The non-compact normed space of norms on a finite-dimensional Banach space
aa r X i v : . [ m a t h . M G ] O c t THE NON-COMPACT NORMED SPACE OF NORMS ON AFINITE-DIMENSIONAL BANACH SPACE
APOORVA KHARE
Abstract.
We discuss a new pseudometric on the space of all norms on a finite-dimensional vectorspace (or free module) F k , with F the real, complex, or quaternion numbers. This metric arisesfrom the Lipschitz-equivalence of all norms on F k , and seems to be unexplored in the literature. Weinitiate the study of the associated quotient metric space, and show that it is complete, connected,and non-compact. In particular, the new topology is strictly coarser than that of the Banach–Mazurcompactum. For example, for each k > {k·k p : p ∈ [1 , ∞ ] } maps isometricallyand monotonically to [0 , log k ] (or [0 ,
1] by scaling the norm), again unlike in the Banach–Mazurcompactum.Our analysis goes through embedding the above quotient space into a normed space, and revealsan implicit functorial construction of function spaces with diameter norms (as well as a variant ofthe distortion). In particular, we realize the above quotient space of norms as a normed space.We next study the parallel setting of the – also hitherto unexplored – metric space S ([ n ]) of allmetrics on a finite set of n elements, revealing the connection between log-distortion and diameternorms. In particular, we show that S ([ n ]) is also a normed space. We demonstrate embeddingsof equivalence classes of finite metric spaces (parallel to the Gromov–Hausdorff setting), as well asof S ([ n − S ([ n ]). We conclude by discussing extensions to norms on an arbitrary Banachspace and to discrete metrics on any set, as well as some questions in both settings above. Contents
1. The metric space of norms: definition and main result 12. Diameter norms and an endofunctor 43. Distances between p -norms; proof of the main result 84. The normed space of metrics on a finite set 115. Norms on arbitrary Banach spaces; concluding remarks and questions 14References 161. The metric space of norms: definition and main result
It is a folklore result that all norms on a finite-dimensional (real or complex) normed linear spaceare topologically equivalent – i.e., Lipschitz – with respect to one another. The space of norms haslong been studied using the Banach–Mazur pseudometric. Our goal in this work is to explain anew, strictly coarser topology on the space of norms on R k – the equivalence classes are now givenby dilations – which leads us to a non-compact quotient metric space S k ( R ). The Banach–Mazurcontinuum turns out to be a (compact) quotient of this space; we will see for instance that the twotopologies agree on the sets of p -norms for p ∈ [1 ,
2] and [2 , ∞ ], but not for p ∈ [1 , ∞ ].We then study the space S k ( R ) by working in a broader context of function spaces with diameternorms. As we explain below, (a) this function space construction is functorial and applies in aspecial case to the setting of S k ( R ); (b) we deduce that the metric on S k ( R ) is in fact a norm; and Date : November 1, 2018.
Key words and phrases.
Norms, metric space, diameter seminorm, distortion, Banach–Mazur compactum,Gromov–Hausdorff distance. (c) we interpret this new metric/norm through a variant of the distortion between metric spaces.(d) We also apply this functorial framework to deduce similar structural properties of the metricspace of all metrics on each finite set (see Section 4), and of families of norms on an arbitrary Banachspace. Hence the present paper, as we were surprisingly unable to find these results recorded inthe literature.We begin by setting notation. Fix an integer k > F over R that is adivision ring (equivalently, F lacks zerodivisors), that is, F = R , C , or H . We will denote dim R F by d ; also let 1 , i (and j, k ) denote the standard R -basis elements in C (or H ). Recall the conjugationoperation α α ∗ in F , which is the unique R -linear anti-involution that fixes 1 and acts asmultiplication by − { i, j, k } ∩ F . Now a norm on F k is a function N : F k → R satisfying thefollowing properties for all x , y ∈ F k and α ∈ F :(1) Positivity: N ( x ) >
0, with equality if and only if x = 0.(2) Homogeneity: N ( α x ) = | α | N ( x ), where | α | := √ αα ∗ will be termed the absolute value of α ∈ F (to distinguish it from the norm). Recall | · | is multiplicative on F .(3) Sub-additivity: N ( x + y ) N ( x ) + N ( y ).Denote the space of all norms on F k by N ( F k ). Here are some basic properties of this space,some of which are used below. Lemma 1.1.
For F = R , C , or H , and an integer k > , the space N ( F k ) is closed under thefollowing operations: • Addition. • Multiplication by R > . (Thus, N ( F k ) is a convex cone.) • Pointwise limits, as long as the limiting function is positive except at . • Pre-composing by continuous additive maps A : F k → F k with trivial kernel – equivalently,real-linear maps A ∈ GL dk ( R ) under some identification of F k with R dk . In other words, A ∈ GL dk ( R ) , N ∈ N ( F k ) = ⇒ ( x N ( A x )) ∈ N ( F k ) . Notice that there are also other ways to construct norms, e.g. adding norms on subspaces of F k to a given norm in N ( F k ). See Equation (3.7) below for an example.The next result is standard for F = R , and easily extends to C or H . Lemma 1.2.
All norms in N ( F k ) are Lipschitz-equivalent, i.e., for any two norms N, N ′ ∈ N ( F k ) there exist constants < m M such that m · N ( x ) N ′ ( x ) M · N ( x ) , ∀ x ∈ F k . (1.3)For completeness we include a proof-sketch, in a slightly more general setting that is relevant tothe present work (below). Proof.
This follows from two observations – (i) F k is a finite-dimensional vector space over R , and(ii) N ( F k ) ⊂ N ( R dk ) under the R -linear homeomorphism F ∼ = R d , where d = dim R F . Theseobservations reduce the situation to the well-known case of F = R , where we remind that the resultagain follows from two observations: (a) Every norm is bounded above by a positive multiple of thesup-norm; one obtains this by working on the boundary ∂C of the cube C = [ − , d . (b) Given acompact metric space ( X, d ) (such as X = ∂C ), all continuous maps in C ( X, (0 , ∞ )) are pairwise‘Lipschitz equivalent’. A more general statement is that given any set X , all set maps : X → (0 , ∞ )with image bounded away from 0 and ∞ are pairwise ‘Lipschitz equivalent’. (cid:3) The preceding result is well-known. A less well-known result (which we were unable to find inthe literature) is the following construction, which was mentioned by V.G. Drinfeld in a lecture atthe University of Chicago in the early 2000s:
HE NON-COMPACT NORMED SPACE OF NORMS ON A FINITE-DIMENSIONAL BANACH SPACE 3
Proposition 1.4.
Say that two norms
N, N ′ on F k are equivalent , written N ∼ N ′ , if N ′ ≡ αN for some positive real number α . Then the space S k ( R ) := N ( R k ) / ∼ is a metric space, with metric d S k ( R ) ([ N ] , [ N ′ ]) := log( M N,N ′ /m N,N ′ ) , (1.5) where M = M N,N ′ , m = m N,N ′ denote the largest and smallest Lipschitz constants respectively, inEquation (1.3) . Once formulated, the result is shown in a straightforward manner. Informally, ‘the space ofmetrics forms a metric space’. The reader may also recognize (1.5) as a variant of the log-distortion between metrics; see Section 4 for more on this.
Remark 1.6.
We now record the connection between the space S k ( F ) and the Banach–Mazurcompactum (see e.g. [16]), where two k -dimensional Banach spaces U, V over F = R or C havedistance log inf {k T k · k T − k : T ∈ GL ( U, V ) } . Now if two norms are proportional, and thus represent the same point in S k ( R ), then they also dothe same in the Banach–Mazur compactum: just note that k T k · k T − k = 1 for T the identity mapon R k . It follows that the Banach–Mazur compactum is a quotient of S k ( R ). One consequence ofour main result (Theorem 1.8 below) is that the topology in S k ( R ) is strictly coarser.The space S k ( F ) does not seem to be known to experts, nor is it defined or analyzed in theliterature; we initiate its study in the present work. In light of the preceding remark, we hope thatsubsequent, continued analysis of S k ( F ) will also yield additional information about the Banach–Mazur compactum.We begin with an immediate consequence of the above observation that N ( F k ) ⊂ N ( R dk ): in asense, it suffices to work with F = R (as we do below): Corollary 1.7.
The space S k ( F ) := N ( F k ) / ∼ is a closed metric subspace of S dk ( R ) , with commonmetric given by (1.5) .Proof. We show that S k ( F ) is closed in S dk ( R ). Suppose [ N l ] → [ N ] in S dk ( R ), with N l ∈ N ( F k ) ∀ l and N ∈ N ( R dk ). Without loss of generality, rescale the N l and assume via (1.3) that N l N : F k \ { } → [1 , M l ] , ∀ l > M l → l → ∞ . But then, N l → N pointwise on F k . In particular, given α ∈ F and nonzero x ∈ F k , N ( α x ) N ( x ) = lim l →∞ N l ( α x ) N l ( x ) = | α | , and from this it follows that N ∈ N ( F k ) as desired. (cid:3) We now state the main result of the present work (with the caveat that this result is placed in amore general, functorial framework introduced in the following section). It implies as a consequencethat d S k ( F ) is not just a metric, but also a norm: Theorem 1.8.
For F = R , C , H , the space S k ( F ) is a complete, path-connected metric subspace ofa real Banach space. It is a singleton set for k = 1 , and unbounded for k > . (In less formal terms: ‘the space of norms lies in a normed linear space.’) A second consequence isthat in dimensions two and higher, the space of equivalence classes of norms is not compact. APOORVA KHARE
Acknowledgments.
This work is partially supported by Ramanujan Fellowship SB/S2/RJN-121/2017 and MATRICS grant MTR/2017/000295 from SERB (Govt. of India), by grant F.510/25/CAS-II/2018(SAP-I) from UGC (Govt. of India), and by a Young Investigator Award from the InfosysFoundation. I thank Terence Tao for valuable discussions – especially about the final section –as well as Gautam Bharali, Javier Cabello S´anchez, Hariharan Narayanan, and M. Amin Sofi forseveral useful comments that helped improve previous versions of this manuscript. Part of thiswork was carried out during a visit to UCLA, to which I am thankful for its hospitality.2.
Diameter norms and an endofunctor
The goal of this section and the next is (to proceed toward) proving Theorem 1.8. While it ispossible to provide a direct proof, our construction of a family of Banach spaces that each encompass S k ( F ) (as asserted in Theorem 1.8) turns out to be part of a broader functorial setting – which wewill use below in more than one setting. Thus we explain this setting in the present section, andcomplete the proof in Section 3.The most primitive framework we consider is that of an abelian topological semigroup ( G , + , d G )with an associative, commutative binary operation + : G × G → G and a translation-invariantmetric d G , i.e., d G ( x + z, y + z ) = d G ( x, y ) , ∀ x, y, z ∈ G . (2.1)Notice that in such a semigroup, one does not necessarily have inverses (i.e., ‘negatives’) or theidentity element e = 0 G . However, the following is easily shown (see e.g. [11] for details). • the semigroup always has at most one idempotent 2 e = e ; • if such an e exists, it is the unique identity element in (the monoid) G ; • if G does not contain an idempotent, one can formally attach such an idempotent e withmetric d G ( e, z ) := d G ( z, z ) , d G ( e, e ) = 0, and this creates the unique smallest monoid (withtranslation-invariant metric) containing G .Examples of such semigroups G abound in the literature, the most prominent being Banachspaces. However, there are several ‘intermediate’ classes of such abelian semigroups, includingmonoids (i.e. ‘ N ∪ { } -modules’), groups (i.e. Z -modules), torsion-free divisible groups (i.e. Q -modules), and normed linear spaces (i.e. R -modules). More generally, one can consider metric R -modules, where R ⊂ R is a unital subring. As a further variant, one has the subclass of R -normed R -modules (see Definition 2.2).Our first goal is to show that the seminorm defined in Theorem 2.3 below endows all G -valuedfunction spaces with the structure of a (pseudo-)metric. In fact, we show that this holds on a morestructural level. For instance, it is clear that if G is a monoid, then so is the corresponding functionspace; and this holds for the finer structures mentioned above as well. What we also show is thatthe function space construction is also compatible with ‘good’ homomorphisms.A systematic way to carry out this bookkeeping is that each of the above classes of semigroups isin fact a category, and the function space construction is a covariant endofunctor for each of thesecategories. This is now explained. Definition 2.2.
Let R ⊂ R denote any unital subring.(1) Let Semi denote the category of abelian topological semigroups ( G , + , d G ) with translation-invariant metric d G , as above, and whose morphisms are semigroup homomorphisms thatare Lipschitz.(2) Let Mon denote the full subcategory of
Semi , whose objects are monoids.(3) Let R - Mod denote the subcategory of
Semi , whose objects are R -modules such that multi-plication by scalars in R are Lipschitz maps, and whose morphisms are Lipschitz R -modulemaps.(4) Let R - NMod denote the full subcategory of R - Mod , whose objects are R -normed R -modules G . In other words, d G (0 , ra ) = | r | d G (0 , a ) for all r ∈ R and a ∈ G . HE NON-COMPACT NORMED SPACE OF NORMS ON A FINITE-DIMENSIONAL BANACH SPACE 5 (5) Let R - Mod ⊂ R - Mod (note the unconventional notation) denote the full subcategory of R - Mod whose objects are complete metric spaces; and similarly define R - NMod ⊂ R - NMod .For each of these classes of objects, we now prove that the following function space constructionis functorial:
Theorem 2.3.
Suppose C is one of the categories in Definition 2.2 (i.e., Semi , Mon , R - Mod , . . . , R - NMod ). Given a set X and an object M ∈ C , let F b ( X, M ) denote the set of bounded functions f : X → M , and define d ( f, g ) := sup x,x ′ ∈ X d M ( f ( x ) + g ( x ′ ) , f ( x ′ ) + g ( x )) . (2.4) For every set X that is not a singleton: (1) d is a pseudometric on F b ( X, M ) . (2) If one defines f ∼ g to mean d ( f, g ) = 0 , then ∼ is an equivalence relation, and the quotientspace assignment M F b ( X, M ) / ∼ is a covariant isometric endofunctor of the category C .Here, we term a functor F : C → C ′ to be isometric if: (a) all Hom -spaces in C , C ′ are metricspaces, and (b) F : Hom( C , C ) → Hom( F ( C ) , F ( C )) is an isometry for all objects C , C ∈ C . Remark 2.5.
As we show in Section 3, the connection to norms arises from the fact that the space S k ( F ) ⊂ S dk ( R ) embeds into F b ( X, R ) / ∼ for some compact subset X ⊂ F k . Thus by Theorem 2.3for C = R - NMod , the metric on the space of norms arises as a norm in a Banach space.For more categorical consequences and ramifications related to Theorem 2.3, we refer the readerto e.g. [11]. Also notice that if X is a singleton then F b ( X, M ) ≃ M , whence F b ( X, M ) / ∼ is thetrivial semigroup (or Banach space). In this case the above result is true, except perhaps for theword ‘isometric’. Remark 2.6.
For general abelian metric semigroups M ∈ C as in Theorem 2.3, an example offunctions f, g ∈ F b ( X, M ) with distance zero is to choose and fix m ∈ M , and take g ( x ) ≡ m + f ( x )on all of X . If M is moreover a monoid and f ≡ M , then these are the only examples.In the further special case when M is an abelian metric group, the functions g ≡ m + f turnout to be the only examples of equivalent functions, for all f ∈ F b ( X, M ). Moreover, the (pseudo-)metric defined above has a more accessible interpretation as a diameter seminorm : d ( f, g ) = N ( f − g ) , where N ( f ) := sup x,x ′ ∈ X d M ( f ( x ) , f ( x ′ )) = diam(im( f )) . (2.7)In particular, the equivalence relation f ∼ g amounts to f − g being a constant function. For(abelian) monoids ( M, M ), essentially this last assertion also follows if one restricts to the set ofbounded functions f : X → M satisfying: 0 M ∈ im( f ). More precisely, if d ( f, g ) = 0 for suchfunctions, there exists m ∈ M such that − m ∈ M and g ≡ m + f on M . Proof of Theorem 2.3.
We begin by showing that (1) holds for all abelian semigroups M , and thenturn to (2) for each successively smaller category. To show (1), we will show the triangle inequality;note this also proves the transitivity of ∼ and hence that ∼ is an equivalence relation on F b ( X, M ).Given x, y ∈ X , d M ( f ( x ) + g ( y ) , f ( y ) + g ( x ))= d M ( f ( x ) + g ( y ) + h ( y ) , f ( y ) + g ( x ) + h ( y )) d M ( f ( x ) + g ( y ) + h ( y ) , f ( y ) + g ( y ) + h ( x )) (2.8)+ d M ( f ( y ) + g ( y ) + h ( x ) , f ( y ) + g ( x ) + h ( y )) d ( f, h ) + d ( h, g ) . APOORVA KHARE
As this inequality holds for all x, y ∈ X , the triangle inequality follows. This proves (1), and asa consequence, F b ( X, M ) / ∼ is always a metric space under the metric (2.4). That this metric istranslation-invariant (2.1) is straightforward.We now claim that (2) holds, first for the category Semi . Indeed, one defines the (bounded)function f + g pointwise for f, g ∈ F b ( X, M ). We claim that if f ∼ f ′ and g ∼ g ′ (i.e., they havedistances zero between them) in F b ( X, M ), then f + g ∼ f ′ + g ′ . This follows because d M (( f + g )( x ) + ( f ′ + g ′ )( y ) , ( f + g )( y ) + ( f ′ + g ′ )( x ))= d M ([ f ( x ) + f ′ ( y )] + [ g ( x ) + g ′ ( y )] , [ f ( y ) + f ′ ( x )] + [ g ( y ) + g ′ ( x )]) = 0 , which in turn follows from the equalities: f ( x )+ f ′ ( y ) = f ( y )+ f ′ ( x ) and g ( x )+ g ′ ( y ) = g ( y )+ g ′ ( x ),for all x, y ∈ X .Thus, + is well-defined on F b ( X, M ) / ∼ . Similarly, one verifies that if f n → f and g n → g in F b ( X, M ) / ∼ , then f n + g n → f + g . Next, given a semigroup morphism ϕ : M → N , post-composing by ϕ defines a map ϕ ◦ − of semigroups : F b ( X, M ) → F b ( X, N ); and if d ( f, g ) = 0 in F b ( X, M ), then d ( ϕ ◦ f, ϕ ◦ g ) = 0 in F b ( X, N ). Thus ϕ induces a well-defined map[ ϕ ] : F b ( X, M ) / ∼ → F b ( X, N ) / ∼ . (2.9)Finally, we verify that the given functor induces isometries on Hom-spaces. The first sub-step isto claim that every Hom-space Hom( M, N ) in
Semi is itself a semigroup, with translation-invariantmetric given by: d ( η, ϕ ) := sup m = m ′ ∈ M d N ( η ( m ) + ϕ ( m ′ ) , ϕ ( m ) + η ( m ′ )) d M ( m, m ′ ) . We only show that if d ( η, ϕ ) = 0 then η = ϕ ; the remainder of the claim is straightforward (withthe triangle inequality following similarly to (2.8)). Indeed, if d ( η, ϕ ) = 0, then: d ( η, ϕ ) = 0 = ⇒ d M ( η ( m ) + ϕ (2 m ) , η (2 m ) + ϕ ( m )) = 0 ∀ m ∈ M = ⇒ η ≡ ϕ. This proves the above claim.We now show that the map ϕ [ ϕ ] is an isometry[ − ] : Hom C ( M, N ) → Hom C ( F b ( X, M ) / ∼ , F b ( X, N ) / ∼ ) . Suppose η, ϕ : M → N are semigroup morphisms, and let d ( η, ϕ ) = L . If L = 0 then the precedingcomputation shows η ≡ ϕ and hence [ η ] = [ ϕ ]. Otherwise, we compute from first principles: d ([ η ] , [ ϕ ]) = sup [ f ] =[ g ] ∈ F b ( X,M ) / ∼ d ( η ◦ [ f ] + ϕ ◦ [ g ] , ϕ ◦ [ f ] + η ◦ [ g ]) d ([ f ] , [ g ])= sup [ f ] =[ g ] ∈ F b ( X,M ) / ∼ d ([ f ] , [ g ]) sup x,x ′ ∈ X d N ( η ( m ) + ϕ ( m ′ ) , ϕ ( m ) + η ( m ′ )) , where m := f ( x ) + g ( x ′ ) and m ′ := f ( x ′ ) + g ( x ). Now note that d N ( η ( m ) + ϕ ( m ′ ) , ϕ ( m ) + η ( m ′ )) d ( η, ϕ ) d M ( f ( x ) + g ( x ′ ) , f ( x ′ ) + g ( x )) L · d ([ f ] , [ g ])for all x, x ′ ∈ X . It follows that d ([ η ] , [ ϕ ]) L = d ( η, ϕ ).To show the reverse inequality, suppose m l = m ′ l , l ∈ N are sequences in M such that thesequences d l := d N ( η ( m l ) + ϕ ( m ′ l ) , ϕ ( m l ) + η ( m ′ l )) d M ( m l , m ′ l )are non-decreasing to L = d ( η, ϕ ) as l → ∞ . We now use that X is not a singleton, whence for afixed element x ∈ X , we consider f l | X ≡ m l , g l | X \ x ≡ m l , g l ( x ) := m ′ l , l ∈ N . HE NON-COMPACT NORMED SPACE OF NORMS ON A FINITE-DIMENSIONAL BANACH SPACE 7
Clearly d ([ f l ] , [ g l ]) = d M ( m l , m ′ l ), whence for any x ∈ X \ x , we compute from above: d ([ η ] , [ ϕ ]) > sup l ∈ N d N ( η ( f l ( x ) + g l ( x )) + ϕ ( f l ( x ) + g l ( x )) , η ( f l ( x ) + g l ( x )) + ϕ ( f l ( x ) + g l ( x ))) d ([ f l ] , [ g l ])= sup l ∈ N d N ( η ( m l ) + ϕ ( m ′ l ) , ϕ ( m l ) + η ( m ′ l )) d M ( m l , m ′ l )= sup l ∈ N d l = L = d ( η, ϕ ) . This proves the theorem for metric semigroups, i.e. for C = Semi . We next impose the additionalstructure in each smaller subcategory one by one, and show the result for the remaining C . Clearly,if M is a monoid, then so is F b ( X, M ) / ∼ , with identity f ≡ M . Now the result is easily verifiedfor C = Mon . (Note that all morphisms are automatically monoid maps.)Next suppose C = R - Mod . One checks that if f ∼ g then rf ∼ rg , where rf ∈ F b ( X, M ) isdefined in the usual (pointwise) fashion. Also, multiplication by r is Lipschitz on F b ( X, M ) / ∼ ifit is so on M itself. Now the result is easily verified in this setting. Finally, if M is also R -normed,then one checks that so is F b ( X, M ) / ∼ .This shows the result for all categories except for R - Mod , R - NMod . For these latter cases, it sufficesto show the claim that if M is a complete abelian metric group then so is F b ( X, M ) / ∼ . We beginby isolating the main component of this argument into a standalone result (together with somerelated preliminaries). Lemma 2.10.
Fix a set X and an abelian metric semigroup ( M, + , d M ) . (1) F b ( X, M ) is a metric space under the sup-norm d ∞ ( f, g ) := sup x ∈ X d M ( f ( x ) , g ( x )) . Moreover, F b ( X, M ) is complete if and only if M is complete. (2) The quotient map of metric spaces : F b ( X, M ) → F b ( X, M ) / ∼ is Lipschitz of norm at most . (3) If M is moreover a group, there exists a section Φ x : F b ( X, M ) / ∼ → F b ( X, M ) which isa sub-contraction: d ∞ (Φ x ([ f ]) , Φ x ([ g ])) d ([ f ] , [ g ]) , ∀ x ∈ X, [ f ] , [ g ] ∈ F b ( X, M ) , (2.11) and such that the image of Φ x is precisely the set of functions vanishing at x .Proof. (1) is well-known, and (2) is standard using: d M ( f ( x ) + g ( x ′ ) , f ( x ′ ) + g ( x )) d M ( f ( x ) + g ( x ′ ) , g ( x ) + g ( x ′ )) + d M ( g ( x ) + g ( x ′ ) , g ( x ) + f ( x ′ )) d ∞ ( f, g ) , ∀ f, g ∈ F b ( X, M ) , x, x ′ ∈ X. To show (3), choose any representative f of [ f ] ∈ F b ( X, M ) / ∼ , recalling by Remark 2.6 that f isunique up to translation by an element of M . Now the ‘Kuratowski’ map Φ x ([ f ]) := f ( x ) − f ( x )satisfies (2.11). (We also point out for completeness some related observations at the start of [4,Section 3].) (cid:3) Returning to the proof of the above claim, suppose [ f n ] ∈ F b ( X, M ) / ∼ is Cauchy, with M complete. For any fixed x ∈ X , this implies by Lemma 2.10(3) that Φ x ([ f n ]) is Cauchy, whenceit converges in the d ∞ metric to some bounded map f by Lemma 2.10(1). Hence, [ f n ] → [ f ] byLemma 2.10(2). (cid:3) APOORVA KHARE
We now refine the above categorical construction, when the domain X is additionally equippedwith a topology: Corollary 2.12.
Given a topological space X and an abelian metric semigroup ( M, + , d M ) , define C b ( X, M ) to be the set of bounded continuous functions : X → M . Now fix a unital subring R ⊂ R . (1) If M is in fact an abelian group, then C b ( X, M ) / ∼ is a closed subobject of F b ( X, M ) / ∼ . (2) With notation as in Theorem 2.3 (for any of the categories C ), C b ( X, M ) forms a pseu-dometric subspace of F b ( X, M ) , whence C b ( X, M ) / ∼ is a subobject of F b ( X, M ) / ∼ in C .Moreover, M C b ( X, M ) / ∼ is also a covariant endofunctor of C , which is isometric if X is not a singleton. Note that the case of X compact Hausdorff and M = R was studied in greater detail in [4], andis part of a broader program to study isometries and linear isomorphisms of spaces of continuousfunctions. See e.g. [1, 4, 10], and the references therein. Proof.
To show (1), if f n : X → M are continuous and [ f n ] → [ f ] for some f ∈ F b ( X, M ), thenΦ x ([ f n ]) → Φ x ([ f ]) uniformly by (2.11), whence Φ x ([ f ]) is continuous and hence so is f .For the categories whose objects are not all complete, the assertion (2) follows from Theorem 2.3and the continuity of the R -module operations. For the categories C = R - Mod , R - NMod , one furtheruses the previous part and that F b ( X, M ) / ∼ is complete by Theorem 2.3. (cid:3) Distances between p -norms; proof of the main result In this section we prove Theorem 1.8, deriving part of it from the functorial framework discussedin the previous section.3.1. p -norm computations. Part of the proof of Theorem 1.8 works with the p -norms on F k ; thus,we begin by providing ‘more standard’ models for certain sets of such norms. Given p ∈ [1 , ∞ ),define for x = ( x , . . . , x k ) ∈ F k its p -norm: k ( x , . . . , x k ) k p := ( | x | p + · · · + | x k | p ) /p , (3.1)and also define k ( x , . . . , x k ) k ∞ := max j | x j | .As is well-known in the Banach–Mazur framework for F = R or C [6], if p, q ∈ [1 , ∞ ] and2 − p, − q have the same sign, then the norms k · k p and k · k q on F k have Banach–Mazur distance | /p − /q | · log( k ). However, this does not usually hold for 1 p < < q ∞ – for instanceif k = 2 then k · k and k · k ∞ denote the same point in the Banach–Mazur compactum. In thepresent setting of S ′ k ( F ), the p -norms share the above behavior for p ∈ [1 ,
2] and [2 , ∞ ], but differin the metric structure for [1 , ∞ ]: Proposition 3.2.
Let S ′ k ( F ) ⊂ S k ( F ) denote the equivalence classes of the norms {k · k p : 1 p ∞} . Then the map f : S ′ k ( F ) → [0 , log k ] , given by f ( k · k p ) := log kp for p ∈ [1 , ∞ ) and f ( k · k ∞ ) := 0 ,is an isometric bijection. Thus the p -norms behave ‘uniformly well’: S ′ k ( R ) ∼ = S ′ k ( C ) ∼ = S ′ k ( H ) ∼ = [0 , log k ]. Proof.
For 1 p < q < ∞ , H¨older’s inequality implies k − /p k x k p k − /q k x k q for all x ∈ F k ,and equality is attained at the vectors with all equal coordinates. For the other way, we claimthat k x k p > k x k q , with equality along the coordinate axes. Indeed, by rescaling one may assume k x k p = 1, whence | x j | p j . Thus | x j |
1, and it follows that k x k qq = X j | x j | q X j | x j | p = 1 , whence k x k q k x k p . From this it follows that d S k ( F ) ( k · k p , k · k q ) = log k /p − /q . HE NON-COMPACT NORMED SPACE OF NORMS ON A FINITE-DIMENSIONAL BANACH SPACE 9
Finally, it is evident that k x k ∞ k x k p k /p k x k ∞ for all p ∈ [1 , ∞ ) and x ∈ F k , with equalityattained in the same two cases as above. Thus, d S k ( F ) ( k · k p , k · k ∞ ) = log k /p . This concludes theproof. (cid:3) Example 3.3.
As another example, notice using Lemma 1.1 that given a non-negative measure µ supported on [1 , ∞ ), the function N µ ( x ) := Z ∞ k x k p dµ ( p )is a norm, if convergent on F k . Now the same reasoning as in the above proof shows that for allsuch µ > d S k ( F ) ( N µ , k · k q ) = log R q k /p dµ ( p ) k /q R q dµ ( p ) log( k )(1 − q − ) , (3.4)where sup(supp µ ) < q < ∞ .The above provide examples of subsets of S k ( F ) with bounded diameter. However, this does notalways happen, and we now mention such an example, which also serves to show the ‘unbounded-ness’ assertion in the main result. Example 3.5.
Given p ∈ [1 , ∞ ], q ∈ [0 , ∞ ), and an integer 1 j k , define N p,q,j ( x ) := k x k p + q | x j | , x ∈ F k (3.6)and consider the family of such norms for a fixed p : S k,p := { N p,q,j : q ∈ [0 , ∞ ) , j ∈ [ k ] } . (3.7)(one verifies easily that these are norms). We now claim that akin to Proposition 3.2 for the p -norms, the family S k,p can also be realized as a more familiar metric subspace of a Banach space: Proposition 3.8.
Suppose k > . The subset S k,p defined in (3.7) isometrically embeds into R k with the ℓ -norm, via N p,q,j log(1 + q ) e j . The image of S k,p is the union of the non-negativecoordinate semi-axes.Proof. Notice that N p,q,j ( x ) (1 + q ) N p,q ′ ,j ′ ( x ) for all q, q ′ > j = j ′ ∈ { , . . . , k } , withequality attained at least for x = x e j (here, e , . . . , e n comprise the standard basis of F k ). Itfollows that d S k ( F ) ( N p,q,j , N p,q ′ ,j ′ ) = log(1 + q ) + log(1 + q ′ ) . One next shows that for a fixed j ∈ { , . . . , k } ,0 q q ′ < ∞ = ⇒ N p,q,j ( x ) N p,q ′ ,j ( x ) q ′ q N p,q,j ( x ) , ∀ x ∈ F k , with equality possible on F k \{ } in either inequality (note that equality in the lower bound requires k > d S k ( F ) ( N p,q,j , N p,q ′ ,j ) = log(1 + q ′ ) − log(1 + q ). This concludes the proof. (cid:3) Proof of the main result.
With Proposition 3.8 and the functorial analysis in the previoussection in hand, we can show our main result.
Proof of Theorem 1.8.
Begin by fixing any compact subset X ⊂ F k \ { } ∼ = R dk \ { } satisfying: ∀ x ∈ F k \ { } , ∃ α x ∈ F × such that α x x ∈ X. (3.9)(For instance, X could be the unit sphere S dk − .) The bulk of the proof involves showing the claimthat the space S k ( F ) is a closed metric subspace of the Banach space C ( X, R ) / ∼ = C b ( X, R ) / ∼ (see Corollary 2.12, noting that X is compact). In particular, S k ( F ) is complete.To show the claim, we construct the embedding Ψ : N ( F k ) → C ( X, R ) as follows: given a norm N ∈ N ( F k ), define Ψ( N ) := log N | X ∈ C ( X, R ). Since N is a norm, it is uniquely determined by its restriction to X , whence Ψ is injective. Moreover, the respective notions of ∼ are compatiblevia taking the logarithm, whence Ψ induces an injection [Ψ] : S k ( F ) ֒ → C ( X, R ) / ∼ of metricspaces. It is easily verified that [Ψ] is an isometry; recall here that the metric on S k ( F ) is given by: d ([ N ] , [ N ′ ]) := log( M N,N ′ /m N,N ′ ) (see Equations (1.5) and (1.3)).It remains to show closedness. Suppose N l are norms on F k such that [log N l | X ] → [ f ] in C ( X, R ) / ∼ under the metric in (1.5) (recall Corollary 2.12 here). As above, one can chooserepresentative norms N l on F k and a function f ∈ C ( X, R ) in their equivalence classes, such thatfor all l >
0, (log N l ) − f : X → [0 , ǫ l ] , ǫ l > , (3.10)with ǫ l → + as l → ∞ . In particular, N l → exp( f ) pointwise on X . Define N ( ) := 0 , N ( x ) := | α x | − exp( f ( α x x )) , ∀ x ∈ F k \ { } , where α x comes from the defining property of X . The above claim is proved if one shows that N isa norm on F k . First note that N is indeed well-defined: if α x and β x are such that α x x , β x x ∈ X for some x , then | α x | − exp( f ( α x x )) = lim l →∞ | α x | − N l ( α x x ) = lim l →∞ | β x | − N l ( β x x ) = | β x | − exp( f ( β x x )) . Next, N is homogeneous: given any scalar β ∈ F and vector x ∈ F k \ { } , N ( β x ) N ( x ) = | α β x | − exp f ( α β x β x ) | α x | − exp f ( α x x ) = lim l →∞ | α β x | − N l ( α β x β x ) | α x | − N l ( α x x ) = lim l →∞ | β | = | β | . Finally, observe that N is sub-additive, i.e., N ( x + y ) N ( x ) + N ( y ) for x , y ∈ F k . Indeed, thisis immediate if any of x , y , x + y is zero, so assume this does not happen, and compute: N ( x + y ) = | α x + y | − lim l →∞ N l ( α x + y ( x + y )) | α x + y | − lim l →∞ N l ( α x + y x ) + N l ( α x + y y )= | α x + y | − lim l →∞ (cid:18) | α x + y || α x | N l ( α x x ) + | α x + y || α y | N l ( α y y ) (cid:19) = N ( x ) + N ( y ) . The closedness of S k ( F ) now follows, whence by Theorem 2.3 and Corollaries 1.7 and 2.12, we havea chain of inclusions with closed images S k ( F ) ֒ → S dk ( R ) ֒ → C b ( X, R ) / ∼ ֒ → F b ( X, R ) / ∼ ;The above claim now follows; hence S k ( F ) is complete. Next, Lemma 1.1 implies that N ( F k ) isconvex, hence path-connected, whence so is S k ( F ). Moreover, clearly S ( F ) is a point. Finally,assuming k >
1, Proposition 3.8 shows that S k ( F ) is unbounded. (cid:3) The above proof shows that the metric space of norms embeds into C ( X, R ) / ∼ for many differentcompact topological subspaces X ⊂ F k (see (3.9)). We conclude by exploring these embeddings ingreater detail, fixing F = R for convenience. Specifically, if X = S k − denotes the unit sphere, thenunder the embedding S k ( R ) ֒ → B := C ( S k − , R ) / ∼ , the origin in the Banach space B is preciselythe image of the 2-norm k · k . More generally, we have such an identification of the origin for every norm – but not for every space X . Proposition 3.11.
Given a norm N : R k → R and any radius r > , let X N,r denote the sphere X N,r := { x ∈ R k : N ( x ) = r } . Now fix a norm N on R k , as well as a subset X ⊂ R k such that every nonzero vector is a positivereal multiple of a point in X . Then the following are equivalent: HE NON-COMPACT NORMED SPACE OF NORMS ON A FINITE-DIMENSIONAL BANACH SPACE 11 (1) S k ( R ) embeds as a closed subset in the Banach space C ( X, R ) / ∼ for some topological space X via [ N ′ ] [log N ′ | X ] ; and this embedding maps the equivalence class [ N ] to the origin. (2) X = X N,r for some r > .However, there exist compact sets X ⊂ R k such that the image of the embedding S k ( R ) ֒ → C ( X, R ) / ∼ avoids the origin.Proof. If (1) holds, then N | X must be constant, whence X ⊂ X N,r for some r >
0. Moreover, if x ∈ X N,r , then α x ∈ X for some α >
0. Since N ( α x ) = αr , it follows that α = 1 and hence X = X N,r .Conversely, suppose (2) holds. Since all norms on R k are equivalent, the space X N,r is compactand hence satisfies (3.9), whence the proof of Theorem 1.8 applies to it. In particular, the imageof N under the embedding : S k ( R ) ֒ → C ( X N,r , R ) / ∼ is a constant function, whose image under ∼ is the trivial class.Finally, given any point y = 0, define the sphere X y := { x ∈ R k : k x − y k = 1 + k y k } . It is clear that is in the ‘interior’ of the sphere. Also notice that for every unit direction v ∈ S k − ,there exists a unique α > α v ∈ X y . Indeed, from the conditions k α v − y k = 1 + k y k , α > , one derives: α = p h v , y i + h v , y i . In particular, X y satisfies (3.9) and hence the proof ofTheorem 1.8 applies to it. However, no norm maps via the embedding S k ( R ) ֒ → B := C ( X y , R ) / ∼ to the origin in the Banach space B . Indeed, if [ N ] B , then the norm N would restrict to aconstant on X y . But this is false: X y intersects the line R y at the two points ( k y k ± p k y k ) y ,and as these are not negatives of one another, evaluating N yields unequal values. (cid:3) The normed space of metrics on a finite set
We now study a parallel setting to the metric space of norms on F k , in which the above functorialapproach is also applicable. Given a finite set [ n ] := { , . . . , n } with n >
2, it is possible toimpose a pseudometric on the space of metrics on [ n ] in the same way as above: given metrics ρ, ρ ′ : X × X → [0 , ∞ ), define d [ n ] ( ρ, ρ ′ ) := log max j = k ρ ′ ( x j , x k ) ρ ( x j , x k ) − log min j = k ρ ′ ( x j , x k ) ρ ( x j , x k ) ;note that exp ◦ d [ n ] is termed the distortion in metric geometry and computer science.We cite the well-known surveys [13, 15] for further details and reading on the numerous applica-tions of distortion and metric geometry to computer science, combinatorics, and other fields. Alsonote that there is a different, well-studied pseudometric on the space of metrics on [ n ], or moregenerally on all compact metric spaces. This is the Gromov–Hausdorff metric (which is not com-parable to d [ n ] , as we see below). Nevertheless, the metric d [ n ] , as well as the connection betweendiameter norms and (log-)distortion, do not seem to be studied or recorded in the literature. Thismotivates the present section.We begin with a result that is parallel to Theorem 1.8 for S k ( F ), and again follows from theabove functorial analysis. In particular, it shows that the metric d [ n ] is also a diameter norm: Theorem 4.1.
Fix an integer n > . (1) The map d [ n ] is a pseudometric on the space of metrics on [ n ] , with equivalence classesprecisely consisting of proportional metrics. (2) The quotient metric space S ([ n ]) is a complete, path-connected, metric subspace of theBanach space R ( [ n ]2 ) / ∼ = C ( (cid:0) [ n ]2 (cid:1) , R ) / ∼ with the diameter norm, where (cid:0) [ n ]2 (cid:1) denotesthe discrete set of two-element subsets of [ n ] . (3) The space S ([ n ]) is a singleton if n = 2 , and unbounded otherwise.Proof. We only point out why for n > S ([ n ]) is unbounded. Indeed, for each m > X m := { , , . . . , n − n + m + 1 } be the (induced) metric subspace of ( R , | · | ), and let ρ n,m :[ n ] × [ n ] → R be the metric induced by the unique rank/order-preserving map : X m → [ n ] ⊂ R .Compare ρ n,m to the discrete metric ρ ( x, y ) = 1 − δ x,y : the log-distortion between them is log( n + m ),and m can grow without bound. (cid:3) Remark 4.2.
The metric on S ([ n ]) – henceforth denoted by d S ([ n ]) – is not comparable to thewell-studied Gromov–Hausdorff metric on compact metric spaces: d GH ( X , X ) := inf Z,ι ,ι d H ( ι ( X ) , ι ( X )) , where one runs over all metric spaces Z and isometric embeddings : ι j : X j ֒ → Z ; and where d H denotes the Hausdorff distance in Z . To see why d GH is not comparable to the above metric d S ([ n ]) ,for any n > n ], say { a, b } 6 = { c, d } ⊂ [ n ]. Now definemetrics ρ, ρ ′ on [ n ] via: ρ ( a, b ) = ρ ′ ( c, d ) = 1 / ρ, ρ ′ are 1. Thesetwo metric spaces are clearly isometric under a ↔ c, b ↔ d , and all other points left unchanged.However, the metrics are not proportional. Going the other way, proportional but unequal metricson [ n ] do not admit an isometry between them.Our next few results are meant to help better understand the metric space S ([ n ]). The followingresult parallels Proposition 3.8, and illustrates how a certain one-parameter family of metrics on[ n ] can be understood through a more standard model: Proposition 4.3.
Fix integers n > and j (cid:0) n (cid:1) , as well as any bijection to identify the setof pairs (cid:0) [ n ]2 (cid:1) (i.e. edges between points in [ n ] ) with the set (cid:2)(cid:0) n (cid:1)(cid:3) = { , . . . , (cid:0) n (cid:1) } . Given a ∈ (0 , ,let ρ j,a : [ n ] × [ n ] → R denote the metric in which all nonzero distances in [ n ] are , except for thedistance corresponding to the edge j , which is a . Define S ′ ([ n ]) := { ρ j,a : j ∈ (cid:2)(cid:0) n (cid:1)(cid:3) , a ∈ (0 , } . Then S ′ ([ n ]) (or its set of equivalence classes) embeds isometrically into R ( n ) with the ℓ -norm,via: ρ j,a (log a ) e j . The image is the union of the non-positive coordinate semi-axes. The set S ′ ([ n ]) comprises the metrics in which [ n ] may be viewed as a weighted graph with alledge weights but one equal and at least as large as the remaining edge weight. Proof.
Write N := (cid:0) n (cid:1) for convenience. Viewing each metric ρ on [ n ] as a function : [ N ] → (0 , ∞ ),say ( ρ (1) , . . . , ρ ( N ) ) T , it follows that d S ([ n ]) ( ρ j,a , ρ j ′ ,a ′ ) = log max k N ρ ( k ) j,a ρ ( k ) j ′ ,a ′ − log min k N ρ ( k ) j,a ρ ( k ) j ′ ,a ′ . Now if j = j ′ then the distance is | log a − log a ′ | , else the distance is − log a − log a ′ . (cid:3) Remark 4.4.
Notice that the function ρ j,a is a metric on [ n ] if and only if a ∈ (0 , ρ j,a and ρ j ′ ,a ′ for a, a ′ ∈ (0 , j = j ′ then we once againobtain | log a − log a ′ | , but if j = j ′ then we have: d S ([ n ]) ( ρ j,a , ρ j ′ ,a ′ ) = ( | log a | + | log a ′ | , if either a, a ′ > a, a ′ {| log a | , | log a ′ |} , otherwise . We next study embeddings in S ([ n ]) of metric spaces. In doing so, we are motivated by recentwork [7], where it was shown that every finite metric space of size at most (cid:0) n (cid:1) embeds isometricallyinto the Gromov–Hausdorff space of isometry classes of n -element metric spaces. In other words, a HE NON-COMPACT NORMED SPACE OF NORMS ON A FINITE-DIMENSIONAL BANACH SPACE 13 representative from the Gromov–Hausdorff equivalence class of every metric space of size at most (cid:0) n (cid:1) embeds isometrically into Gromov–Hausdorff space. The following result is parallel in spirit,for the space S ([ n ]): Theorem 4.5.
Let ( X, d ) be a finite metric space, and n > an integer such that X (cid:0) n (cid:1) . Thenthere exists an equivalent metric space to ( X, d ) – i.e., a rescaling of d – that admits an isometricembedding into S ([ n ]) . Note that the result fails to hold for n = 2, since S ([2]) is a singleton.Before proving Theorem 4.5, we recall that its Gromov–Hausdorff analogue in [7] was statedusing the smallest n such that | X | (cid:0) n (cid:1) . While our variant does not a priori have this extrarestriction, we point out that the two versions are equivalent for S ([ n ]), because of the followingresult: Proposition 4.6.
For all n > , the metric space S ([ n ]) isometrically embeds into S ([ n + 1]) .Proof. Given a finite metric space (
X, d ) with | X | = n , embed it into a metric space X ⊔ { n + 1 } ,where n + 1 is an additional point with distance diam( X ) from every x ∈ X . A straightforwardcomputation (perhaps rescaling both diameters to 1 for convenience) now shows that this definesan isometry from S ([ n ]) into S ([ n + 1]). (cid:3) We now prove the above theorem.
Proof of Theorem 4.5.
In fact we will construct the embedding : (
X, α · d ) ֒ → S ([ n ]) for a specific α >
0, using several ‘natural’ tools. We begin by describing these tools.Observe that every metric on [ n ] can be viewed as an element of (0 , ∞ ) N where N = (cid:0) n (cid:1) . Viataking logarithms, the space S ([ n ]) is in bijection with the set Ψ([ n ]) of all tuples ( ψ ij ) T ∈ R ( [ n ]2 ) ∼ = R N (here i < j ) such thatexp( ψ ij ) + exp( ψ jk ) > exp( ψ ik ) , ∀ i < j < k n. (4.7)Note that rescaling the metric by α > ψ ij by log α . In other words,Ψ([ n ]) sits inside R N / ∼ , where ∼ denotes additive translations by scalar multiples of (1 , . . . , T .The next observation is that for an integer p >
0, the space ( R p , k·k ∞ ) is isometrically isomorphicas a Banach space to R p +1 / ∼ with the diameter norm diam, where ∼ denotes quotienting byadditive translation by multiples of (1 , . . . , T . More generally, given integers 0 < p < q , the mapΨ p,q : ( x , . . . , x p ) T ( x , . . . , x p , × ( q − p ) ) T + R (1 , . . . , Tq is an isometric linear embedding of ( R p , k · k ∞ ) into ( R q / ∼ , diam). For this reason, note in theprevious paragraph that the bijection of setslog[ − ] : ( S ([ n ]) , d S ([ n ]) ) → (Ψ([ n ]) , diam)is in fact an isometry of metric spaces.Finally, we recall the Fr´echet embedding [5], which maps an N -element metric space ( X = { x , . . . , x N − } , d ) isometrically into R N − with the sup-norm, via: x j ( d ( x , x j ) , . . . , d ( x N − , x j )) T for 0 j N −
1. Let us denote this embedding by
F r : X → R | X |− .With these ingredients in hand, we claim: Proposition 4.8.
Fix an integer n > and a metric space ( X, d ) such that | X | N = (cid:0) n (cid:1) .If X has diameter at most log 2 , then the composite map Ψ | X |− ,N ◦ F r : (
X, d ) ֒ → ( R | X |− , k · k ∞ ) ֒ → ( R N / ∼ , diam) has image inside the metric space Ψ([ n ]) ≃ S ([ n ]) . The converse holds if X is a three-element set. Notice that Proposition 4.8 implies Theorem 4.5, by rescaling the metric on X by (log 2) / diam( X ).(The case of | X | = 2 is straightforward.) (cid:3) To complete the proof, it remains to show the preceding proposition.
Proof of Proposition 4.8.
If diam X log 2, then we claim for each x ∈ X that any three coordi-nates of the Fr´echet tuples ( d ( x ′ , x )) x ′ ∈ X satisfy (4.7). Indeed, if x , x , x ∈ X thenexp d ( x , x ) exp d ( x , x ) + exp d ( x , x ) . This shows Ψ | X | ,N ◦ F r ( X ) ⊂ Ψ([ n ]). Conversely, let X = { x, y, z } ; then we are assuming thatΨ | X | ,N ◦ F r ( X ) = { ( d ( x, y ) , d ( x, z ) , , . . . , T , (0 , d ( y, z ) , , . . . , T , ( d ( z, y ) , , , . . . , T } is contained in Ψ([ n ]). Hence the last of the three points in R N satisfies (4.7), which in turn implies:exp d ( z, y ) . The same argument using the other two Fr´echet embeddings of X shows that diam X log 2. (cid:3) Norms on arbitrary Banach spaces; concluding remarks and questions
Parallel settings.
As shown in Section 2, diameter norms offer a unified and functorial frame-work, which subsumes and explains both Theorem 1.8 about norms on F k , as well as Theorem 4.1about metrics on [ n ]. This treatment also applies more generally, and we begin this final section bystating (without proofs, and for completeness) two parallel results: in an arbitrary Banach spaceand in a class of discrete metrics on an arbitrary set. Proposition 5.1.
Let B be an arbitrary Banach space over F = R or C , and let X ⊂ B \ { } bea subset such that for all = x ∈ B , there exists α x ∈ F × such that α x x ∈ X . Then the set ofequivalence classes of norms on B (i.e., up to scaling by (0 , ∞ ) ) whose restriction to X is boundedaway from , ∞ can be isometrically realized as a complete, path-connected metric subspace of theBanach space C b ( X, R ) / ∼ with the diameter norm. Notice that if moreover B is finite-dimensional and X is compact then this reduces to Theo-rem 1.8. Similarly, for any set X we have the following extension of Theorem 4.1: Proposition 5.2.
For any nonempty set X of size at least , the equivalence classes (again byscaling) of metrics d on X bounded away from , ∞ outside the diagonal – i.e., such that < inf x = x ′ ∈ X d ( x, x ′ ) sup x = x ′ ∈ X d ( x, x ′ ) < ∞ form a complete, path-connected unbounded metric subspace of the Banach space C b ( (cid:0) X (cid:1) , R ) / ∼ .Here (cid:0) X (cid:1) denotes the discrete set of pairs of elements in X . Further questions.
Following Theorems 1.8 and 4.1 studying the norms on F k and themetrics on [ n ] respectively, it may be interesting to further explore the spaces S k ( F ) and S ([ n ]);exploring the former may provide additional insights into the Banach–Mazur compactum quotientspace. Thus, we conclude with some observations and questions in both of the above settings.(1) Are there more standard mathematical (geometric) models with which one can identify themetric spaces S k ( F ) and S ([ n ])? What can one say about their geometric properties?(2) What are the automorphism groups of these spaces? (Depending on the category underconsideration, one may wish to study homeomorphisms, isometries, . . . ) For instance, bythe final assertion in Lemma 1.1, S k ( F ) is equipped with the group P GL dk ( R ) of isometries,under a real-linear identification of F k with R dk . An additional observation (by TerenceTao in recent discussions) is that S k ( F ) also carries an isometric involution, which arisesfrom considering dual norms. A parallel observation is that the space S ([ n ]) is equippedwith an obvious symmetry group S n of automorphisms. (In contrast, the Gromov–Hausdorff It is easy to verify here that for A ∈ GL dk ( R ), d S k ( F ) ( N ( A · − ) , N ′ ( A · − )) = d S k ( F ) ( N ( · ) , N ′ ( · )). HE NON-COMPACT NORMED SPACE OF NORMS ON A FINITE-DIMENSIONAL BANACH SPACE 15 space has no isometries [9].) One may also consider local isometries of S ([ n ]) and S k ( F ), aspreviously done for the Gromov–Hausdorff space in [8]. Remark 5.3.
For completeness we point out that individual norms can indeed be un-changed under precomposing by elements of
P GL dk ( R ). For instance, for the k · k -normone has the image of O dk ( R ), while for p ∈ [1 , ∞ ] \ { } , results of Banach [2] and Lam-perti [12] show that ‘generalized permutation matrices’ are isometries of k·k p . These consistof the products of permutation matrices with diagonal orthogonal or unitary matrices for F = R or C respectively. (When F = R , this is precisely the Weyl group of type B or C ,i.e. the hyperoctahedral group S ≀ S n of signed permutations.)At the same time, say for F = R there is no nontrivial matrix A ∈ GL k ( R ) , A R × · Id,whose precomposition fixes all of S k ( F ). Indeed, using a k · k p -norm for p = 2, by theprevious paragraph A must be a nonzero scalar multiple of some signed permutation matrix A ′ ∈ S ≀ S n , say A = cA ′ . Suppose the nonzero entries of A ′ correspond to the (signed)permutation σ ∈ S n . Now let N ( x ) := P kj =1 j | x j | for x ∈ R k . If e , . . . , e k comprise thestandard basis elements of R k , and N ( A x ) ≡ c ′ N ( x ) on R k for some c ′ >
0, then N ( A e j ) = c ′ N ( e j ) ∀ j = ⇒ c ′ j = | c | σ − ( j ) , ∀ j. Multiplying these inequalities yields: | c | = c ′ . Now evaluating at e j + e σ − ( j ) yields: j + σ − ( j ) = σ − ( j ) + σ − ( j ) , ∀ j ∈ [ k ] . Hence σ has order at most 2. Using this and evaluating at e j + 2 e σ − ( j ) yields: j + 2 σ − ( j ) = σ − ( j ) + 2 j, ∀ j and we conclude that σ = Id. Finally, suppose two diagonal entries of A are unequal,say a = c ′ , a = − c ′ . Define the norm N ( x ) := k x k + | x + x | , and evaluate it at x = (1 , , , . . . , T : 0 = c ′ N ( x ) − N ( A x ) = 4 c ′ − c ′ . Since c ′ >
0, our supposition must therefore be false, concluding the proof. (cid:3) (3) How does the space S k ( F ) relate to S k +1 ( F )? Observe by Proposition 3.2 that the p -normsisometrically map to the p -norms, provided one rescales the metric/norm on each S k ( F ) bylog( k ). Alternately, without rescaling any of the norms on S k ( F ), is it possible to computethe fibers of ‘the’ restriction map : S k +1 ( F ) → S k ( F )?On a related note (say with F = R for convenience), is this restriction map a surjection?I.e., is there a “Hahn–Banach” extension of every norm on R k to one on R k +1 , say minimallyincreasing/without increasing the (log-)distortion relative to some reference norm?(4) Notice that the previous question has a variant for S ([ n ]) with a positive answer, by Propo-sition 4.6. Moreover, the fibers of the restriction of norms from [ n + 1] to [ n ] are solutionsets to finite systems of inequalities. It may be interesting to study the structures of thesesolution spaces.(5) To understand the ‘sizes’ and growth of balls in these spaces, one can also explore theirmetric entropy. Recall for a metric space X and a radius r >
0, the metric entropy of E ⊂ X is the largest number of points in E that are r -separated. This is related to theinternal and external covering numbers and the packing number of E ; we refer the readerto [18] for a detailed introduction to these ideas.(6) What is the smallest Banach space inside which these spaces (or distinguished subsetstherein) can be isometrically embedded? Of course if we restrict to finite subsets X thenthe classic observation of Fr´echet [5] shows that ( X, d ) isometrically embeds into R | X |− with the supnorm (and into R | X |− if | X | >
4) – see the discussion prior to Proposition 4.8.
If instead of the supnorm one is interested in Euclidean space embeddings – for subsets of S k ( F ) or for S ([ n ]) – the classic paper of Schoenberg [17] (following related works in metricgeometry by Menger, Fr´echet, von Neumann, and others) provides the following result forfinite metric spaces X : Theorem 5.4 (Schoenberg [17], 1935) . Fix integers n, r > , and a finite set X = { x , . . . , x n } together with a metric d on X . Then ( X, d ) isometrically embeds into R r (with the Euclidean distance/norm) but not into R r − if and only if the n × n matrix A := ( d ( x , x j ) + d ( x , x k ) − d ( x j , x k ) ) nj,k =1 (5.5) is positive semidefinite of rank r . We also refer the reader to [17] for more general results for separable X , and [3, 14] formore recent, well-known variants with constraints on the ‘embedding dimension’ r .(a) We end with some examples and comments in each of the two settings, starting with S ([ n ]). Note that Proposition 4.3 shows an isometric embedding into R ( n ) with the 1-norm for a subset of S ([ n ]). It would be interesting to explore into what Banach spacecan the larger subset of norms explored in Remark 4.4 be isometrically embedded.Note that this is the restriction of the following metric on the union of the X, Y -axes: d (( x, , (0 , y )) := ( k ( x, y ) k , if xy > k ( x, y ) k ∞ , otherwise , and d restricted to the X or Y axis is the usual Euclidean distance. Can this metricspace be (better) understood in terms of an isometrically embedding into a Banachspace?Another question is if Theorem 4.5 can be strengthened, to characterize the finitemetric spaces on at most (cid:0) n (cid:1) elements, which can be embedded isometrically – i.e.,without scaling the metric – into S ([ n ]).(b) Here are some examples of ‘finite-dimensional embeddings’ for infinite subsets of S k ( F ).Recall from Proposition 3.8 that for each p ∈ [1 , ∞ ], the family of norms { N p,q,j : q ∈ [0 , ∞ ) , j ∈ [ k ] } isometrically embeds into R k with the 1-norm. Next, by Proposition 3.2the p -norms isometrically embed inside a one-dimensional real normed space (in fact,inside [0 , log k ]). On the other hand for the p -norms, one can show that the image S ′ k ( R ) of the p -norms in C ( R k \ { } , R ) / ∼ (akin to (3.9)) has affine hull of infinite– in fact uncountable – dimension. However, this is a consequence of the specificembedding and not an intrinsic property of S ′ k ( R ). Thus, it is not clear what is thesmallest (dimensional) Banach space containing an isometric copy of S k ( F ). References [1] Jes´us Araujo,
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