Abstract and Applied Analysis

Abstract and Applied Analysis / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 562507 | 8 pages | https://doi.org/10.1155/2009/562507

Some Properties of Spaces

Academic Editor: Stevo Stevic
Received27 Jan 2009
Revised26 Mar 2009
Accepted21 Apr 2009
Published14 Jun 2009

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 [1] 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 [2] 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 [3]):

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., [4]). The classical Mazur-Ulam theorem in [5] 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., [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 onto an open subset of another normed space can be extended to an affine isometry from to . In 1987, Tingley [8] 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 [9], 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 [10] and [11]. See [12] 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 [9].

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 [3]).

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 [5]), 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).

References

  1. Y. Yılmaz, “Structural properties of some function spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 59, no. 6, pp. 959–971, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. Y. Yılmaz, “Relative bases in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 2012–2021, 2009. View at: Publisher Site | Google Scholar
  3. S. Rolewicz, Metric Linear Spaces, PWN-Polish Scientific, Warsaw, Poland, D. Reidel, Dordrecht, The Netherlands, 2nd edition, 1984. View at: MathSciNet
  4. R. E. Megginson, An Introduction to Banach Space Theory, vol. 183 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1998. View at: Zentralblatt MATH | MathSciNet
  5. S. Mazur and S. Ulam, “Sur les transformations isometriques d'espaces vectoriels normes,” Comptes Rendus de l'Académie des Sciences, vol. 194, pp. 946–948, 1932. View at: Google Scholar | Zentralblatt MATH
  6. T. M. Rassias, “Properties of isometric mappings,” Journal of Mathematical Analysis and Applications, vol. 235, no. 1, pp. 108–121, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. P. Mankiewicz, “On extension of isometries in normed linear spaces,” Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 20, pp. 367–371, 1972. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  8. D. Tingley, “Isometries of the unit sphere,” Geometriae Dedicata, vol. 22, no. 3, pp. 371–378, 1987. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. G. Ding, “The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space,” Science in China Series A, vol. 45, no. 4, pp. 479–483, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. G. An, “Isometries on unit sphere of (lβn),” Journal of Mathematical Analysis and Applications, vol. 301, no. 1, pp. 249–254, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. X. Fu, “Isometries on the space s,” Acta Mathematica Scientia Series B, vol. 26, no. 3, pp. 502–508, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  12. G. Ding, “The isometric extension problem in the unit spheres of lp(Γ)(p>1) type spaces,” Science in China, vol. 32, no. 11, pp. 991–995, 2002. View at: Google Scholar

Copyright © 2009 Xiaohong Fu and Songxiao Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

515 Views | 307 Downloads | 2 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.