Orthogonal Symmetries and Reflections in Banach Spaces

Let X be a Banach space. We introduce a concept of orthogonal symmetry and reflection in X. 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 spaceX by considering sections over subspaces. The results are then applied to the space C(I) of continuous functions on a compact set I. We obtain some nontrivial symmetries of the unit ball of C(I). We also show that, under natural symmetry conditions, every odd function is orthogonal to every even function inX. We conclude with some suggestions for further investigations.


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  ∈  ( = ( +   )/2), such that, in addition, if we move  and   "away from " an equal distance, the two new points  1 and    1 still satisfy the same conditions; namely, ‖  1 ‖ = ‖ 1 ‖ and  1 and   1 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.

Weak Symmetry in Banach Spaces
Throughout this section,  is a Banach space and  is a closed subspace of .(1) 2

Journal of Mathematics
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.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   () ̸ = 0, in which case the set of weak-reflection points of  is given by {  :   is a weak-refection point of } = 2  () − . ( (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   () ̸ = 0. Conversely, suppose that   () ̸ = 0 for some  ∈ .Let ℎ  ∈   () and let   fl 2ℎ  − . ( 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 ker  denotes the kernel of the linear functional .
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 Corollary 2.4 in [5]; hence  is not weakly symmetric with respect to .An example of a nonreflexive Banach space is the Banach space BV[0, 1] of functions of bounded variation on the closed interval [0, 1] [6].On the other hand, every reflexive subspace  of a Banach space  is proximinal; see Corollary 2.5 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.

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  1 and   1 still satisfy the same conditions.With this in mind, we introduce the following definition of symmetry.
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  ⊥  () = {ℎ  } is a singleton, then we denote ℎ  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 ℎ  fl ( +   )/2 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 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  1 and  2 .Let   denote the reflection with respect to   ,  = 1, 2. For each  ∈ , we have   () −   =   − ; hence It follows that By induction we obtain that Similarly we show that Now 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, weakreflection 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 ‖ − ℎ  ‖ = (, Hence ‖−ℎ  ‖ = (, ) and the proof of Part (i) is complete.
Hence (2ℎ  − ℎ) ∈   () for every ℎ ∈   ().This implies that ℎ  is indeed a center of   ().It follows, by Lemma 11 and since   () is bounded, that ℎ  is the unique center   of   ().Hence the orthogonal projection of  onto  is unique and is given by It follows from Remark 9 that   fl 2 ⊥  () −  is a reflection point of  and that it is unique since, again by Remark 9, (  +)/2 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 ‖(, )‖ ∞ fl max{||, ||}, 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  fl (, ) ∈ , 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  fl {(, ) ∈  : ‖(, )‖ ≤ 1} ; (33) 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  1 ,  2 ,  1 , and  2 in , we have Now, let  = (, ) and  = (, ) be two arbitrary elements in .
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 (, ) ∈ ,   (, ) = {(0, )} .
Hence (42) does not hold and consequently (, ) does not have a reflection point if  ̸ = 0. 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  fl span{, } 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  fl ℎ + ||( −   ) ∈  +  for some ℎ ∈  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 ) ∈   (49) but the weak reflection point Hence   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.(53) 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    fl  − ℎ is a reflection point of  with respect to , that  is symmetric with respect to , and that Hence since    +    = 0.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 dim  ≥ 2. 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 2dimensional subspace of , let  be a nonzero element in , and let  fl span {} fl { :  ∈ } . (57) 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 ℎ  ∈  such that (−ℎ  ) ⊥ .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 .

Symmetries of the Space 𝐶(𝐼)
Let  be a nonempty compact subset of the set of real numbers ,  containing at least two elements, and let () be the 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 (−1, 1), by a 2-dimensional subspace  fl span{, }, where ( ̸ =0) ∈ (−1, 1) and ( ̸ = 0)∈ (−1, 1), then Theorem 23 tells us that the two lines   fl span{} and   fl span{} 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  fl span{, } of () into two identical halves modulo reflection with respect to .
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 ,  ≥ 2. 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, 1 ≤  ≤ ∞, then  is orthogonally symmetric with respect to all subspaces  spanned by the coordinate-vectors,  fl span{  :  ∈ }, where  is a nonempty finite or infinite subset of positive integers and   is the usual coordinate vector with 1 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.

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  +   2 ∈ ,          −   2         =  (, ) .
Proof.Let  ∈ .Then  = ℎ + , for some ℎ ∈  and  ∈ .Now let    fl ℎ − .Then we have, since  ⊥  and  − ℎ =  ∈ , Definition 2. Let  be a closed subspace of a Banach space .An element ℎ  ∈  is called a best approximation from  of an element  ∈  if it satisfies      − ℎ      =  (, ) fl inf {‖ − ℎ‖ : ℎ ∈ } .(2) 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   :  → 2  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 ().
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 .