#### Abstract

We provide a representation of elements of the space for a locally convex space and and determine its continuous dual for normed space and . In particular, we study the extension and characterization of isometries on space, when 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 be a Hausdorff locally convex space, let be a family of seminorms on determining its topology and, let be a set. We say that belongs to if and only if

for each in , where . Obviously, is a Hausdorff locally convex space with the seminorms for each in . When , Yilmaz in  investigated some structural properties of the function space for a Hausdorff locally convex space and obtained the continuous duals of and for a normed space . It should be mentioned that  is a powerful tool in the detailed investigation of mentioned function spaces.

Let be a real space with the -norm and with an unconditional basis . The norm is called symmetric if, for any permutation and for an arbitrary sequence of numbers equal either to 1 or to −1, the following equality holds (see ):

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

is an isometry of the onto itself.

Let and be normed spaces. A mapping is called an isometry if for all (see, e.g., ). The classical Mazur-Ulam theorem in  describes the relation between isometry and linearity and states that every onto isometry between two normed spaces with is linear. So far, this has been generalized in several directions (see, e.g., ). One of them is the study of the isometric extension problem.

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

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

It is very difficult to answer this question, even in two dimensional cases. In the same paper, Tingley proved that if and are finite-dimensional Banach spaces and is a surjective isometry, then for all . In , Ding gave an affirmative answer to Tingley problem, when and are Hilbert spaces. In the case and are metric vector spaces, the corresponding extension problem was investigated in  and . See  for some related results.

In this paper we obtain some structural properties of for . We mainly provide a representation of the elements of space and obtain continuous duals of for a normed space , where . We also study the extension and characterization of isometries on space, when 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 .

#### 2. Some Results of Spaces

In this section we obtain some structural properties of the function space . 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 ).

Lemma 2.1. Let be a real infinite-dimensional with a basis and with a symmetric norm . Then either 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 be a Hausdorff locally convex space, let be a family of seminorms on determining its topology, and let be a set. Then each is represented by where is defined by

Proof. We denote by the family of all finite subsets of the index set . We write if the net converges to . Define for a finite subset of . We must prove that the net converges to in . By the definition of , we have For (where denotes a base of neighborhoods of the origin of ), there exist and such that Since for each , then for , we can find such that Hence, setting , we have for each . This implies . That is

Remark 2.3. If is a normed space and denotes the norm of , it holds that and .

Theorem 2.4. Let be a normed space and let be a set. Then for each , there exists such that and , where and .

Proof. By Theorem 2.2, is represented by If , then Define by . Next, we prove that
Let be an arbitrary finite subset of . Since Bishop and Phelps showed that the norm-attainers are dense in for every Banach space when (the symbol denotes a field that can be either and ), there exists in the closed unit ball of such that for each . Let us write in the polar form, that is, and define the function from to by Obviously, . Therefore, for this , we have
Thus Since is an arbitrary finite subset of , we have and so . Moreover, by Hölder inequality, we have from which we get Combining (2.15) and (2.18) yields . Thus we define a linear isometry with . To prove that is surjective. Indeed, for , there exists defined on such that that is, . By Mazur-Ulam theorem (see ), is a linear isometry from onto , thus The proof of this Theorem is finished.

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

Proof. Suppose that is an unconditional basis for with . Let By Theorem 2.2, if then is represented by that is is a basis for . Note that is an unconditionally convergent series in and that is an unconditional basis for . Thus is an unconditional basis for . by the definition of norm on and symmetry of norm on it follows that For any permutation of positive integers , we have thus has symmetric norm. By Lemma 2.1, either 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 and be normed spaces and let be an isometric operator mapping into . If for any and any then can be isometrically extended to the whole space. Furthermore, when is surjective, can be linearly and isometrically extended to the whole space.

Proof. Set It is easy to see that for all . In particular, when either or is zero element, we have Thus, it suffices to prove (3.3) whenever
Suppose, on the contrary, there exist such that and Define a function on by The facts that is a continuous function, and assure that there exists such that (by the intermediate value theorem). Let . We see that , and lie on a straight line and . Hence a contradiction. Thus can be isometrically extended to the whole space, and is an extension of .
If is surjective, then the conclusion follows easily from the Mazur-Ulam Theorem.

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

Proof. Since is an isometry, we have for all in that that is, and thus we have The last equality gives that Thus holds for all in . 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).