝑋 has the following property:
there are two constants 0<𝜀<1 and 0<𝛿<1 such that {𝑥_𝑛}⊂𝑆_𝑋𝑥_𝑛⇀𝑥⟨𝑥_𝑛−𝑥,𝑓_𝑛⟩≥𝜀,forsome{𝑓_𝑛}⊂𝑆_𝑋^∗satisfying𝑓_𝑛∈∇_𝑥𝑛,∀𝑛}⇒‖𝑥‖≤1−𝛿, (2.4) then 𝑋 has weak normal structure.

Proof. Suppose that 𝑋 does not have weak normal structure. Then there exists a sequence {𝑥_𝑛} in 𝑋 such that the following properties are satisfied (see [11]): (i)diam{𝑥_𝑛}=1;(ii)𝑥_𝑛⇀0; (iii)‖𝑥_𝑛−𝑥‖→1 for all 𝑥∈co{𝑥_𝑛}. In particular, since 0∈co{𝑥_𝑛}, we have ‖𝑥_𝑛‖→1.
We now show that for each 0<𝜀<1 and 0<𝛿<1, there are an element 𝑧∈𝑋 and sequences {𝑧_𝑛}⊂𝑆_𝑋 and {𝑓_𝑛}⊂𝑆_𝑋^∗ such that (i)𝑧_𝑛⇀𝑧; (ii)⟨𝑧_𝑛−𝑧,𝑓_𝑛⟩≥𝜀 and ⟨𝑧_𝑛,𝑓_𝑛⟩=1 for all 𝑛; (iii)‖𝑧‖>1−𝛿. To see this, let 0<𝛿<1 and 0<𝜀<1 be given. We may assume that ‖𝑥₁‖>1−𝛿. For each 𝑛, let 𝑔_𝑛∈∇_{𝑥𝑛−(1/2)𝑥1}. This implies ⟨𝑥_𝑛−(1/2)𝑥₁,𝑔_𝑛⟩=‖𝑥_𝑛−(1/2)𝑥₁‖→1. We observe that 1=⟨𝑥_𝑛−12𝑥₁,𝑔_𝑛⟩=12⟨𝑥_𝑛−𝑥₁,𝑔_𝑛1⟩+2⟨𝑥_𝑛,𝑔_𝑛⟩≤12‖𝑥_𝑛−𝑥₁1‖+2‖𝑥_𝑛‖→1. (2.5) In particular, ⟨𝑥_𝑛−𝑥₁,𝑔_𝑛⟩→1 and ⟨𝑥_𝑛,𝑔_𝑛⟩→1.
By Lemma 2.4, there are sequences {𝑧_𝑛}⊂𝑆_𝑋 and {𝑓_𝑛}⊂𝑆_𝑋^∗ such that 𝑓_𝑛∈∇_𝑧𝑛,∀𝑛,‖𝑧_𝑛−(𝑥_𝑛−𝑥₁)‖→0,‖𝑓_𝑛−𝑔_𝑛‖→0. (2.6) Put 𝑧=−𝑥₁. It is clear that 𝑧_𝑛⇀𝑧. Moreover, we have |⟨𝑧_𝑛−𝑧,𝑓_𝑛⟩−⟨𝑥_𝑛,𝑔_𝑛⟩|≤|⟨𝑧_𝑛−𝑧,𝑓_𝑛⟩−⟨𝑧_𝑛−𝑧,𝑔_𝑛⟩|+|⟨𝑧_𝑛−𝑧,𝑔_𝑛⟩−⟨𝑥_𝑛,𝑔_𝑛⟩|≤‖𝑧_𝑛−𝑧‖‖𝑓_𝑛−𝑔_𝑛‖+‖𝑧_𝑛−(𝑥_𝑛−𝑥₁)‖‖𝑔_𝑛‖→0. (2.7) Consequently, ⟨𝑧_𝑛−𝑧,𝑓_𝑛⟩→1>𝜀.
By discarding terms from the beginning of the sequence {𝑧_𝑛}, we obtain a contradiction with the assumption. This finishes the proof.

Corollary 2.6. A Banach space 𝑋 has weak normal structure if 𝑋 is semi-UKK.

Corollary 2.7 (see [12]). Every wUKK space has weak normal structure. Recall that a Banach space 𝑋 is wUKK if there are two constants 𝜀>0 and 𝛿>0 such that {𝑥_𝑛}⊂𝐵_𝑋(wUKK)∶𝑥_𝑛⇀𝑥sep{𝑥_𝑛}≥𝜀}⇒‖𝑥‖≤1−𝛿. (2.8)

Corollary 2.8 (see [13]). Every 𝑈-space has weak normal structure.

2.3. Stability Results under Taking Finite Direct Sums

In this subsection, we give a necessary and sufficient condition for the direct sum of finitely many Banach spaces to be semi-KK and semi-UKK. Let us recall some definitions.

Let 𝑍 be a finite dimensional normed space (ℝ^𝑁,‖⋅‖_𝑧), which has a monotone norm; that is, ‖(𝑎₁,…,𝑎_𝑁)‖_𝑧≤‖(𝑏₁,…,𝑏_𝑁)‖_𝑧 (2.9) if 0≤𝑎_𝑖≤𝑏_𝑖 for each 𝑖=1,…,𝑁. We write (𝑋₁⊕⋯⊕𝑋_𝑁)_𝑍 for the 𝑍-direct sum of the Banach spaces 𝑋₁,…,𝑋_𝑁 equipped with the norm ‖(𝑥₁,…,𝑥_𝑁)‖=‖(‖𝑥₁‖_𝑋1,…,‖𝑥_𝑁‖_𝑋𝑁)‖_𝑍, (2.10) where 𝑥_𝑖∈𝑋_𝑖 for each 𝑖=1,…,𝑁.

One should notice that in defining (𝑋₁⊕⋯⊕𝑋_𝑁)_𝑍, we only need to know the behavior of the 𝑍-norm on ℝ^𝑁₊. Consequently, we can and do assume that the 𝑍-norm is absolute; that is, ‖(𝑎₁,…,𝑎_𝑁)‖_𝑍=‖(|𝑎₁|,…,|𝑎_𝑁|)‖_𝑍,∀(𝑎₁,…,𝑎_𝑁)∈ℝ^𝑁. (2.11)

The following fact can be proved easily but plays an important role in this paper.

Lemma 2.9. Suppose that 𝑋₁,…,𝑋_𝑁 are Banach spaces. Then each element 𝑓 in the dual (𝑋₁⊕⋯⊕𝑋_𝑁)^∗_𝑍 of the 𝑍-direct sum (𝑋₁⊕⋯⊕𝑋_𝑁)_𝑍 is identified with the element (𝑥^∗₁,…,𝑥^∗_𝑁) in the 𝑍^∗-direct sum (𝑋^∗₁⊕⋯⊕𝑋^∗_𝑁)_𝑍^∗ such that ⟨(𝑥₁,…,𝑥_𝑛),𝑓⟩=⟨𝑥₁,𝑥^∗₁⟩+⋯+⟨𝑥_𝑁,𝑥^∗_𝑁⟩ (2.12) for all (𝑥₁,…,𝑥_𝑁)∈(𝑋₁⊕⋯⊕𝑋_𝑁)_𝑍. Moreover, ‖𝑓‖=‖(‖𝑥^∗₁‖_𝑋∗1,…,‖𝑥^∗_𝑁‖_𝑋∗𝑁)‖_𝑍∗. (2.13)

Recently, Dowling et al. [14] proved the following theorem.

Theorem 2.10. Let 𝑋₁,…,𝑋_𝑁 be Banach spaces. Then (𝑋₁⊕⋯⊕𝑋_𝑁)_𝑍 is KK (UKK, resp.) if and only if for each 1≤𝑗≤𝑁, (1)𝑋_𝑗 is KK (UKK, resp.), and (2)either 𝑋_𝑗 is Schur or 𝑍 is strictly monotone in the 𝑗th coordinate.

Recall that a Banach space 𝑋 is a Schur space if weak and norm sequential convergences coincide in 𝑋, and 𝑍 is strictly monotone in the 𝑗th coordinate if ‖(𝑎₁,…,𝑎_𝑛)‖_𝑍<‖(𝑏₁,…,𝑏_𝑛)‖_𝑍, (2.14) where 0≤𝑎_𝑖≤𝑏_𝑖 for each 𝑖=1,…,𝑛 and 0≤𝑎_𝑗<𝑏_𝑗. Note that by the triangle inequality and the assumption that the 𝑍-norm is absolute, 𝑍 is strictly monotone in the 𝑗th coordinate if and only if ‖(𝑎₁,…,𝑎_𝑛)‖_𝑍<‖(𝑏₁,…,𝑏_𝑛)‖_𝑍, (2.15) where 0≤𝑎_𝑖=𝑏_𝑖 for each 𝑖≠𝑗 and 0=𝑎_𝑗<𝑏_𝑗.

We first define a generalization of Schur spaces.

Definition 2.11. A Banach space 𝑋 is a semi-Schur space if {𝑥_𝑛}⊂𝑋(semi-Schur)∶𝑥_𝑛⇀𝑥{𝑓_𝑛}⊂𝑆_𝑋^∗satisfying𝑓_𝑛∈∇_𝑥𝑛,∀𝑛}⇒‖𝑥_𝑛‖−⟨𝑥,𝑓_𝑛⟩→0. (2.16)

The following two propositions follow easily from the definition of semi-Schur spaces and semi-KK spaces.

Proposition 2.12. A Banach space 𝑋 is semi-Schur if and only if {𝑥_𝑛}⊂𝑆_𝑋𝑥_𝑛⇀𝑥{𝑓_𝑛}⊂𝑆_𝑋^∗satisfying𝑓_𝑛∈∇_𝑥𝑛,∀𝑛}⇒‖𝑥_𝑛‖−⟨𝑥,𝑓_𝑛⟩→0. (2.17)

Proposition 2.13. A Banach space 𝑋 satisfies semi-KK property if and only if {𝑥_𝑛𝑥}⊂𝑋_𝑛⇀𝑥‖𝑥_𝑛‖→‖𝑥‖{𝑓_𝑛}⊂𝑆_𝑋^∗satisfying𝑓_𝑛∈∇_𝑥𝑛,∀𝑛}⇒⟨𝑥,𝑓_𝑛⟩→‖𝑥‖. (2.18)

We say that 𝑍 has property (S-𝑗) where 1≤𝑗≤𝑁 if whenever 𝑎₁,…,𝑎_𝑁,𝑏₁,…,𝑏_𝑁≥0 and 0≤𝑎^′_𝑗<𝑎_𝑗 satisfy ‖(𝑎₁,…,𝑎^𝑗th′_𝑗,…,𝑎_𝑁)‖_𝑍=‖(𝑎₁,…,𝑎_𝑗,…,𝑎_𝑁)‖_𝑍=1,‖(𝑏₁,…,𝑏_𝑗,…,𝑏_𝑁)‖_𝑍∗=𝑎₁𝑏₁+⋯+𝑎_𝑗𝑏_𝑗+⋯+𝑎_𝑁𝑏_𝑁=1, (2.19) it follows that 𝑏_𝑗=0.

Theorem 2.14. Suppose that 𝑋₁,…,𝑋_𝑁 are Banach spaces. Then the direct sum (𝑋₁⊕⋯⊕𝑋_𝑁)_𝑍 is semi-KK if and only if for each 𝑗=1,…,𝑁(a)𝑋_𝑗 is semi-KK and (b)either 𝑋_𝑗 is semi-Schur or 𝑍 has property (S-𝑗).

Proof. Sufficiency. Suppose that there exists 𝑗∈{1,…,𝑁} such that 𝑋_𝑗 is not semi-Schur and 𝑍 does not have property (S-𝑗). For convenience, we may assume that 𝑗=1. Since 𝑋₁ is not semi-Schur, there exist sequences {𝑥¹_𝑛}⊂𝑆_𝑋1, {𝑓¹_𝑛}⊂𝑆_𝑋^∗₁, and a number 𝜀₀>0 such that 𝑥¹_𝑛⇀𝑥¹∈𝐵_𝑋1,𝑓¹_𝑛∈∇_𝑥1𝑛,⟨𝑥¹,𝑓¹_𝑛⟩≤1−𝜀₀,∀𝑛. (2.20) Since 𝑍 does not have property (S-1), there exist numbers 𝑎₁,…,𝑎_𝑁,𝑏₁,…,𝑏_𝑁≥0 such that the following properties are satisfied: (i)‖(0,𝑎₂,…,𝑎_𝑁)‖_𝑍=‖(𝑎₁,𝑎₂,…,𝑎_𝑁)‖_𝑍=1; (ii)‖(𝑏₁,…,𝑏_𝑁)‖_𝑍^∗=𝑎₁𝑏₁+⋯+𝑎_𝑁𝑏_𝑁=1; (iii)𝑎₁>0 and 𝑏₁>0. For 𝑗=2,…,𝑁, let 𝑥^𝑗∈𝑆_𝑋𝑗 and 𝑓^𝑗∈∇_𝑥𝑗. Now we put 𝐱=(𝑎₁𝑥¹,𝑎₂𝑥²,…,𝑎_𝑁𝑥^𝑁𝐱),_𝑛=(𝑎₁𝑥¹_𝑛,𝑎₂𝑥²,…,𝑎_𝑁𝑥^𝑁𝐟),_𝑛=(𝑏₁𝑓¹_𝑛,𝑏₂𝑓²,…,𝑏_𝑁𝑓^𝑁). (2.21) It is clear that {𝐱_𝑛} is a sequence of norm one elements converging weakly to 𝐱 and 𝐟_𝑛∈∇_𝐱𝑛 for all 𝑛. Moreover, by the monotonicity of ‖⋅‖_𝑍, we have ‖𝐱‖=1. Finally, we show that ⟨𝐱,𝐟_𝑛⟩↛1. To see this, we consider ⟨𝐱,𝐟_𝑛⟩=𝑎₁𝑏₁⟨𝑥¹,𝑓¹_𝑛⟩+𝑎₂𝑏₂⟨𝑥²,𝑓²⟩+⋯+𝑎_𝑁𝑏_𝑁⟨𝑥^𝑁,𝑓^𝑁⟩≤(1−𝜀₀)𝑎₁𝑏₁+𝑎₂𝑏₂+⋯+𝑎_𝑁𝑏_𝑁. (2.22) It then follows from 𝑎₁>0 and 𝑏₁>0 that (1−𝜀₀)𝑎₁𝑏₁+⋯+𝑎_𝑁𝑏_𝑁<1. This shows that (𝑋₁⊕⋯⊕𝑋_𝑁)_𝑍 is not semi-KK and hence the first half of the proof is done.
Necessity. Suppose that the conditions (a) and (b) hold. Put 𝐴={𝑗∶𝑋_𝑗isasemi-Schurspace},𝐵={𝑗∶𝑍hasproperty(S-𝑗)}. (2.23) Then, by (b), 𝐴∪𝐵={1,…,𝑁}. Let {𝐱_𝑛} be a sequence of norm one elements in (𝑋₁⊕⋯⊕𝑋_𝑁)_𝑍 converging weakly to a norm one element 𝐱∈(𝑋₁⊕⋯⊕𝑋_𝑁)_𝑍 and {𝐟_𝑛} a sequence of norm one elements in ((𝑋₁⊕⋯⊕𝑋_𝑁)_𝑍)^∗=(𝑋^∗₁⊕⋯⊕𝑋^∗_𝑁)_𝑍^∗ such that 𝑓_𝑛∈∇_𝑥𝑛 for all 𝑛. For convenience, let us write 𝐱=(𝑥¹,…,𝑥^𝑁),𝐱_𝑛=(𝑥¹_𝑛,…,𝑥^𝑁_𝑛),𝐟_𝑛=(𝑓¹_𝑛,…,𝑓^𝑁_𝑛), (2.24) where 𝑥^𝑗,𝑥^𝑗_𝑛∈𝑋_𝑗 and 𝑓^𝑗_𝑛∈𝑋^∗_𝑗 for all 𝑗 and 𝑛. We prove that ⟨𝑥¹,𝑓¹_𝑛⟩+⋯+⟨𝑥^𝑁,𝑓^𝑁_𝑛⟩→1. (2.25) Notice that 1=‖(𝑥¹_𝑛,…,𝑥^𝑁_𝑛)‖=⟨𝐱_𝑛,𝐟_𝑛⟩=⟨𝑥¹_𝑛,𝑓¹_𝑛⟩+⋯+⟨𝑥^𝑁_𝑛,𝑓^𝑁_𝑛⟩≤‖𝑥¹_𝑛‖‖𝑓¹_𝑛‖+⋯+‖𝑥^𝑁_𝑛‖‖𝑓^𝑁_𝑛‖≤‖(𝑥¹_𝑛,…,𝑥^𝑁_𝑛)‖‖(𝑓¹_𝑛,…,𝑓^𝑁_𝑛)‖^∗=1. (2.26) Then ⟨𝑥^𝑗_𝑛,𝑓^𝑗_𝑛⟩=‖𝑥^𝑗_𝑛‖‖𝑓^𝑗_𝑛‖ and ‖𝑥¹_𝑛‖‖𝑓¹_𝑛‖+⋯+‖𝑥^𝑁_𝑛‖‖𝑓^𝑁_𝑛‖=1 (2.27) for all 𝑛. In order to show that (2.25) holds, it suffices to show that ⟨𝑥^𝑗,𝑓^𝑗_𝑛⟩−‖𝑥^𝑗_𝑛‖‖𝑓^𝑗_𝑛‖→0,as𝑛→∞ (2.28) for all 𝑗=1,…,𝑁.
Let us note the following facts: (i)𝑓^𝑗_𝑛/‖𝑓^𝑗_𝑛‖∈∇_𝑥^𝑗_𝑛 provided that 𝑓^𝑗_𝑛≠0; (ii)𝑥^𝑗_𝑛⇀𝑥^𝑗 as 𝑛→∞ for all 𝑗. We first prove that (2.28) holds for all 𝑗∈𝐴. To see this, we note that if 𝑓^𝑗_𝑛=0, then ⟨𝑥^𝑗,𝑓^𝑗_𝑛⟩=‖𝑥^𝑗_𝑛‖‖𝑓^𝑗_𝑛‖=0. Now, we assume that 𝑓^𝑗_𝑛≠0 for all 𝑛. It follows then that 𝑓^𝑗_𝑛/‖𝑓^𝑗_𝑛‖∈∇_𝑥^𝑗_𝑛 and hence from the semi-Schur property of 𝑋_𝑗 that ⟨𝑥^𝑗,𝑓^𝑗_𝑛/‖𝑓^𝑗_𝑛‖⟩−‖𝑥^𝑗_𝑛‖→0.
Passing to a subsequence, we may assume that the following limits: lim_𝑘‖𝑥^𝑗_𝑛𝑘‖,lim_𝑘‖𝑓^𝑗_𝑛𝑘‖exist. (2.29) Notice that ‖(lim_𝑘‖𝑥¹_𝑛𝑘‖,…,lim_𝑘‖𝑥^𝑁_𝑛𝑘‖)‖_𝑍=‖(lim_𝑘‖𝑓¹_𝑛𝑘‖,…,lim_𝑘‖𝑓^𝑁_𝑛𝑘‖)‖_𝑍∗=1,lim_𝑘‖𝑥¹_𝑛𝑘‖lim_𝑘‖𝑓¹_𝑛𝑘‖+⋯+lim_𝑘‖𝑥^𝑁_𝑛𝑘‖lim_𝑘‖𝑓^𝑁_𝑛𝑘‖=1. (2.30) We next show that (2.28) holds for all 𝑗∈𝐵. Let us split the proof into two cases.
Case 1. There exists 𝑗∈𝐵 such that ‖𝑥^𝑗‖<lim_𝑘‖𝑥^𝑗_𝑛𝑘‖. (2.31) In this case, it follows from the property (S-𝑗) that lim_𝑘‖𝑓^𝑗_𝑛𝑘‖=0. (2.32) This implies that ⟨𝑥^𝑗,𝑓^𝑗_𝑛𝑘⟩−‖𝑥^𝑗_𝑛𝑘‖‖𝑓^𝑗_𝑛𝑘‖→0.Case 2. ‖𝑥^𝑗‖=lim_𝑘‖𝑥^𝑗_𝑛𝑘‖ for all 𝑗∈𝐵. Again, if 𝑓^𝑗_𝑛𝑘=0, then ⟨𝑥^𝑗,𝑓^𝑗_𝑛⟩=‖𝑥^𝑗‖‖𝑓^𝑗_𝑛‖=0. Now we may assume that 𝑓^𝑗_𝑛𝑘≠0 for all 𝑘. This implies that 𝑓^𝑗_𝑛𝑘/‖𝑓^𝑗_𝑛𝑘‖∈∇_𝑥^𝑗_𝑛𝑘 and hence it follows from the semi-KK property of 𝑋_𝑗 that ⟨𝑥^𝑗,𝑓^𝑗_𝑛𝑘/‖𝑓^𝑗_𝑛𝑘‖⟩−‖𝑥^𝑗‖→0. In particular, ⟨𝑥^𝑗,𝑓^𝑗_𝑛𝑘⟩−‖𝑥^𝑗_𝑛𝑘‖‖𝑓^𝑗_𝑛𝑘‖→0.From both cases, we have proved that every subsequence of the sequence {⟨𝐱,𝐟_𝑛⟩} has a further subsequence {⟨𝐱,𝐟_𝑛𝑘⟩} such that ⟨𝐱,𝐟_𝑛𝑘⟩→1. Hence ⟨𝐱,𝐟_𝑛⟩→1, as desired.

Using the proof of the preceding theorem and the fact that the property (S-𝑗) is a uniform property, we obtain the following result.

Theorem 2.15. Suppose that 𝑋₁,…,𝑋_𝑁 are Banach spaces. Then (𝑋₁⊕⋯⊕𝑋_𝑁)_𝑍 is semi-UKK if and only if for each 𝑗=1,…,𝑁, (a)𝑋_𝑗 is semi-UKK and (b)either 𝑋_𝑗 is semi-Schur or 𝑍 has property (S-𝑗).

Finally, we use the characterization above and Theorem 2.10 to construct a Banach space which is semi-UKK but not UKK.

Example 2.16 (A Banach space which is semi-UKK but not UKK). Let 𝑍 be a two-dimensional space ℝ² equipped with the norm |(𝛼,𝛽)|=𝛼|𝛽|if|𝛼|≤|𝛽|,₂+𝛽²2|𝛼|,if|𝛼|>|𝛽|. (2.33) It follows that 𝑍 is a (uniformly) smooth space, and its unit sphere consists of (i)two half unit circles: the first one is a right half centered at (1,0) and the second is a left half centered at (−1,0)(ii)two horizontal line segments joining the points (−1,1) and (1,1) and the points (−1,−1) and (1,−1), respectively. Furthermore, 𝑍 has properties (S-1) and (S-2) but is not strictly monotone in the first-coordinate. Let 𝑋=(ℓ₂⊕ℝ)_𝑍. Then 𝑋 is semi-UKK but not UKK. The latter follows since 𝑍 is not strictly monotone in the first coordinate and ℓ₂ does not have the Schur property.

3. 𝑈-Spaces and Uniformly Alternatively Convex or Smooth Spaces

In this section, we discuss some properties of uniformly alternatively convex or smooth spaces which was introduced by Kadets et al. [7].

Definition 3.1. A Banach space 𝑋 is uniformly alternatively convex or smooth if {𝑥_𝑛},{𝑦_𝑛}⊂𝑆_𝑋(UACS)∶‖𝑥_𝑛+𝑦_𝑛‖→2⟨𝑥_𝑛,𝑓_𝑛⟩→1forsome{𝑓_𝑛}⊂𝑆_𝑋^∗}⇒⟨𝑦_𝑛,𝑓_𝑛⟩→1. (3.1)

Remark 3.2. It is not hard to see that 𝑋 is UACS if and only if {𝑥_𝑛},{𝑦_𝑛}⊂𝑆_𝑋‖𝑥_𝑛+𝑦_𝑛‖→2{𝑓_𝑛}⊂𝑆_𝑋^∗satisfying𝑓_𝑛∈∇_𝑥𝑛,∀𝑛}⇒⟨𝑦_𝑛,𝑓_𝑛⟩→1. (3.2) Consequently, 𝑋 is UACS if and only if it is a 𝑈-space.

It is proved by Gao and Lau ([13, Theorem 4.4]) that every UACS space has uniform normal structure which is the same result of Theorem 3.1 of Sirotkin [15]. In fact, this result is recently strengthened by Saejung in [16]. Recall that a Banach space 𝑋 has uniform normal structure if there exists a constant 0<𝑐<1 such that for every bounded closed convex subset 𝐶 of 𝑋 containing more than one point there exists a point 𝑥₀∈𝐶 such that sup{‖𝑥₀−𝑧‖∶𝑧∈𝐶}<𝑐⋅diam𝐶.

Moreover, it was Lau ([4, Theorem 2.4]) who proved that 𝑋 is UACS if and only if its dual space 𝑋^∗ is UACS. By Sirotkin's result ([15, Theorem 2.3]), we have the following theorem.

Theorem 3.3. Let (𝑆,Σ,𝜇) be a complete measure space and 𝑋 be a Banach space. Then the following statements are equivalent: (i)𝐿^𝑝(𝜇,𝑋) is UACS for some (and hence all) 1<𝑝<∞; (ii)𝐿^𝑝(𝜇,𝑋^∗) is UACS for some (and hence all) 1<𝑝<∞; (iii)𝑋 is UACS; (iv)𝑋^∗ is UACS. In particular, if 𝑋 is UACS, then both 𝐿^𝑝(𝜇,𝑋) and 𝐿^𝑝(𝜇,𝑋^∗) have uniform normal structure.

Recall that 𝐿^𝑝(𝜇,𝑋), where 1<𝑝<∞, is the Lebesgue-Bochner function space of 𝜇-equivalence classes of strongly measurable functions 𝑓∶𝑆→𝑋 with ∫_𝑆‖𝑓(𝑡)‖^𝑝𝑑𝜇<∞, endowed with the norm ‖𝑓‖=(∫_𝑆‖𝑓(𝑡)‖^𝑝𝑑𝜇)^1/𝑝 (for more detail, see [17, 18]).

Acknowledgment

The first author is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.