Some Properties of 𝑙^𝑝(𝐴,𝑋) Spaces

Abstract

We provide a representation of elements of the space 𝑙^𝑝(𝐴,𝑋) for a locally convex space 𝑋 and 1≤𝑝<∞ and determine its continuous dual for normed space 𝑋 and 1<𝑝<∞. In particular, we study the extension and characterization of isometries on 𝑙^𝑝(N,𝑋) space, when 𝑋 is a normed space with an unconditional basis and with a symmetric norm. In addition, we give a simple proof of the main result of G. Ding (2002).

1. Introduction

Let 𝑋 be a Hausdorff locally convex space, let 𝑅 be a family of seminorms on 𝑋 determining its topology and, let 𝐴 be a set. We say that 𝑥 belongs to 𝑙^𝑝(𝐴,𝑋) if and only if

_𝑎∈𝐴[(𝑟∘𝑥)(𝑎)]^𝑝<∞ (1.1) for each 𝑟 in 𝑅, where 1≤𝑝<+∞. Obviously, 𝑙^𝑝(𝐴,𝑋) is a Hausdorff locally convex space with the seminorms (∑_𝑎∈𝐴[(𝑟∘𝑥)(𝑎)]^𝑝)^1/𝑝, for each 𝑟 in 𝑅. When 𝑝=1, Yilmaz in [1] investigated some structural properties of the function space 𝑙¹(𝐴,𝑋) for a Hausdorff locally convex space 𝑋 and obtained the continuous duals of 𝑙¹(𝐴,𝑋) and 𝑐₀(𝐴,𝑋) for a normed space 𝑋. It should be mentioned that [2] is a powerful tool in the detailed investigation of mentioned function spaces.

Let 𝑋 be a real 𝐹 space with the 𝐹-norm ‖𝑥‖ and with an unconditional basis {𝑒_𝑛}. The norm ‖𝑥‖ is called symmetric if, for any permutation {𝑝_𝑛} and for an arbitrary sequence {𝜀_𝑛} of numbers equal either to 1 or to −1, the following equality holds (see [3]):

‖𝑡₁𝑒₁+⋯+𝑡_𝑛𝑒_𝑛+⋯‖=‖𝜀₁𝑡₁𝑒_𝑝1+⋯+𝜀_𝑛𝑡_𝑛𝑒_𝑝𝑛+⋯‖. (1.2)

As follows from the definition of symmetric norms, the operator 𝑉 defined by the formula

𝑉(𝑡₁𝑒₁+⋯+𝑡_𝑛𝑒_𝑛+⋯)=𝜀₁𝑡₁𝑒_𝑝1+⋯+𝜀_𝑛𝑡_𝑛𝑒_𝑝𝑛+⋯ (1.3) is an isometry of the 𝑋 onto itself.

Let 𝐸 and 𝐹 be normed spaces. A mapping 𝑉∶𝐸→𝐹 is called an isometry if ‖𝑉𝑥−𝑉𝑦‖=‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐸 (see, e.g., [4]). The classical Mazur-Ulam theorem in [5] describes the relation between isometry and linearity and states that every onto isometry 𝑉 between two normed spaces with 𝑉(0)=0 is linear. So far, this has been generalized in several directions (see, e.g., [6]). One of them is the study of the isometric extension problem.

Mankiewicz in [7] showed that an isometry which maps a connected subset of a normed space 𝑋 onto an open subset of another normed space 𝑌 can be extended to an affine isometry from 𝑋 to 𝑌. In 1987, Tingley [8] posed the problem of extending an isometry between unit spheres as follows.

Let 𝐸 and 𝐹 be two real Banach spaces. Suppose that 𝑉₀ is a surjective isometry between the two unit spheres 𝑆₁(𝐸) and 𝑆₁(𝐹). Is 𝑉₀ necessarily a restriction of a linear or affine transformation to 𝑆₁(𝐸)?

It is very difficult to answer this question, even in two dimensional cases. In the same paper, Tingley proved that if 𝐸 and 𝐹 are finite-dimensional Banach spaces and 𝑉₀∶𝑆₁(𝐸)→𝑆₁(𝐹) is a surjective isometry, then 𝑉₀(𝑥)=−𝑉₀(−𝑥) for all 𝑥∈𝑆₁(𝐸). In [9], Ding gave an affirmative answer to Tingley problem, when 𝐸 and 𝐹 are Hilbert spaces. In the case 𝐸 and 𝐹 are metric vector spaces, the corresponding extension problem was investigated in [10] and [11]. See [12] for some related results.

In this paper we obtain some structural properties of 𝑙^𝑝(𝐴,𝑋) for 1<𝑝<∞. We mainly provide a representation of the elements of 𝑙^𝑝(𝐴,𝑋) space and obtain continuous duals of 𝑙^𝑝(𝐴,𝑋) for a normed space 𝑋, where 1<𝑝<∞. We also study the extension and characterization of isometries on 𝑙^𝑝(𝐍,𝑋) space, when 𝑋 is a normed space with an unconditional basis and with a symmetric norm. Finally, we give a simple proof of an isometric extension theorem of [9].

2. Some Results of 𝑙^𝑝(𝐴,𝑋) Spaces

In this section we obtain some structural properties of the function space 𝑙^𝑝(𝐴,𝑋)(1≤𝑝<∞). For this purpose, we need a lemma that will be used in the proofs of our main results. We begin with the following well-known result (see [3]).

Lemma 2.1. Let 𝑋 be a real infinite-dimensional 𝐹-space with a basis {𝑒_𝑛} and with a symmetric norm ‖𝑥‖. Then either 𝑋 is a Hilbert space or each isometry is of type (1.3).

Now we are in position to state and prove the main results in this section.

Theorem 2.2. Let 𝑋 be a Hausdorff locally convex space, let 𝑅 be a family of seminorms on 𝑋 determining its topology, and let 𝐴 be a set. Then each 𝑥∈𝑙^𝑝(𝐴,𝑋)(1≤𝑝<∞) is represented by 𝑥=Σ_𝑎∈𝐴(𝐼_𝑎∘𝑥)(𝑎), (2.1) where 𝐼_𝑎∶𝑋→𝑙^𝑝(𝐴,𝑋) is defined by 𝐼_𝑎(𝑡)(𝑏)=𝑡,𝑏=𝑎,0,𝑏≠𝑎,𝑏∈𝐴. (2.2)

Proof. We denote by ℱ the family of all finite subsets of the index set 𝐴. We write 𝑥=Σ_𝑎∈𝐴(𝐼_𝑎∘𝑥)(𝑎) if the net (∑_𝑎∈𝐹(𝐼_𝑎∘𝑥)(𝑎)∶𝐹∈ℱ) converges to 𝑥. Define 𝑆_𝐹(𝑥)=_𝑎∈𝐹(𝐼_𝑎∘𝑥)(𝑎) (2.3) for a finite subset 𝐹 of 𝐴. We must prove that the net (𝑆_𝐹(𝑥)∶ℱ) converges to 𝑥 in 𝑙^𝑝(𝐴,𝑋). By the definition of 𝑆_𝐹(𝑥), we have 𝑆_𝐹(𝑥)(𝑎)=𝑥(𝑎),𝑎∈𝐹,0,𝑎∈𝐴⧵𝐹. (2.4) For 𝑈∈𝒩₀(𝑙^𝑝(𝐴,𝑋)) (where 𝒩₀(𝑙^𝑝(𝐴,𝑋)) denotes a base of neighborhoods of the origin of 𝑙^𝑝(𝐴,𝑋)), there exist 𝜀>0 and 𝑟₁,𝑟₂,…,𝑟_𝑛∈𝑅 such that 𝑈⊇^𝑛_𝑖=1{𝑧∶_𝑎∈𝐴[(𝑟_𝑖∘𝑧)(𝑎)]^𝑝<𝜀}. (2.5) Since ∑_𝑎∈𝐴[(𝑟∘𝑥)(𝑎)]^𝑝<∞ for each 𝑟∈𝑅, then for 𝑖(1≤𝑖≤𝑛), we can find 𝐹_𝑖∈ℱ such that _{𝑎∈𝐴⧵𝐹𝑖}[(𝑟_𝑖∘𝑥)(𝑎)]^𝑝<𝜀. (2.6) Hence, setting 𝐹₀∶=⋃^𝑛_𝑖=1𝐹_𝑖, we have _𝑎∈𝐴[(𝑟_𝑖∘[𝑥−𝑆_𝐹(𝑥)])(𝑎)]^𝑝=_{𝑎∈𝐴⧵𝐹}[(𝑟_𝑖∘𝑥)(𝑎)]^𝑝<𝜀 (2.7) for each 𝐹⊇𝐹₀. This implies 𝑥−𝑆_𝐹(𝑥)∈𝑈. That is 𝑥=Σ_𝑎∈𝐴(𝐼_𝑎∘𝑥)(𝑎).

Remark 2.3. If 𝑋 is a normed space and ‖‖_𝑝 denotes the norm of 𝑙^𝑝(𝐴,𝑋), it holds that ‖𝐼_𝑎(𝑡)‖_𝑝=‖𝑡‖ and ‖𝐼_𝑎‖=1.

Theorem 2.4. Let 𝑋 be a normed space and let 𝐴 be a set. Then for each 𝑓∈𝑙^𝑝(𝐴,𝑋)^′, there exists 𝜓∈𝑙^𝑞(𝐴,𝑋^′) such that 𝑓(𝑥)=_𝑎∈𝐴𝜓(𝑎)[𝑥(𝑎)], (2.8) and 𝑙^𝑝(𝐴,𝑋)^′=𝑙^𝑞(𝐴,𝑋^'), where 1/𝑝+1/𝑞=1 and 1<𝑝<∞.

Proof. By Theorem 2.2, 𝑥∈𝑙^𝑝(𝐴,𝑋) is represented by 𝑥=_𝑎∈𝐴𝐼_𝑎[𝑥(𝑎)]. (2.9) If 𝑓∈𝑙^𝑝(𝐴,𝑋)^′, then 𝑓(𝑥)=_𝑎∈𝐴𝑓∘𝐼_𝑎[𝑥(𝑎)]. (2.10) Define 𝜓∶𝐴→𝑋^' by 𝜓(𝑎)=𝑓∘𝐼_𝑎. Next, we prove that 𝜓∈𝑙^𝑞(𝐴,𝑋^').
Let 𝐹 be an arbitrary finite subset of 𝐴. Since Bishop and Phelps showed that the norm-attainers are dense in 𝐵(𝑋,𝑌) for every Banach space 𝑋 when 𝑌=𝔽 (the symbol 𝔽 denotes a field that can be either ℝ and ℂ), there exists 𝜉(𝑎) in the closed unit ball of 𝑋 such that ‖𝜓(𝑎)‖=|𝜓(𝑎)[𝜉(𝑎)]| (2.11) for each 𝑎∈𝐹. Let us write 𝜓(𝑎)[𝜉(𝑎)] in the polar form, that is, 𝜓(𝑎)[𝜉(𝑎)]=𝑒^𝑖𝜃𝑎|𝜓(𝑎)[𝜉(𝑎)]|, (2.12) and define the function 𝑥 from 𝐴 to 𝑋 by 𝑥(𝑎)=‖𝜓(𝑎)‖^𝑞−1𝑒^{−𝑖𝜃𝑎}𝜉(𝑎),if𝑎∈𝐹and𝜓(𝑎)[𝜉(𝑎)]≠0,0,if𝑎∉𝐹or𝜓(𝑎)[𝜉(𝑎)]=0. (2.13) Obviously, 𝑥∈𝑙^𝑝(𝐴,𝑋). Therefore, for this 𝑥, we have
|𝑓(𝑥)|=|_𝑎∈𝐴𝜓(𝑎)[𝑥(𝑎)]|=|_𝑎∈𝐹‖𝜓(𝑎)‖^𝑞−1𝑒^{−𝑖𝜃𝑎}𝑒^𝑖𝜃𝑎=|𝜓(𝑎)[𝜉(𝑎)]||_𝑎∈𝐹‖𝜓(𝑎)‖^𝑞≤‖𝑓‖‖𝑥‖≤‖𝑓‖(_𝑎∈𝐹(‖𝜓(𝑎)‖^𝑞−1)^𝑝)^1/𝑝=‖𝑓‖(_𝑎∈𝐹‖𝜓(𝑎)‖^𝑞)^1/𝑝. (2.14) Thus (_𝑎∈𝐹‖𝜓(𝑎)‖^𝑞)^1/𝑞≤‖𝑓‖<∞. (2.15) Since 𝐹 is an arbitrary finite subset of 𝐴, we have ‖𝜓‖=(_𝑎∈𝐴‖𝜓(𝑎)‖^𝑞)^1/𝑞≤‖𝑓‖<∞, (2.16) and so 𝜓∈𝑙^𝑞(𝐴,𝑋^'). Moreover, by Hölder inequality, we have |𝑓(𝑥)|≤_𝑎∈𝐴‖𝜓(𝑎)‖‖𝑥(𝑎)‖≤(_𝑎∈𝐴‖𝜓(𝑎)‖^𝑞)^1/𝑞(_𝑎∈𝐴‖𝑥(𝑎)‖^𝑝)^1/𝑝=‖𝜓‖‖𝑥‖, (2.17) from which we get ‖𝑓‖≤‖𝜓‖. (2.18) Combining (2.15) and (2.18) yields ‖𝑓‖=‖𝜓‖. Thus we define a linear isometry 𝑇∶𝑙^𝑝(𝐴,𝑋)^'→𝑙^𝑞(𝐴,𝑋^') with 𝑇𝑓=𝜓. To prove that 𝑇 is surjective. Indeed, for 𝜓∈𝑙^𝑞(𝐴,𝑋^'), there exists 𝑓 defined on 𝑙^𝑝(𝐴,𝑋) such that 𝑓(𝑥)=_𝑎∈𝐴𝜓(𝑎)[𝑥(𝑎)], (2.19) that is, 𝑇𝑓=𝜓. By Mazur-Ulam theorem (see [5]), 𝑇 is a linear isometry from 𝑙^𝑝(𝐴,𝑋)^' onto 𝑙^𝑞(𝐴,𝑋^'), thus 𝑙^𝑝(𝐴,𝑋)^'=𝑙^𝑞(𝐴,𝑋^'). (2.20) The proof of this Theorem is finished.

Theorem 2.5. Let 𝑋 be a normed space with an unconditional basis and with a symmetric norm. Then 𝑙^𝑝(𝐍,𝑋) is also a normed space with an unconditional basis and with a symmetric norm. Moreover, either 𝑙^𝑝(𝐍,𝑋) is a Hilbert space or each isometry is of type (1.3).

Proof. Suppose that {𝑒_𝑘} is an unconditional basis for 𝑋 with ||𝑒_𝑘||=1. Let 𝑒_ik=(𝑜,…,𝑒_𝑘,𝑜,…)_𝑖thplace. (2.21) By Theorem 2.2, if 𝑥(𝑖)=Σ^∞_𝑘=1𝑎_ik𝑒_𝑘 then 𝑥∈𝑙^𝑝(𝐍,𝑋) is represented by 𝑥=_{𝑖∈𝑁𝑘∈𝑁}𝑎_ik𝑒_ik, (2.22) that is {𝑒_ik}_{𝑖∈𝐍,𝑘∈𝐍} is a basis for 𝑙^𝑝(𝐍,𝑋). Note that 𝑥=∑_{𝑖∈𝑁,𝑘∈𝑁}𝑎_ik𝑒_ik is an unconditionally convergent series in 𝑙^𝑝(𝐍,𝑋) and that {𝑒_𝑘} is an unconditional basis for 𝑋. Thus {𝑒_ik}_{𝑖∈𝐍,𝑘∈𝐍} is an unconditional basis for 𝑙^𝑝(𝐍,𝑋). by the definition of norm on 𝑙^𝑝(𝐍,𝑋) and symmetry of norm on 𝑋 it follows that ‖𝑎_ik𝑒_ik‖=(‖𝑎_ik𝑒_ik‖^𝑝)^1/𝑝=(|𝑎_ik|^𝑝)^1/𝑝. (2.23) For any permutation of positive integers {𝑝_ik}, we have ‖𝜀_ik𝑎_ik𝑒_𝑝ik‖=(|𝑎_ik|^𝑝)^1/𝑝, (2.24) thus 𝑙^𝑝(𝐍,𝑋) has symmetric norm. By Lemma 2.1, either 𝑙^𝑝(𝐍,𝑋) is a Hilbert space or each isometry is of type (1.3).

3. A Simple Proof of an Isometric Extension Result in Hilbert Space

Lemma 3.1. Let 𝐸 and 𝐹 be normed spaces and let 𝑉₀ be an isometric operator mapping 𝑆₁(𝐸) into 𝑆₁(𝐹). If for any 𝜆∈𝐑 and any 𝑥,𝑦∈𝑆₁(𝐸),‖𝑉₀𝑥−|𝜆|𝑉₀𝑦‖≤‖𝑥−|𝜆|𝑦‖, (3.1) then 𝑉₀ can be isometrically extended to the whole space. Furthermore, when 𝑉₀ is surjective, 𝑉₀ can be linearly and isometrically extended to the whole space.

Proof. Set 𝑉𝑥=‖𝑥‖𝑉₀(𝑥‖𝑥‖),if𝑥≠0,0,if𝑥=0. (3.2) It is easy to see that ‖𝑉𝑥−𝑉𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐸. In particular, when ‖𝑥‖=‖𝑦‖ either 𝑥 or 𝑦 is zero element, we have ‖𝑉𝑥−𝑉𝑦‖=‖𝑥−𝑦‖. (3.3) Thus, it suffices to prove (3.3) whenever ‖𝑥‖>‖𝑦‖>0.
Suppose, on the contrary, there exist 𝑥₀,𝑦₀∈𝐸 such that ‖𝑥₀‖>‖𝑦₀‖>0 and ‖𝑉𝑥₀−𝑉𝑦₀‖<‖𝑥₀−𝑦₀‖. Define a function on 𝐑 by 𝜑(𝜆)=‖𝑥₀+𝜆(𝑦₀−𝑥₀)‖. (3.4) The facts that 𝜑(𝜆) is a continuous function, 𝜑(1)=‖𝑦₀‖<‖𝑥₀‖ and lim_𝜆→+∞𝜑(𝜆)=+∞ assure that there exists 𝜆₀∈(1,+∞) such that 𝜑(𝜆₀)=‖𝑥₀‖ (by the intermediate value theorem). Let 𝑧₀=𝑥₀+𝜆₀(𝑦₀−𝑥₀). We see that 𝑥₀,𝑦₀, and 𝑧₀ lie on a straight line and ‖𝑧₀‖=‖𝑥₀‖. Hence ‖𝑧₀−𝑥₀‖=‖𝑧₀−𝑦₀‖+‖𝑦₀−𝑥₀‖>‖𝑉𝑧₀−𝑉𝑦₀‖+‖𝑉𝑥₀−𝑉𝑦₀‖≥‖𝑉𝑧₀−𝑉𝑥₀‖=‖𝑧₀−𝑥₀‖, (3.5) a contradiction. Thus 𝑉₀ can be isometrically extended to the whole space, and 𝑉 is an extension of 𝑉₀.
If 𝑉₀ is surjective, then the conclusion follows easily from the Mazur-Ulam Theorem.

Theorem 3.2. Suppose that 𝐸 and 𝐹 are Hilbert spaces and 𝑉₀ is a surjective isometric operator mapping 𝑆₁(𝐸) onto 𝑆₁(𝐹). Then 𝑉₀ can be linearly and isometrically extended to the whole space.

Proof. Since 𝑉₀ is an isometry, we have for all 𝑥,𝑦 in 𝑆₁(𝐸) that ⟨𝑉₀(𝑥)−𝑉₀(𝑦),𝑉₀(𝑥)−𝑉₀(𝑦)⟩=⟨𝑥−𝑦,𝑥−𝑦⟩, (3.6) that is, 2−⟨𝑉₀(𝑥),𝑉₀(𝑦)⟩−⟨𝑉₀(𝑦),𝑉₀(𝑥)⟩=2−⟨𝑥,𝑦⟩−⟨𝑦,𝑥⟩, (3.7) and thus we have ⟨𝑉₀(𝑥),𝑉₀(𝑦)⟩+⟨𝑉₀(𝑦),𝑉₀(𝑥)⟩=⟨𝑥,𝑦⟩+⟨𝑦,𝑥⟩. (3.8) The last equality gives that ⟨𝑉₀(𝑥),𝑉₀(𝑥)⟩−𝜆⟨𝑉₀(𝑥),𝑉₀(𝑦)⟩−𝜆⟨𝑉₀(𝑦),𝑉₀(𝑥)⟩+𝜆²⟨𝑉₀(𝑦),𝑉₀(𝑦)⟩=1+𝜆²−𝜆⟨𝑉₀(𝑥),𝑉₀(𝑦)⟩−𝜆⟨𝑉₀(𝑦),𝑉₀(𝑥)⟩=1+𝜆²−𝜆⟨𝑥,𝑦⟩−𝜆⟨𝑦,𝑥⟩=⟨𝑥,𝑥⟩−𝜆⟨𝑥,𝑦⟩−𝜆⟨𝑦,𝑥⟩+𝜆²⟨𝑦,𝑦⟩. (3.9) Thus ‖𝑉₀(𝑥)−𝜆𝑉₀(𝑦)‖=‖𝑥−𝜆𝑦‖ (3.10) holds for all 𝜆 in 𝐑. Now we can apply Lemma 3.1 to obtain the desired result.

Acknowledgments

The authors of this paper are supported by the NSF of Guangdong Province (no. 7300614).