#### Abstract

Let be a Banach space. We introduce a concept of orthogonal symmetry and reflection in . We then establish its relation with the concept of best approximation and investigate its implication on the shape of the unit ball of the Banach space by considering sections over subspaces. The results are then applied to the space of continuous functions on a compact set . We obtain some nontrivial symmetries of the unit ball of . We also show that, under natural symmetry conditions, every odd function is orthogonal to every even function in . We conclude with some suggestions for further investigations.

#### 1. Introduction

When we try to imagine or picture a reflection of a point in a Banach space with respect to, say, a line passing through the origin, we tend to put ourselves in the context of a Euclidean space and think of a “mirror reflection” of the point , that is, of a point that satisfies ; and are equidistant from a point (, such that, in addition, if we move and “away from ” an equal distance, the two new points and still satisfy the same conditions; namely, and and are equidistant from ; see Definition 6. It would be indeed nice and convenient if such a symmetry always existed, given its implications on the geometry of the unit ball of . This would be a valuable asset that could help in establishing results in that otherwise may prove to be difficult. Moreover, there are many results in the literature that rely either directly or indirectly on the geometry of the unit ball of a Banach space. It may be interesting to revisit these and investigate the presence of such symmetries and their consequences. For some recent results in this direction, we refer the readers to [1, 2]. We also note that the investigation of many concepts within Banach spaces is highly active. For some recent results along these lines we refer the readers for examples to [3, 4].

However, we all know that, in general, these “mirror symmetries” do not hold when we work inside a Banach space. So naturally, one should wonder about the types of spaces that do possess such symmetries. Our aim in this paper is to give necessary and sufficient conditions for this to be true and to investigate the consequences of these symmetries on the geometry of the unit ball of the Banach space; see Section 3. Before doing so, we investigate in Section 2 a weaker version of symmetry, which we term as “weak symmetry”; see Definition 1. We will establish a link between weak symmetry and the concept of best approximation (see Definition 2) and establish some of its properties and characteristics. In Section 4, we apply our results to the space of continuous functions on a compact subset of . We are able to obtain some nontrivial geometric regularity for the unit ball of . We also show that every odd function is orthogonal to every even function (Theorem 23). We conclude at the end of the paper by suggesting some directions for further investigations.

#### 2. Weak Symmetry in Banach Spaces

Throughout this section, is a Banach space and is a closed subspace of .

*Definition 1. *Let be a closed subspace of a Banach space . An element is called a weak-reflection point with respect to of the element if If every element in a nonempty subset of admits a weak-reflection point with respect to , then we say that is weakly symmetric with respect to .

We also need the following definition.

*Definition 2. *Let be a closed subspace of a Banach space . An element is called a best approximation from of an element if it satisfies If every element in admits a best approximation from then we say that is proximinal in . The set of all best approximations of an element in is denoted by . The set-valued map is called the metric projection of onto . If is a singleton for every , then is said to be Chebyshev in . If is a singleton for some , then we denote the best approximation of also by .

It is clear that, in general, an element may have more than one weak-reflection point with respect to . In fact, we have the following characterization of uniqueness.

Lemma 3. *Let be a closed subspace of a Banach space . Then we have the following.**(i) An element admits a weak-reflection point with respect to if and only if , in which case the set of weak-reflection points of is given by **(ii) An element admits a unique weak-reflection point with respect to if and only if is a singleton.**(iii) Every element in admits a unique weak-reflection point with respect to if and only if is Chebyshev in .*

*Proof. *It follows directly from Definitions 1 and 2 that if an element admits a weak-reflection point with respect to then hence . Conversely, suppose that for some . Let and let Then one easily checks that is a weak-reflection point of . This completes the proof of Part (i). Parts (ii) and (iii) follow directly from Part (i). This ends the proof.

To explain the use of the term “weak” symmetry in Definition 1, we observe the following.

Lemma 4. *Let be a closed subspace of a Banach space . The following statements are equivalent.**(i) The Banach space is weakly symmetric with respect to **(ii) The subspace is proximinal in .**(iii) For every , there exists an element and a linear functional , where is the continuous dual space of , such that where denotes the kernel of the linear functional .*

*Proof. *(i) (ii): this follows directly from Lemma 3.

(ii) (iii): this follows from Lemma 3 and Theorem1.1 in [5].

It follows from the previous lemma that one can find a Banach space and a closed subspace of such that is not weakly symmetric with respect to . In fact, it is well known that every nonreflexive Banach space admits a nonproximinal hyperplane ; see Corollary2.4 in [5]; hence is not weakly symmetric with respect to . An example of a nonreflexive Banach space is the Banach space of functions of bounded variation on the closed interval [6]. On the other hand, every reflexive subspace of a Banach space is proximinal; see Corollary2.5 in [5]; hence a Banach space is always weakly symmetric with respect to its reflexive subspaces. In particular, every Banach space is weakly symmetric with respect to its finite dimensional subspaces. The following follows from the previous observations.

*Remark 5. *A Banach space is weakly symmetric with respect to all of its closed subspaces if and only if it is reflexive.

#### 3. Orthogonal Symmetry in Banach Spaces

The notions of weak-reflection and weak symmetry that were discussed above are quite different from the “usual” notions of reflection and symmetry. As mentioned above, when we think about a reflection point of a point with respect to a subspace of a Banach space , we tend to visualize (and would like it to be true) points and that are “mirror images” of each other with respect to , that is, that are equidistant from the origin and from , such that if we move them “away from ” an equal distance, the two new points and still satisfy the same conditions. With this in mind, we introduce the following definition of symmetry.

*Definition 6. *Let be a closed subspace of a Banach space . An element is called a reflection point with respect to of the element if If every element in a nonempty subset of admits a reflection point with respect to , then we say that is symmetric with respect to .

Setting in the second condition in Definition 6, we obtain immediately the following.

*Remark 7. *If is a reflection point with respect to of an element , then

We recall the following notion of orthogonality, which was first introduced by Roberts [7].

*Definition 8. *Two elements and of a Banach space are said to be orthogonal (we write ) if, for every , Two nonempty subsets and of are said to be orthogonal, and we write , if every element of is orthogonal to every element of . Given , we write if for every .

If is a closed subspace of a Banach space and , then we denote the set of orthogonal projections of onto by : If is a singleton, then we denote also by .

Note that this definition of orthogonality is symmetric in the sense that For a generalization of this notion of orthogonality to the case of orthogonal sequences in complex Banach spaces, we refer the reader to [8].

The following observation gives an alternative way of thinking about the notion of symmetry introduced in Definition 6 in terms of orthogonality.

*Remark 9. *Let be a closed subspace of a Banach space . An element is a reflection point with respect to of the element if and only if is an orthogonal projection of the point onto , that is, if and only if

To address the question of uniqueness of reflection points (equivalently of orthogonal projections), we recall the following definition.

*Definition 10. *Let be a nonempty subset of a Banach space . An element is called a center of if is symmetric with respect to , that is, if Note that is symmetric with respect to if and only if is symmetric with respect to the origin.

First we take a look at the uniqueness of centers. We have the following.

Lemma 11. *If is a bounded nonempty subset of a Banach space , then admits at most one center.*

*Proof. *Suppose admits two distinct centers and . Let denote the reflection with respect to , . For each , we have ; hence It follows that By induction we obtain that Similarly we show thatNow let be fixed and consider the two sequences of points in defined by It follows from (16) and (17) that This implies that is unbounded, which contradicts the assumption in the lemma. This ends the proof.

The following follows from the proof of the previous lemma.

Corollary 12. *Let be a nonempty subset of a Banach space . If admits two distinct centers, then is unbounded.*

We note that there are various notions of centers in the literature and that they all play an important role in approximation theory and in geometric functional analysis in general.

We have seen above in Section 2 that, in general, weak-reflection points are not unique and that a weak-reflection point of an element with respect to a closed subspace is unique if and only if is a singleton. The situation is quite different in the case of symmetry. The following theorem shows, in particular, that the concept of symmetry is stronger than the concept of weak symmetry and that reflection points and orthogonal projections, when they exist, are unique.

Theorem 13. *Let be a closed subspace of a Banach space . Then we have the following.**(i) If is symmetric with respect to , then is weakly symmetric with respect to .**(ii) Every element admits at most one orthogonal projection onto . Moreover, if an element admits an orthogonal projection onto then the set of best approximations admits a unique center , admits a unique reflection point with respect to , and the reflection point of is given by **(iii) Every element in admits at most one reflection point with respect to . Moreover, if an element admits a reflection point with respect to , then set of best approximations admits a unique center , admits a unique orthogonal projection onto , and the orthogonal projection of onto is given by *

*Proof. *(i) Suppose that is symmetric with respect to . Let be given and let be a reflection point of . We will show that is a weak-reflection point of . Indeed we have, by Remark 9, Hence it suffices to show that . For every we have, since , It follows, since is a convex function of , that Hence and the proof of Part (i) is complete.

(ii) Let be fixed and suppose that admits an orthogonal projection onto ; hence Then, by Remark 9, admits a reflection point and hence, by Part (i) and Lemma 3, . We now show that is a center of . We need to show that , for every . Let be given. It follows, since and since , that Hence for every . This implies that is indeed a center of . It follows, by Lemma 11 and since is bounded, that is the unique center of . Hence the orthogonal projection of onto is unique and is given by It follows from Remark 9 that is a reflection point of and that it is unique since, again by Remark 9, must be an orthogonal projection of onto . This completes the proof of Part (ii).

(iii) This follows directly from Part (ii) and Remark 9. This completes the proof of the theorem.

The following follows immediately from Theorem 13.

Corollary 14. *A Banach space is symmetric with respect to a closed subspace if and only if every element admits an orthogonal projection onto .*

*Remark 15. *For every element , is always a convex subset of the subspace of ; hence the center and the orthogonal projection always belong to whenever they exist.

It is important to note that does not have to be Chebyshev in in order for to be symmetric with respect to . Indeed we have the following.

*Example 16. *Consider the Banach space where , and let Then we have, for every , hence is proximinal but not Chebyshev in . Also, one can easily verify that is symmetric with respect to and that, for every , the unique center of and the unique reflection point of are given by

We also point out that the existence of a center of does not guarantee the existence of a reflection point (or of an orthogonal projection) of . Indeed we have the following.

*Example 17. *Let and be as in Example 16 but, instead of , consider the norm Note that is the Minkowski functional associated with the closed convex and symmetric subset of given by hence indeed it is a norm on . To verify this, all we need to check is the triangle inequality as the other properties are trivial: First, recall that we always have, for all , hence, for all , Also, for all , , , and in , we have Now, let and be two arbitrary elements in .

If , then we have, by (35) and (36),

If , then one of the factors is positive and one is negative, say It follows, by (35), that where the last inequality follows from the fact that and we can also easily obtain a similar inequality for . Going back to our example, it is clear that, for every , Hence is Chebyshev in and, for each , admits a unique center, namely, . It follows from Theorem 13 that if an element in admitted a reflection point with respect to , then we would have and, by Remark 9, we would also have or, in other words, But clearly this is not possible if . Indeed, in this case and while Hence (42) does not hold and consequently does not have a reflection point if .

We now highlight the effect of symmetry on the geometry of the unit ball of the Banach space by looking at its 1-dimensional sections over the subspace , that is, by looking at sections of by subspaces of in which is of codimension 1.

Theorem 18. *If a Banach space is symmetric with respect to a proper closed subspace , then its unit ball is symmetric with respect to . Moreover, given any point , the subspace divides the unit ball of into two halves which are identical modulo reflection with respect to . The two halves of are given by where is the unique center of .*

*Proof. *Since is symmetric with respect to , it follows immediately from Remark 7 that is symmetric with respect to . Now let be given. It follows from Remark 9 and Theorem 13 that where is the reflection point of . If , then we have Hence . Therefore is symmetric with respect to . Since is a hyperplane in , divides into two halves, and . Clearly, if for some and , then the reflection point of is given by This follows from Remark 9 and from the fact that . Hence is the reflection of . Similarly we show that is the reflection of . This ends the proof.

One should note that the orthogonal symmetry of the unit ball given by Theorem 18 is not always true in Banach spaces. It follows from Definition 6 and Remark 9 that it holds if and only if the unit ball of is symmetric with respect to in the sense of Definition 6. In particular, it is possible for a Banach space to be weakly symmetric with respect to a closed subspace , while the unit ball is neither symmetric nor weakly symmetric with respect to . Indeed we have the following.

*Example 19. *Consider again the example given in Example 17 above. Then is weakly symmetric with respect to , since is Chebyshev in . It is easy to verify that but the weak reflection pointHence is neither weakly symmetric nor symmetric with respect to .

We now consider the case where is the direct sum of two mutually orthogonal subspaces.

Theorem 20. *Suppose that a Banach space is the direct sum of two mutually orthogonal closed subspaces and : Then is symmetric with respect to both and and we have, for each , where and are the reflections of with respect to and , respectively.*

*Proof. *Let . Then , for some and . Now let . Then we have, since and , Hence, by Remark 9, is a reflection point of with respect to and, by Theorem 13, Since was arbitrary in , is symmetric with respect to . Similarly we show that is a reflection point of with respect to , that is symmetric with respect to , and that Hence since . This ends the proof.

The reader may be wondering about the situation when a Banach space is symmetric with respect to all of its closed subspaces. For one thing, it follows from Remark 5 and Theorem 13 that must be reflexive. Also it is clear that this condition cannot be sufficient, as symmetry and weak symmetry are not equivalent. It turns out that the situation is possible only if is isometric to a Hilbert space. We have the following stronger result.

Theorem 21. *Let be a Banach space satisfying . Then is symmetric with respect to all of its 1-dimensional closed subspaces if and only if it is isometric to a Hilbert space.*

*Proof. *Suppose is symmetric with respect to all of its 1-dimensional closed subspaces. Let be an arbitrary 2-dimensional subspace of , let be a nonzero element in , and let We first show that admits a nonzero orthogonal element in . Let be linearly independent of . Since is 1-dimensional, is symmetric with respect to by assumption. Therefore there exists such that . It follows, since and , that Therefore we have proved that in every 2-dimensional subspace of every nonzero element admits a nonzero orthogonal element. It follows by a theorem of James [9] that is isometric to a Hilbert space. This ends the proof.

*A Final Remark*

*Remark 22. *Since, by Remark 7, the concept of symmetry given by Definition 6 is norm-preserving, it follows immediately that in Theorem 18 we may replace the unit balls by balls or spheres of radius .

#### 4. Symmetries of the Space

Let be a nonempty compact subset of the set of real numbers , containing at least two elements, and let be the Banach space of real-valued continuous functions defined on , where the norm of an element is given by Given and , we let be the element of defined by Note that whenever . If is symmetric with respect to some , we denote by and the closed subspaces of given by When , we let

With these definitions in mind, we have the following.

Theorem 23. *If is symmetric with respect to some , then and is the direct sum of and : hence is symmetric with respect to both and . Moreover, for each , we have where and are the reflections of with respect to and , respectively.*

*Proof. *First we prove that is orthogonal to : Let and be given. Since is an even function, is an odd function, and ; we have for every . Hence Now, let be given. Then Note that It follows that and, since , that The theorem now follows from Theorem 20. This completes the proof.

Note that Theorems 23 and 18 shed some light on the shape and geometry of the unit ball of the Banach space , which otherwise are not easy to visualize. In general the only guaranteed symmetry in a Banach space is symmetry with respect to the origin. But here, with the help of Theorems 23 and 18, one can actually visualize sections of the unit ball of . Indeed, if we consider, for example, a 2-dimensional section of the unit ball, say of , by a 2-dimensional subspace , where (≠0) and (≠0), then Theorem 23 tells us that the two lines and divide the 2-dimensional unit sphere of into four identical quarters modulo reflections with respect to and Note that this characteristic is not always true in Banach spaces. More generally, the following follows immediately from Theorems 23 and 18.

Corollary 24. *Let be symmetric with respect to some and let represent one of the two subspaces and . Then, for every , divides each of the unit sphere and the unit ball of the subspace of into two identical halves modulo reflection with respect to . The two halves of and are given by, respectively, *

We note that there are symmetries in other than those mentioned above. Indeed, let denote the cardinality of and, for each , let be defined by

Then we have the following.

Proposition 25. *Let be symmetric with respect to some , , and let . Choose any two elements and from the unit sphere of such that Now let , , and . Then Moreover, the hyperplanes and of together divide each of the unit sphere and the unit ball of into four identical quarters modulo reflections with respect to and . The four quarters of the unit sphere and the unit ball of are given by, respectively, *

*Proof. *First note that, since , one can indeed choose two elements and from the unit sphere of such that Let , , and let . Then is the disjoint union of , , and . Note that, since , and are nonempty. Since is symmetric with respect to and , it follows that First, we show that , . Let be fixed. Then we have, since , , and have disjoint supports, Since is nonempty and symmetric with respect to , so is the closure of . It follows from Theorem 23 that and hence the restrictions of and to are orthogonal in . This implies that It follows that hence and consequently , as required.

To prove that , it suffices to show that . But this follows directly from the fact that and .

Note that and are linearly independent. Therefore, since , Also It follows from Theorem 18 that the hyperplanes and of together divide the unit sphere and the unit ball of into four identical quarters modulo reflections with respect to and . The four quarters of the unit sphere of are Similarly we obtain the four quarters of the unit sphere of .

We conclude with some suggestions for further investigations:(1)With Theorem 18 in mind, one may look at -dimensional or infinite-dimensional sections of the unit ball of the Banach space over the subspace , . For this one may need to consider the extension of the notion of orthogonality introduced in [8].(2)Instead of a general Banach space, we may consider specific Banach spaces and identify the subspaces such that is symmetric with respect to . For instance, it is not difficult to show that if is the usual sequence-space, , then is orthogonally symmetric with respect to all subspaces spanned by the coordinate-vectors, , where is a nonempty finite or infinite subset of positive integers and is the usual coordinate vector with in the th-place and zeros elsewhere. Still, are there other subspaces with respect to which is symmetric?(3)As mentioned above in Introduction, there are many results in the literature that rely either directly or indirectly on the geometry of the unit ball of a Banach space. It may be interesting to revisit these and investigate the presence of orthogonal symmetries and their consequences.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.