EXT /05/2000

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1 On the Construction of Selfpolar and Selfpolar Hilbertian Norms on Inner Product Spaces EXT /05/2000 Gerald Hofmann Klaus Luig May 6, 2000 Abstract Noticing that the selfpolar Hilbertian norms are the appropriate substitutes for the canonical norms when a pre-hilbert space is replaced by a èpossibly indeæniteè inner product space, the Aronszajn-Schatten iteration of selfpolar norms is discussed. Furthermore, we discuss the question of how many non-equivalent selfpolar and selfpolar Hilbertian norms exist on ènon-degenerateè inner product spaces of countably inænite dimension by constructing a large subclass of such norms and deciding which of them are equivalent. As a preparation for this discussion, we give a complete and constructive classiæcation of all selfpolar and all selfpolar Hilbertian norms on two-dimensional Minkowski space. Introduction Spaces with indeænite inner product were ærst applied to quantum æeld theory èqftè by Dirac and Pauli about 70 years ago ècf. ë6ë, ë2ëè. Since the Gupta- Bleuler formalism of QED èquantum electro-dynamicsè it is clear that the use of indeænite metric is indispensable. As explained by Kugo, Nakanishi, Ojima, and Strocchi, now the use of indeænite metric is quite natural as the framework of QFT, and especially, of the operator formalism of gauge theories and quantum gravity ècf. ë7ë, ë8ë, ë9ë, ë25ëè. In those theories the Hilbert space of state vectors is replaced by some Krein space èë4ë, ë5ëè. Considering a pre-hilbert èunitaryè space E; ë:; :ë èi.e., the inner product ë:; :ë is positive deæniteè, a canonical norm pèxè = p ëx; xë; è.è x 2 E, exists on E. Recall that the following apply:

2 jëx;yëj èaè p is the unique selfpolar norm on E;ë:; :ë, i.e., pèxè = sup, 06=y2E pèyè x 2 E ècf. ë7, pp ëè. èbè p satisæes the parallelogram identity. ècè the completed hull E eçèpè endowed with the continuously extended inner product ë:; :ë ç is a Hilbert space, where çèpè denotes the topology deæned by the norm p. Motivated by the ærst paragraph, it is of interest to ænd the corresponding substitutes when the pre-hilbert space is replaced by some non-degenerate inner product space E;è:; :è, èi.e., E is a real or complex vector space equipped with a bilinear symmetric respective sesquilinear Hermitian form è:; :è which is non-degenerate; in the latter case it is assumed that the inner product è:; :è is antilinear in its ærst variableè. If the inner product è:; :è is indeænite, then è.è does not deæne a norm, and thus, no canonical norm is available. The role of the canonical norm è.è is now played by the selfpolar Hilbertian norms q on E, i.e., it holds qèxè = jèx; yèj sup ; è.2è 06=y2E qèyè and furthermore, there is a positive deænite inner product ë:; :ë one æ E such that p qèxè = ëx; xë ; è.3è x 2 E. Recall also that a norm is Hilbertian èi.e., è.3è appliesè if, and only if, it satisæes the parallelogram identity ècf. ë5, p. 24ëè. The above èaè, èbè show that the selfpolar Hilbertian norms are the natural generalization of the canonical norm when the class of pre-hilbert spaces is generalized to inner product spaces. The signiæcance of selfpolar and selfpolar Hilbertian norms is due to the following: èiè If q is a selfpolar norm on a non-degenerate inner product space E;è:; :è, then èiè the inner product è:; :è extends çèqè-continuously onto e E æ e E, èiiè the inner product space e E;è:; :è ç is non-degenerate. èiiè If q is a selfpolar Hilbertian norm on a non-degenerate inner product space E;è:; :è, then E;è:; e :è ç is a Krein space èë, Th.èiiiè, Prop. 2ëè, i.e., ee;ë:; :ëèqè:è = p ë:; :ëè is a Hilbert space and the Gram operator J deæned by èx; yè ç =ëx; Jyë, x; y 2 e E is a symmetry èj = J æ = J, è. Notice also that ècè is generalized from the case of positive deænite inner products ë:; :ë to èpossibly indeæniteè inner products in the following: 2

3 èc'è if q is a Hilbertian selfpolar norm on E, then the completed hull e E çèqè endowed with the continuously extended inner product è:; :è ç is a Krein space, ècf. ë2ëè. In generalization of the positive deænite case, where the canonical norm is uniquely deæned, the following is possible in the general case of an inner product space: èiè there is no selfpolar norm on E;è:; :è ècf. ë5, Example III.3.2ëè. èiiè there is exactly one normed self-polar topology ç on E; è:; :è, i.e., a selfpolar norm p deæning ç exists, and furthermore, every selfpolar norm on E deænes a topology equivalent to ç ècf. ë2ë, and references cited thereè. èiiiè there is a whole family of selfpolar norms deæning non-equivalent topologies on E; è:; :è èsuch a family of selfpolar norms was explicitely constructed by Araki in ë2ëè. Along these lines, it is known that if E; è:; :è is non-degenerate and quasi-deænite èi.e. quasi-positive or quasi-negative, where quasi-positive è-negativeè means that E has no negative èpositiveè deænite subspace of inænite dimensionè, then all selfpolar norms are equivalent to the same selfpolar Hilbertian norm ë5ë. While an operator description for selfpolar Hilbertian norms is developed by Hansen on the special class of inner product spaces which allow a complete Hilbertian majorant èë7ë, see Theorem 3.9è, the present paper is aimed at explicit constructions of as well selfpolar as selfpolar Hilbertian norms on non-degenerate inner product spaces. In section 2 the Aronszajn-Schatten iteration of such norms is given. Our results are a generalization of as well Aronszajn's iteration of selfpolar norms on inner product spaces as of Schatten's iteration of the ç-topology on the tensor product E æ E ècf. ë3ë, ë7ë, ë20ë, ë24ë, ë26ëè. It is shown in Theorem 2.8 that for each positively homogeneous functional p on a non-degenerate inner product space satisfying p 0 ç p èsee Notation 2.è, a selfpolar norm p is constructed by an iteration process starting from p. This iteration can be used for a numerical calculation, and as well a priori as a posteriori estimates are obtained in Lemma 2.2. Furthermore, if p is Hilbertian, then the selfpolar norm p is Hilbertian, too èsee Corollary 2.è. While the Aronszajn-Schatten iteration yields the existence of selfpolar norms, it does not answer the question whether or not all selfpolar norms are equivalent. Furthermore it is of interest whether a given selfpolar norm is equivalent to a Hilbertian selfpolar norm. Section 3 is devoted to these questions. For the class of inner product spaces of countable algebraic dimension and a huge family of selfpolar norms on them an answer is given in Theorem 3.2. These inner product spaces are of interest for the investigation of some special models of gauge-æeld theories èëë, ë8ë, ë9ë, ë0ë, ë3ë, ë4ë, è. 3

4 The analysis of section 3 relies on a complete classiæcation of the selfpolar and selfpolar Hilbertian norms on M 2 í the real Minkowski space of dimension two í only the results of which are given there. The proofs of these results are postponed until section 4, where all selfpolar and selfpolar Hilbertian norms on M 2 are explicitely constructed èsection 4 is to some extent independent of the rest of the paper and may be omitted, if the reader is not interested in the details of the two-dimensional caseè. It turns out that in contrast to the positive deænite case, where exactly one selfpolar Hilbertian norm exists, on M 2 there are as well non-denumerably many selfpolar as non-denumerably many selfpolar Hilbertian norms. The idea of the construction is to describe the norms p on M 2 by the p of their unit balls U p, p are convex curves. Polarization of p in M 2 then transforms to a Legendre-like transformation "*" of convex curves, which isintroduced in section 4.. In order to construct convex curves which are invariant under the *-operation, and thus describe selfpolar norms, it is shown that every arc connecting the two mass shells h := fx =èx 0 ;x è 2M 2 ; x 2 =;x 0 é 0g, h 2 := fx =èx 0 ;x è 2 M 2 ; x 2 =,;x é 0g, and lying completely between them èsee Figure 2è, completes to such a *-invariant curve onm 2. Furthermore, it is shown that every convex curve related to a selfpolar norm is obtained by that construction èsee Theorem 4.7è. The special case of selfpolar Hilbertian norms on M 2 is considered in Proposition Construction of selfpolar norms using the Aronszajn-Schatten iteration Letting E; è:; :è be a non-degenerate inner product space, the following is aimed at deæning selfpolar norms on E. Though p will mostly be a norm in the subsequent applications, it is ærst assumed that p only satisæes pèçxè = jçj pèxè; 0 ç pèxè é è2.è for each x 2 E; ç 2 C. Notice that U p = fx 2 E; pèxè ç g is circled and absorbent, but not necessarily absolutely convex, since it is not assumed that the triangle inequality applies to p. Setting R + = fç 2 R; ç ç 0g, the following is conærmed. 2. Notation. For an inner product space E;è:; :è, let P èresp. e Pè denote the set of all norms èresp. all functionals p : E! R + satisfying è2.è è on E. Itis obvious that Pçe P. 2.2 Notation. Let p; q 2 e P. aè Setting p è0è = p, deæne p ènè èxè := sup jèx; yèj = p èn,è èyèç 4 sup 06=y2E jèx; yèj p èn,è èyè ; è2.2è

5 x 2 E, n =; 2;:::, where for the last expression "0=0" is deæned to be zero. Here p ènè èx 0 è= is possible for certain x 0 2 E. For p èè and p è2è, it will also be written p 0 and p 00, respectively. bè Let p ç q denote that there is a constant 0écé such that pèxè ç cqèxè è2.3è for each x 2 E. Furthermore, let p ç q denote that è2.3è applies, and additionally, there is a sequence fx n g n=, x n 2 E, such that pèx n è= and qèx n è!as n!. cè For a norm p let çèpè denote the locally convex topology deæned py p. For two l.c. topologies ç j èj =; 2è, let ç ç ç 2 èresp. ç ç ç 2 è denote that ç 2 is æner èresp. strictly ænerè than ç. Considering the dual pair èe; Eè, where the pairing is given by the inner product è:; :è, the weak topology ç is deæned by the following set of seminorms x; y 2 E. x! q y èxè =jèx; yèj ; Some properties of p ènè è:è are collected in the following. Since the proofs are straightforward, they are omitted. 2.3 Lemma. Let p è0è è:è =pè:è, qè:è 2 e P on a non-degenerate inner product space E; è:; :è. Then, aè p ènè satisfy the triangle inequality and homogenity, i.e., p ènè èx + yè ç p ènè èxè+p ènè èyè; p ènè èçxè =jçj p ènè èxè for each x; y 2 E; ç 2 C ;n2n; bè for 0 éç2 R, it follows èçpè 0 èxè =ç, p 0 èxè; x2 E, cè if pèxè ç qèxè for each x 2 E, then p 0 èyè ç q 0 èyè for each y 2 E, c'è if p ç q, then q 0 ç p 0, dè it holds jèx; yèjçp ènè èxè p èn,è èyè è2.4è for each x; y 2 E; n 2 N; where the product on the right-hand side of è2.4è is deæned to be inænite whenever one factor is inænite. eè it holds for each x 2 E, fè the following are equivalent: pèxè ç p è2è èxè =p è4è èxè =:::; p èè èxè = p è3è èxè =p è5è èxè =::: 5

6 èiè p 0 èxè é for each x 2 E, èiiè 0 ép ènè èzè é for each 0 6= z 2 E and n =; 2;:::: If furthermore p 2P, then èiè èand consequently also èiièè is equivalent to èiiiè çèpè ç ç: For Hilbertian norms, the following holds. 2.4 Lemma. If p is a Hilbertian norm on a non-degenerate inner product space satisfying ç ç çèpè, then all p ènè ;n=; 2;:::; are Hilbertian norms, too. Proof. Notice ærst that Lemma 2.3fè yields that all p ènè are norms on E, n 2 N. Assume then that there is an s 2 N such that p èsè is a Hilbertian norm. Then there is a positive deænite inner product ë:; :ë èsè on E such that p p èsè èxè = ëx; xë èsè ;x2 E: Noticing that E; ë:; :ë èsè is a pre-hilbert space, consider the completed hull E eçèp èsè è endowed with the continuously extented inner product, which is also denoted by ë:; :ë èsè. E; e ë:; :ë èsè then is a Hilbert space. Consider now the linear functionals ' y è:è =èy; :è; y2 E; deæned on E, and notice that their continuity relative toçèp èsè è follow from j' y èxèj = jèy; xèjçp ès,è èyè p èsè èxè =C y p èsè èxè ; where è2.4è with n = s was applied, and C y := p ès,è èyè é by Lemma 2.3fè. Considering then the çèp èsè è-continuous extensions f' y è:è of' y è:è onto the Hilbert space E; e ë:; :ë èsè, it follows kf' y k = k' y k ; where k:k denotes the norm of a linear functional. The Theorem of Frçechet-Riesz yields as well the existence of an element ~z 2 e E such that f' y è~xè =ë~z; ~xë èsè ; ~x 2 e E,askf'y k = k~zk èsè ; k:k èsè := p ë:; :ë èsè ; for the norm kf' y k of the linear functional f' y è:è. The latter now yields k~zk èsè = kf' y k = k' y k = jèy; xèj sup 06=x2E p èsè èxè = pès+è èyè : è2.5è Since the above mapping y 7! ~z is linear and k:k èsè is Hilbertian, è2.5è implies that the parallelogram identity applies to p ès+è. Hence, p ès+è is Hilbertian. Recalling that p = p è0è is assumed to be Hilbertian, the lemma under consideration is veriæed. Letting E; è:; :è be a non-degenerate inner product space, the present is aimed at constructing selfpolar norms on E; è:; :è. This construction is based on an iteration process, where the iteration uses a function æ : R + æ R +! R + satisfying the properties listed below. 6

7 2.5 Deænition. A function æ : R + æ R +! R + is called selfpolarly normiterating function if èiè æèa; bè =æèb; aè for each a; b 2 R +, èiiè æèa; cè ç æèa; bè, if c ç b and a; b; c 2 R +, èiiiè æèa; bè ç p ab for each a; b 2 R +, èivè æèça; çbè =çæèa; bè for each a; b; ç 2 R +, èvè the equation a =æèa; bè is solvable in R +, and if aé0, then there is the unique solution b = a. The following examples are of some special interest for applications. 2.6 Example. Consider aè æ èa; bè = èa + bè; 2 q bè æ 2 èa; bè = 2 èa2 + b 2 è; cè æ 3 èa; bè = p ab; a; b 2 R +. It is straightforward to check that æ j ;j =; 2; 3; are selfpolarly norm-iterating functions. Recalling Notation 2., let us conærm the following. 2.7 Notation. For a non-degenerate inner product space E; è:; :è, let us put Q := fp 2P; p 0 èxè ç pèxè; x2 Eg ; eq := fp 2 e P; p 0 èxè ç pèxè; x2 Eg : The Aronszajn-Schatten iteration of selfpolar norms is given in the following. 2.8 Theorem. Letting E; è:; :è be a non-degenerate inner product space, p 2 e Q, and æè:; :è a selfpolarly norm-iterating function, deæne recursively p n+ èxè :=æèp n èxè;p 0 n èxèè ; è2.6è p 0 èxè =pèxè; x2 E; n =0; ; 2;:::. Then, aè pèxè := lim n! p n èxè; p èè èxè :=lim n! p 0 nèxè exist, and pèxè = p èè èxè for each x 2 E, bè p 0 èxè ç p 0 èxè çæææç p0 n èxè ç p0 n+ èxè çæææç p èxè çæææç p n+ èxè ç p n èxè çæææç p èxè ç pèxè; x2 E; cè pè:è is a selfpolar norm on E. 7

8 Proof. aè, bè: Assuming that for some n 2 N 0, p n è:è 2 e Q. Then p 0 nèxè ç p n èxè é è2.7è holds for each x 2 E, and consequently, p n+ èxè =æèp n èxè;p 0 nèxèè èæè ç æèp n èxè;p n èxèè èææè ç p n èxè ; è2.8è where è*è follows from è2.7è and Deænition 2.5 èiiè, and è**è is a consequence of Deænition 2.5 èvè. Noticing that p n+ è:è also satisæes è2.è due to Deænition 2.5 èivè, Lemma 2.3cè applies to p n+ è:è and p n è:è, and thus è2.8è yields p 0 nèxè ç p 0 n+èxè : è2.9è Using è2.4è, it follows as well jèx; yèjçp 0 nèxè p n èyè asjèx; yèjçp n èxè p 0 nèyè; and then jèx; yèjç p p 0 nèxè p n èyè p n èxè p 0 nèyè ; è2.0è x; y 2 E; where all factors on the right-hand side of è2.0è are ænite due to è2.7è. Then, ç p 0 jèx; yèj ç è! è+è jèx; yèj n+èxè = sup ç sup p 06=y2E p n+ èyè 06=y2E p 0 nèyè p n èyè è++è ç p p 0 n èxè p n èyè p n èxè p 0 nèyè p p 0 n èyè p n èyè = p p 0 nèxè p n èxè è+è ç æèp n èxè;p 0 n èxèè = p n+èxè ; è2.è x; y 2 E, where both inequalities è+è follow from è2.6è and Deænition 2.5 èiiiè, and è++è is a consequence of è2.0è. Since p 0 è:è =pè:è 2 e Q by assumption of the theorem under consideration, è2.8è, è2.9è and è2.è hold for each n 2 N 0, and thus p 0 èxè ç 0 p0 èxè çæææç p0 n èxè ç p0 n+èxè ç ::: ç p n+ èxè ç p n èxè çæææç p èxè ç p 0 èxè : è2.2è For each x 0 2 E, consider now the two sequences of real numbers èp n èx 0 èè n=0 and èp 0 nèx 0 èè n=0. Since èp n èx 0 èè n=0 is monotonously decreasing and bounded from belowby p 0 0 èx 0è èresp. èp 0 n èx 0èè n=0 is monotonously increasing and bounded from above by p 0 èx 0 èè due to è2.2è, a theorem of classical analysis on the convergence of monotonous sequences yields the existence of real numbers pèx 0 è, p èè èx 0 è such that pèx 0 è = lim n! p nèx 0 è and p èè èx 0 è = lim n! p0 nèx 0 è : For varifying aè and bè, it remains to show that pèxè =p èè èxè; x2 E. If x 0 = 0, then obviously pè0è = p èè è0è = 0 by è2.2è and p 0 è0è = 0. Assuming 8

9 now x 0 6= 0, the assumption pè:è ç p 0 è:è of the theorem under consideration and Lemma 2.3fè èèiè è èiièè yield p 0 èx 0 è é 0 : è2.3è Considering the limit n!in è2.6è, it follows 0 ép 0 0 èx 0è ç pèx 0 è=æèpèx 0 è;p èè èx 0 èè ; and then pèx 0 è=p èè èx 0 èby èvè of Deænition 2.5. cè: For showing that pè:è =p èè è:è is a norm, notice ærst that for each n 2 N 0, p 0 n èx+yè ç p0 n èxè+p0 n èyè; p0 n èçxè =jçj p0 nèxè; x;y2 E; ç 2 C : Taking the limit n!in the above relations, it follows that pè:è is a seminorm. Taking è2.3è and è2.2è into account, it follows that pè:è even is a norm. For showing that pè:è is selfpolar, take p 0 nèxè ç pèxè ç p n èxè from bè. Using Lemma 2.3cè and Lemma 2.3eè, it follows p 0 nèxè ç p 0 èxè ç p 00 nèxè ç p n èxè ; è2.4è x 2 E; n 2 N 0. Considering the limit n!in è2.4è und using pè:è = p èè è:è, pèxè = lim n! p0 nèxè ç p 0 èxè ç lim n! p nèxè =pèxè ; x 2 E, are implied. The proof is complete. Sometimes it will be written p;æè:è instead of pè:è in order to refer explicitly to the function æ used in the iteration process è2.6è. 2.9 Corollary. Let E; è:; :è be a non-degenerate inner product space and p a norm on E. aè If çèp 0 è ç çèpè, then there is a selfpolar norm q on E;è:; :è such that çèp 0 è ç çèqè ç çèpè: bè If çèpè = çèp 0 è, then a selfpolar norm q exists on E;è:; :è such that çèpè =çèqè: Proof. aè Assuming çèp 0 è ç çèpè, there is a constant 0 é c é such that p 0 èxè ç cpèxè; x2 E. Setting qèxè := p cpèxè, it follows çèqè =çèpè and q 0 èxè èæè =è p cè, p 0 èxè ç p cpèxè =qèxè ; x 2 E, where è*è follows from Lemma 2.3bè. Applying Theorem 2.8 to the norm q, there is a selfpolar norm p satisfying q 0 èxè ç qèxè ç qèxè;x2 E. Hence, çèp 0 è=çèq 0 è ç çèqè ç çèqè =çèpè follow. bè is an immediate consequence of aè. 9

10 2.0 Corollary. Let E;è:; :è be a non-degenerate inner product space and p; q norms on E such that çèpè = çèqè and p 0 èxè ç pèxè; q 0 èxè ç qèxè for each x 2 E. For every selfpolarly norm-iterating function æ, take the selfpolar norms p;æ; q;æ from Theorem 2.8. It then follows ç èp;æè =ç èq;æè. Proof. Letting the assumptions of the corollary under consideration be satisæed, çèpè =çèqè implies the existence of constants 0 éc;dé such that cpèxè ç qèxè ç dpèxè; x2 E. Setting ç = maxfc, ;dg, it follows and then by Lemma 2.3bè, cè, ç, pèxè ç qèxè ç çpèxè ; ç, p 0 èxè ç q 0 èxè ç çp 0 èxè ; è2.5è è2.6è x 2 E. Considering the ærst step of the Aronszajn-Schatten iteration, it follows p èxè =æèpèxè;p 0 èxèè èæè ç æèçqèxè;çq 0 èxèè èææè = ç æèqèxè;q 0 èxèè = çq èxè ; and analogously, q èxè ç çp èxè, where è*è follows from è2.5è, è2.6è and Deænition 2.5 èiè, èiiè, and è**è is a consequence of Deænition 2.5 èivè. Hence, ç, p èxè ç q èxè ç çp èxè and ç, p 0 èxè ç q 0 èxè ç çp 0 èxè. Assuming now that there is an n 2 N such that ç, p n èxè ç q n èxè ç çp n èxè ; ç, p 0 nèxè ç q 0 nèxè ç çp 0 nèxè ; è2.7è the same reasoning as above yields ç, p n+ èxè ç q n+ èxè ç çp n+ èxè. Hence è2.7è holds for each n 2 N. Considering now the limit n!in è2.7è, it follows ç, p;æèxè ç q;æèxè ç çp;æèxè; x 2 E. Hence, çèp;æè = çèq;æè. 2. Corollary. Letting p be a Hilbertian norm on a non-degenerate inner product space E;è:; :è satisfying p 0 èxè ç pèxè; x2 E; and using the selfpolarly norm-iterating function æ 2, the Aronszajn-Schatten iteration yields a Hilbertian selfpolar norm p;æ 2 on E;è:; :è. Proof. Assume that there is an n 2 N 0 such that p 0 nèxè ç p n èxè; x2 E; and p n is a Hilbertian norm on E, where p n is taken from è2.6è using æ 2. Noticing that Lemma 2.3fè èèiè è èiiièè yields çèp n è ç ç, it follows from Lemma 2.4 that p 0 n is a Hilbertian norm on E, too. Hence the parallelogram identity applies to both norms p n and p 0 n, and thus, èp n+ èx + yèè 2 +èp n+ èx, yèè 2 =æ 2 èp n èx + yè;p 0 nèx + yèè 2 +æ 2 èp n èx, yè;p 0 nèx, yèè 2 = p n èx + yè 2 + p 0 n èx + yè2 + p n èx, yè 2 + p 0 nèx, yè2 =2èp n èxè 2 + p n èyè 2 + p 0 nèxè 2 + p 0 nèyè 2 è = 2 èæèp n èxè;p 0 nèxèè 2 +æèp n èyè;p 0 nèyèè 2 =2èp n+ èxè 2 + p n+ èyè 2 è ; è2.8è 0

11 x; y 2 E, verifying that the parallelogram identity applies to p n+, too. Hence p n+ is a Hilbertian norm on E, and p 0 èxè ç n+ p n+èxè; x2 E by è2.è. Noticing that the assumption made at the beginning of the proof holds for n = 0 due to the assumptions of the corollary under consideration, è2.8è applies to each n 2 N 0. Considering the limit n!in è2.8è, it follows that the norm satisæes the parallelogram identity, and consequently it is Hilbertian. p;æ 2 Taking æ from Example 2.6, there are simple "a priori" and "a posteriori" estimates for the Aronszajn-Schatten iteration process. 2.2 Lemma. Letting p 2 e Q on a non-degenerate inner product space E;è:; :è, consider the Aronszajn-Schatten iteration p n èxè =æ èp n,èxè;p 0 n,èxèè = 2 èp n,èxè+p 0 n,èxèè ; è2.9è p 0 èxè =pèxè; n=; 2;:::; x2 E. Then, aè jp;æ èxè, p n èxèjç2,n èpèxè, p 0 èxèè èa priori estimateè, bè jp;æ èxè, p n èxèj ç p n èxè, p n,èxè èa posteriori estimateè for each x 2 E. Proof. aè Using Theorem 2.8bè and è2.9è, the a priori estimate follows from jp;æ èxè, p n èxèj ç p n èxè, p 0 nèxè ç 2 èp n,èxè+p 0 èxèè, n, p0 èxè n, = 2 èp n,èxè, p 0 n,èxèè ç ::: ç 2,n èp 0 èxè, p 0 0èxèè ; x 2 E: bè Again using Theorem 2.8bè and è2.9è, the a posteriori estimate follows from jp;æ èxè, p n èxèj ç p n èxè, p 0 nèxè ç p n èxè, p 0 n,èxè = p n èxè, è2p n èxè, p n,èxèè = p n,èxè, p n èxè ; x 2 E: 3 Selfpolar and selfpolar Hilbertian norms on inner product spaces of countable dimension While the existence and construction of selfpolar and selfpolar Hilbertian norms was considered in Theorem 2.8 and Corollary 2., resp., the question of whether or not all selfpolar norms are equivalent is not answered by those results. The aim of the present section is to give, for a certain class of inner product spaces, an explicit construction of inænitely many diæerent selfpolar topologies èin contrast to the deænite case í cf. èaè of the introductionè.

12 3. Inner product spaces of countable dimension Let E be a real vector space of countably inænite algebraic dimension endowed with an inner product è:; :è E so that E;è:; :è E is non-degenerate. If E is quasi-deænite, then all selfpolar norms are equivalent ècf. the introductionè. So, for the following E is supposed not to be quasi-deænite. Since there is an orthonormal basis fe n g n=, èe n ;e m è E = ææ nm ë5, Sect.IV.3ë, it is obvious that E;è:; :è E is isometrically isomorphic to dèrè, the space of all sequences of real numbers with a ænite number of non-zero members, endowed with the inner product, èx 0 ;x ;x 2 ;:::è; èy 0 ;y ;y 2 ;:::è æ d := X k=0 èx 2k y 2k, x 2k+ y 2k+ è: è3.è Letting M 2 denote the two dimensional Minkowski space, i.e. R 2 equipped with the inner product èx; yè :=x 0 y 0,x y, one can think of the abovedèrè; è:; :è d as composed of inænitely many copies of M 2 ; è:; :è. Hence, a large class of selfpolar norms on dèrè; è:;:è d can be constructed starting from sequences of selfpolar norms on M 2 ; è:; :è: 3. Proposition. aè Let p 0 ;p ;p 2 ;::: be asequence of selfpolar norms on M 2 ; è:; :è and deæne pèxè := vu u t X k=0 p 2 k èx2k ;x 2k+ è ; x =èx 0 ;x ;x 2 ;:::è 2 dèrè: è3.2è Then p is a selfpolar norm on dèrè; è:;:è d. If in addition all the p k are Hilbertian, then so is p. bè If p is any selfpolar norm on dèrè; è:; :è d, and U is an isometric or antiisometric automorphism of this space, then p U èxè :=pèu, xè is also a selfpolar norm. If p is Hilbertian, then so is p U. Proof. In order to show in aè that p is selfpolar, ærst observe that jèx; yè d jç pèxè pèyè. Now, given any x 2 dèrè, choose for each k a pair èy 2k ;y 2k+ è such that x 2k y 2k, x 2k+ y 2k+ = p 2 k èx2k ;x 2k+ è and p k èy 2k ;y 2k+ è=p k èx 2k ;x 2k+ è. Collecting all these pairs one obtains a y 2 dèrè satisfying èx; yè d = p 2 èxè and pèyè = pèxè. The rest of the proof is straightforward. Two norms p and q on dèrè, constructed as in Proposition 3. aè from diæerent sequences èp k è and èq k è might or might not be equivalent. There is a simple criterion to decide this question: 2

13 3.2 Lemma. Let p 0 ;p ;p 2 ;::: and q 0 ;q ;q 2 ;::: be two sequences of selfpolar norms on M 2 ; è:; :è and denote by p resp. q the corresponding norms on dèrè as in è3.2è. Set ç + k èp; qè = sup p kèxè 06=x2M 2 q, ç, kèxè k èp; qè = inf p kèxè 06=x2M 2 q. kèxè Then the following are equivalent: èiè p and q deæne the same topology on dèrè. èiiè ç + k èp; qè ç Céfor each k. èiiiè ç, k èp; qè ç cé0 for each k. Proof. Obviously, èiiè applies if and only if p is weaker than q, and èiiiè is equivalent toq being weaker than p. In the present case, both p and q are selfpolar. This means that if one of them is weaker than the other they must be equivalent ë5, Thm. IV.4.2ë. 3.2 Selfpolar norms on M 2 Since the aim is to use Proposition 3. and Lemma 3.2 in order to provide a èpartialè overview of the possible selfpolar and selfpolar Hilbertian topologies on dèrè, the following two sections are devoted to an investigation of the selfpolar and selfpolar Hilbertian norms on M 2. They are described in detail and a complete classiæcation is given. All the proofs of the assertions below and more details will be given in chapter 4. It turns out to be usefull, not to consider a norm p itself, but the p := fx 2M 2 j pèxè =g, i.e. the boundary of the corresponding unit ball U p. Curves in R 2 will be described in polar coordinates, i.e. by functions rè'è, where r gives the èeuclideanè length of a vector, and ' 2 S denotes the angle between the vector and the positive x 0 -axis. 3.3 Observation. There is the following bijection between the norms p on R 2 and the ç-periodic, convex, closed curves r : S! ër ;r 2 ëèr ;r 2 é 0è in R 2 : p 7! r p ; r p è'è :=pç, cos ' sin 'æç, ; '2 S ; r 7! p r ; p r èxè := ç rè'è ; x = ç, cos ' sin 'æ 2 R2 èç ç 0è: è3.3è The homogeneity of the norm includes its reæection invariance, and thus the ç-periodicity ofr p. The restriction of the image of r to some closed interval ër ;r 2 ë reæects both, the fact that the unit ball associated to p must be absorbing and that it must lie in some ænite Euclidean ball. Convexity ænally, which by deænition means that for any two points of the curve the straight line between these points lies completely inside the curve, comes from homogeneity and the triangle inequality for p. 3

14 h 2 t 2 ' 2 è 2 r 2 r 2 ' r r è h t Figure : The choice of èr ;' è and èr 2 ;' 2 è. coordinate axes at æ. The hyperbolae intersect the 3.4 Observation. The set of selfpolar norms on M 2 is invariant under Lorentz transformations. Consider any selfpolar norm p and let æ denote a linear isometry of M 2. Then p æ èxè :=pèæ, xè is a selfpolar norm as well. This remains true if æ is chosen to be an anti-isometry, èæx; æyè =,èx; yè. The curve which is associated to p æ is the image of the curve r p under æ. If, e.g., p is the Euclidean norm, i.e. r p è'è ç describes the unit circle, then under a Lorentz transformation, this circle turns into some ellipse, the axes of which lie along the bisectors of the ærst and second quadrants and for which the product of the half axes equals unity. The construction of selfpolar norms on M 2 which will now be described needs the following two ingredients: Deæne I. The ærst ingredient isapair of angles è' ;' 2 è 2 S æ S subject to the following conditions:, ç 4 é' é ç 4 ; ç 4 é' 2 é 3ç 4 ; è3.4è cos 2 è' + ' 2 è ç cosè2' è jcosè2' 2 èj: è3.5è r := p cos 2' ; r 2 := p ; è jcos 2'2j :=,' ; è 2 := ç, ' 2 : The condition è3.5è is now equivalent tor r 2 jcosè' + ' 2 èjç. Figure shows the situation: Each of the points èr ;' è and èr 2 ;' 2 è lies on one of the hyperbolae èx 0 è 2,èx è 2 = æ. Let us call them h and h 2 as shown in the picture. The tangents of h and h 2 in the given points ècalled t and t 2 resp.è 4

15 t 2 h 2 èr 2 ;' 2 è r0è'è èr ;' è h t Figure 2: The choice of r 0 è'è. are given by their respective normal directions è and è 2. The projections of r and r 2 onto the directions è and è 2 are =r and =r 2 respectively. According to condition è3.5è, the points on the hyperbolae have to be chosen in such away that èr ;' è lies ëbelow" or on t 2 and that èr 2 ;' 2 è lies ëto the left of" or on t. II. The second ingredient for the construction described below is an arbitrary convex arc r 0 :ë' ;' 2 ë! R satisfying r 0 è' è=r ; r 0 è' 2 è=r 2 ; è3.6è è èrè r 0 è' è èlè r 0 è' 2 è ç è 2 : è3.7è èl=rè r 0 è'è denotes the normal direction of the left resp. right derivative of the curve r 0 in the point ' ècf. Deænition 4.3è. The convexity in this case is to be understood in the same way as for closed curves. See Figure 2 for an illustration of the above conditions: The arc r 0 è'è must lie inside the triangle formed by t, t 2 and the line through èr ;' è and èr 2 ;' 2 è. The idea of the construction is, to extend the chosen arc r 0 è'è to a complete closed curve, which will represent the desired selfpolar norm. First, deæne ræ :ë' 2, ç; ' ë! R by ræèèè := sup '2ë' ;' 2ë r 0è'è cosèè + 'è ; è 2 ë' 2, ç; ' ë: 5

16 Now, the completed curve is given by r : S! R rè'è := 8 é é: r 0 è'è ræè'è r 0 è' + çè ræè', çè ; ' 2 ë' ;' 2 ë ; ' 2 ë' 2, ç; ' ë ; ' 2 ë', ç; ' 2, çë ; ' 2 ë' 2 ;' + çë: è3.8è This yields the desired selfpolar norm: 3.5 Proposition èsee Prop. 4.4è. The function rè'è is the representation in polar coordinates of a ç-periodic convex closed curve, i.e., it deænes a norm p r in R 2 ècf. è3.3èè. 3.6 Theorem èsee Thm. 4.5è. p r = p 0 r, p r is selfpolar è 0 : Polarization with respect to M 2 è. It is shown in chapter 4 that every selfpolar norm on M 2 can be constructed in the above described manner: 3.7 Theorem èsee Thm. 4.7è. Set P g := f è' ;' 2 è 2 S æ S j, ç 4 é' é ç 4 ; ç 4 é' 2 é 3ç 4 ; cos 2 è' + ' 2 è é cosè2' è jcosè2' 2 èjg æfç :ë0; ç 2 ë! Rj ç describes a convex arc from è; 0è to è; ç 2 è, completely lying inside the triangle è; 0è; è p 2; ç 4 è; è; ç 2 è g; P e := f è' ;' 2 è 2 S æ S j, ç 4 é' é ç 4 ; ç 4 é' 2 é 3ç 4 ; cos 2 è' + ' 2 è = cosè2' è jcosè2' 2 èjg: There is a bijection from P := P g ëp e èdisjoint unionè into the set of selfpolar norms on M 2 èessentially given by the above constructionè. 3.8 Remark èdescription of the admissible angles ', ' 2 è. The èangle parts of theè parameter sets P g and P e which occur in Theorem 3.7 are depicted in Figure 3. Using the transformation ææ := ' æ è' 2, ç=2è one calculates: cos 2 è' + ' 2 è = sin 2 èæ + è;, cosè2' è cosè2' 2 è = cos 2 èæ,è, sin 2 èæ + è. The condition for P g reads in these coordinates: sin 2 èæ + è é =2 cos 2 èæ,è. For P e : sin 2 èæ + è==2 cos 2 èæ,è. Hence, the points è' ;' 2 è belonging to P g form the interior of the gray lense, the elements of P e are exactly the points of its boundary. 3.3 Selfpolar Hilbertian norms on M 2 In this section it will be determined, which of the selfpolar norms described before are in addition Hilbertian èi.e. which of them are derived from scalar 6

17 ' 2 3ç 4 P e P g, ç 4 ç 4 ' P e Figure 3: The admissible values for è' ;' 2 è ç 4 productsè. The easiest way to do so is to use the following theorem due to Hansen ë7ë. 3.9 Theorem èhansenè. Let E; èæ; æè be a non-degenerate inner product space with an additional scalar product ëæ; æë. The corresponding norm kxk := p ëx; xë is supposed to satisfy jèx; yèj çckxkkyk for some Cé0 èi.e. kæk is to be a majorantè. On the Hilbert space H := e E kæk ècompletion with respect to the norm topologyè the extended inner product èæ; æè ç is assumed tobe non-degenerate. Let p be a norm on E. Then p is Hilbertian and selfpolar if and only if there is a positive and bounded operator T on H; ëæ; æë with the following properties: aè 0 is not an eigenvalue of T, bè JHçDèT, 2 è èthe operator J connects the two inner products: èx; yè =ëx; Jyëè, cè kt 2 xk = kt, 2 Jxk, x 2 E, dè pèxè =kt 2 xk, x 2 E. For two dimensional Minkowski space, E = M 2,ëx; yë =x 0 y 0 + x y,èx; yè = x 0 y 0, x y, J =, 0 0,æ, the theorem reads: The Hilbertian selfpolar norms p have the form pèxè =kt 2 xk, where T is any positive deænite matrix satisfying kt 2 xk = kt, 2 Jxk. The condition for T is in this case equivalent tot = JT, J. 3.0 Proposition. The Hilbertian selfpolar norms on M 2 are given by the formula pèxè = for any æé0. r æç æ ç2, ; ç æ := p 2 èx 0 æ x è èx 2M 2 è; 7

18 3. Remark. The curves pèxè = corresponding to the Hilbertian selfpolar norms p on M 2 are exactly the ellipses which were described in Observation 3.4 èand which can be derived from the unit circle by Lorentz transformationè. Proof of the Proposition. p must have the form p 2 èxè =ëx; T xë =x T Tx, where T satisæes T = JT, J. Let x ;x 2 denote an orthonormal basis of R 2,ëæ; æë, with Tx = t x, Tx 2 = t 2 x 2 èt ;t 2 é 0è. Then TJx = t, Jx,sot, is also an eigenvalue. Now, if t = t 2 then t, = t 2 = t, and so T =l. If t 6= t 2 then Jx, since it is an eigenvector, must be either parallel or perpendicular to x. In the former case, x 2 must also be an eigenvector of J èbecause it is perpendicular to x è and so t, = t and t, 2 = t 2, yielding t = t 2 =. In the latter case, èx ;x è=ëx ;Jx ë = 0, i.e., x èand with it x 2 è is lightlike and thus proportional to è; è or è;,è. The construction of T from its eigenvectors and the eigenvalues t, t 2 = t, yields the given form for pèxè. 3.4 Results for non-quasi-deænite spaces of countable dimension Collecting the results of the previous sections, one ænds the following properties of dèrè; è:;:è d which carry over to E;è:; :è E èas explained at the beginning of section 3.è. 3.2 Theorem. aè There is a non-denumerable set of diæerent selfpolar topologies. bè Even more, thereare non-denumerably many non-equivalent selfpolar Hilbertian norms. cè Every selfpolar norm which is constructed as in Proposition 3. aè and bè is equivalent to some non-selfpolar norm. dè Every selfpolar norm which is constructed as in Proposition 3. aè and bè is equivalent to some selfpolar Hilbertian norm. Proof. It suæces to consider norms of the form è3.2è, because all the properties are invariant under the transformations in 3. bè. These norms are given by sequences èp k è of norms on M 2, i.e. sequences of curves èr pk è ècf. Observation 3.3è. Given two such sequences èp k è and èq k è, the possible equivalence of the corresponding norms on dèrè, p and q, is determined by ç + k èp; qè = sup '2S èor alternatively ç, k èp; qè = inf '2S r qk è'è r pk è í cf. Lemma 3.2. è'è r qk è'è r pk è'è Now, it is easy to construct sequences of "selfpolar" curves èusing the construction of section 3.2è in such a way that the corresponding selfpolar norms on dèrè become mutually non-equivalent. It is also possible to do this using only 8

19 Figure 4: The allowed sector for the Lorentz transformed r pk èsee the proof of Thm. 3.2 dèè. The diagonal lines ètangents of the hyperbolae in èæ p 2; æèè are given by the points è0; æè, èæ p ; 2 0è. the ellipses which correspond to selfpolar Hilbertian norms èremark 3.è. This proves aè and bè. cè For any given norm of the form è3.2è it is possible to modify one or more of the curves r pk and thereby destroy the property of being selfpolar, e.g. by leaving the part of the curve between ' and ' 2 unchanged and deforming the part in ë' 2, ç; ' ë ècf. the construction in section 3.2è. If this is done in such a way that èiiè and èiiiè of Lemma 3.2 hold, then the given norm and the modiæed one are equivalent. dè Let p be any selfpolar norm of the form è3.2è. Denote for every k by èr 2 èkè;' 2 èkèè the point on h 2 which is touched by r pk èsince p k is selfpolar there is exactly one such point, because r pk can be recovered by the construction of section 3.2è í cf. e.g. Figure 2. Deæne q k as that selfpolar Hilbertian norm on M 2 which is described by the ellipse which touches èr 2 èkè;' 2 èkèè. The q k combine to a selfpolar Hilbertian norm q and it remains to show, that p and q are equivalent. To this end ç + k èp; qè èin the above formè has to be considered. Apply for each k to p k and q k that Lorentz transformation which maps the respective ellipse r qk to the unit circle. Since the transformation is linear, this does not change the ç + k, which thus can be calculated using the circle and the transformed r pk. The latter passes through è0; è and, since it is again selfpolar, it can be described by the construction of section 3.2. Now, the conditions of this construction strongly limit the possible course of the curve. It must lie inside the shaded area shown in Figure 4. This is valid for every k, so there are 9

20 bounds on ç + k which are independent ofk. Hence, the equivalence of p and q follows by Lemma Remark. aè It is known that there are selfpolar norms on dèrè; è:; :è d which are not equivalent to any Hilbertian norm. An example of such a norm can be derived from the Araki-Hansen example ë7, Ex. 2.0ë. More precisely, on dèrè endowed with the inner product h:j:i given in ë7, Eq. è2.35èë the `-norm èrestricted to dè is selfpolar, but it is not equivalent to any Hilbertian norm. Remember that dèrè; h:j:i is isometrically isomorphic to dèrè; è:; :è d. bè There are selfpolar norms on dèrè; è:; :è d which are not equivalent toany of the norms of Proposition 3. aè and bè, due to part aè and Theorem 3.2 dè. 4 Selfpolar norms on M 2 In this section the selfpolar norms on two dimensional Minkowski space M 2, i.e. R 2 equipped with the inner product èx; yè :=x 0 y 0, x y, are discussed in detail and a complete classiæcation is given. As pointed out in section 3.2 the idea is not to consider the norms on R 2 themselves but the boundaries of their unit balls í cf. Observation 3.3. In order to exploit the correspondence between norms and curves, a few general notions and facts about curves with the properties listed in Observation 3.3 are needed: 4. Convex curves Using periodicity would be no advantage here, so let for this entire section r : S! ër ;r 2 ëèr ;r 2 é 0è be the representation in polar coordinates of a convex, closed curve inr 2. For simplicity of notation the following conventions with respect to S will be used: aè A statement ' ç ' 2 ç ' 3 ::: involving three or more points ' i 2 S means that these points are lying in the given order anti-clockwise around the circle. The sign é instead of ç means that the points are in addition diæerent from each other. bè Intervals of S are denoted as follows: è' ;' 2 è:=f' 2 S j ' é'é' 2 g and ë' ;' 2 ë:=f' 2 S j ' ç ' ç ' 2 g if ' 6= ' 2.ë'; 'ë :=f'g. 20

21 rè' 0 è ' 0 è gè'è Figure 5: The deænition of a support line ècf. Deænition 4.è. cè For a given vector x 2 R 2,æèxè 2 S denotes the polar angle of x èwith respect to the positive x 0 -axisè. Now, the convexity of r èin the sense of the above deænitionè is equivalent to the following property ofçè'è :=rè'è, : çè' è sinè' 2, ' 3 è+çè' 2 è sinè' 3, ' è+çè' 3 è sinè', ' 2 è ç 0 for ' é' 2 é' 3 ; 0 é' 3, ' éç; as can be seen by an elementary calculation in polar coordinates. Thus, ç is a so-called trigonometrically convex function. In addition it is bounded from above and from below by positive constants. This has í similarly as in the case of convex functions in the usual sense í some very strong implications ècf. ë6ëè which directly carry over from ç to r : aè The function r is continuous, bè At every point, r possesses a left and a right derivative, cè The function r is everywhere diæerentiable, except on a countable set of points. The following deænitions and lemmas are merely adaptations of the discussion of usual convex functions to the present case ècf. ë22, ch.ië or ë23, ch.vëè. So, most of the proofs are only sketched or completely omitted. The reader is, however, invited to make the following statements clear to himself by simply drawing two dimensional sketches. 4. Deænition. The straight line gè'è = rè' 0è cosè' 0, èè cosè', èè ' 2 èè, ç 2 ;è+ ç 2 è; è4.è 2

22 which touches the curve inèrè' 0 è;' 0 è, is called a support line of rè'è in' 0,if its deæning angle è 2 S èits normal directionè satisæes rè'è cosè', èè ç rè' 0 è cosè' 0, èè 8' 2 S ; i.e., if the entire curve rè'è lies on one side of the line ècf. Figure 5è. 4.2 Lemma. The support lines of rè'è in ' 0 are exactly those lines given by è4.è, for which è satisæes " ç ç è " ç ç è d èlè rè'è cos ' æ, ç d' rè'è sin ' 2 ç è ç d èrè rè'è cos ' æ, ç d' rè'è sin ' 2 ; where dèlè d' dèrè and d' '=' 0 '=' 0 denote the left and the right derivative respectively. The subtraction of ç=2 is necessary here, in order to transform the directions of the extremal support lines given by the derivatives into their normal directions ècf. Deænition 4.è. This Lemma motivates the following deænition. 4.3 Deænition. aè èlè r : S h! S èrè r : S! S èl=rè d rè' 0 è:=æ èl=rè rè'è cos ' æ i, ç. 2 d' rè'è sin ' '=' 0 bè : S!fintervals of S g 0 è:=ë@ èlè rè' 0 è;@ èrè rè' 0 èë. 4.4 Lemma. aè Let ' é' 2 é' 3. èlè rè' è èrè rè' è èlè rè' 2 è èrè rè' 2 è èlè rè' 3 è èrè rè' 3 è. bè Let ' ;' 2 2 S. Then S '2ë' ;' =ë@èlè rè' è;@ èrè rè' 2 èë. Sketch of the proof. aè Consider the èsupportè lines given by the normal èl=rè rè' i è and add the lines which pass through the points èrè' è;' è, èrè' 2 è;' 2 è or èrè' 2 è;' 2 è, èrè' 3 è;' 3 è respectively. By deænition of convexity, the arcs between the points must lie ëbeyond" the latter lines. This yields the assertion. Cf. also ë23, Thm. 24.ë. bè follows from aè. Most important for the later application is the notion of the conjugate convex curve r æ : 22

23 4.5 Deænition. r æ èèè := sup ' rè'è cosè',èè ; è 2 S : 4.6 Proposition. The function r æ : S! ër æ ;ræ 2ëèr æ ;ræ 2 é 0è is the representation in polar coordinates of a convex, closed curve in R 2. Sketch of the proof. In this instance it is appropriate to write r æ in the obviously equivalent form r æ èèè = inf '2èè,ç=2;è+ç=2èërè'è cosè', èèë,. Consider the family of lines fg ' èèè =ërè'è cosè', èèë, j ' 2 S g. Now, the curve r æ èèè arises as the inner boundary of that part of the plane, which iscovered by these lines. 4.7 Lemma. aè r æ èèè rè'è cosè', èè ç 8'; è. bè r æ èèè rè'è cosè', èè = èè è Sketch of the proof. aè follows directly from Deænition 4.5. bè is a consequence of Lemma 4.2 and Deænition 4.. The operation æ deænes an involution on the set of curves considered in this section: 4.8 Proposition. r ææ è'è =rè'è 8': Proof. The two parts of Lemma 4.7 imply: = sup r æ èèè cosèè, 'è = rè'è r ææ è'è : è 4.9 Remark ègeometrical interpretation of the involution r $ r æ è. The notion of the conjugate convex curve deæned above is an adaptation of the Legendre transformation for usual convex functions ècf. ë22, 23ëè. This is illustrated by Figure 6. Compare also the formula in Deænition 4.5, which reads in terms of the trigonometrically convex functions ç := r,, ç æ := èr æ è, : ç æ èèè = sup ' with the usual Legendre transformation cosèè, 'è ; çè'è f ë èxè = sup èxy, fèyèè: y 23

24 aè bè è ' rè'è r æ èèè support line in ' èè ë èmè fèxè x support line in x with slope m Figure 6: aè The æ -operation described in this section, bè the usual Legendre transformation ë ècf. Remark 4.9è. 4.0 Lemma èconnection to the polarization of normsè. aè Let 0 denote the polarization with respect to the Euclidean scalar product in R 2. Then r p 0è'è =èr p è æ è'è for all ' and any norm p. bè Now, let 0 denote the polarization with respect to the Minkowski scalar product èx; yè =x 0 y 0, x y. Then r p 0è'è =èr p è æ è,'è for all ' and any norm p. Proof. bè r p 0è'è = p0ç, cos ' æç sin ' = sup è pç, cos è sin è æç, æ ææ ç, cos ' æ, cos è ; ææ sin èæçæ sin ' = sup r p èèè jcosè' + èèj = è èr p è æ è,'è ècf. equation è3.3è and Deænition 4.5è. Part aè can be proven analogously. 4.2 Construction of selfpolar norms on M 2 As already mentioned in section 3.2, the construction which is to be described now, starts from two ingredients ècalled I. and II. thereè. Let, for the present section, these objects be given. Remember also the deænition of the quantities r ;r 2 ;è ;è 2 which are derived from the ingredients. The existence of an arc r 0 with the properties listed in II. is ensured by part cè of the following Lemma: 4. Lemma. aè ' ç è + ç 2 ç ' + ç, ' 2 ç è 2 + ç 2 ç ' 2 + ç. 24

25 bè è ç è 2 ç è + ç ç è 2 + ç. cè Let è 2 denote the normal direction of the line through èr ;' è and èr 2 ;' 2 è. Then è ç è 2 ç è 2. Proof. The restrictions è3.4è are imposed in order to insure aè and bè. Condition è3.5è implies cè. The idea of the construction is, to extend the chosen arc r 0 è'è to a complete closed curve, which will represent the desired selfpolar norm. First, deæne ræ :ë' 2, ç; ' ë! R by ræèèè := sup '2ë' ;' 2ë r 0è'è cosèè + 'è ; è 2 ë' 2, ç; ' ë: 4.2 Lemma. aè ræèèè describes a convex arc. bè ræèèè r 0 è'è cosè' + èè ç 8' 2 ë' ;' 2 ë;è2 ë' 2, ç; ' ë. cè ræèèè r 0 è'è cosè' + èè = èè,è 0 è'è 0 è' è:=ëè ;@ èrè r 0 è' èë 0 è' 2 è:=ë@ èlè r 0 è' 2 è;è 2 ëè. Sketch of the proof. aè Write ræ as ræèèè = inf '2è,è,ç=2;,è+ç=2èëë' ;' 2ëër 0 è'è cosè' + èèë,. Note that due to è3.4è the intersection in this formula is non-empty! Analogous to the situation in Proposition 4.6, the arc ræ is deæned by a family of lines. bè By deænition. cè Like in the proof of Lemma 4.7 bè. For the limit cases ' = ', ' = ' 2 use the fact that,è 2 ëè ;è 2 ë. 4.3 Lemma. ræè' è=r 0 è' è=r ; ræè' 2, çè =r 0 è' 2 è=r 2. Proof.,' = è 0 è' è è ræè' è r 0 è' è cosè2' è = èlemma 4.2 cèè è ræè' è= r èdef. of r è.,è' 2, çè =è 2 0 è' 2 è è ræè' 2, çè r 0 è' 2 è jcosè2' 2 èj = èlemma 4.2 cèè è ræè' 2, çè =r 2 èdef. of r 2 è. 25

26 Now, the completed curve is given by r : S! R rè'è := 8 é é: r 0 è'è ræè'è r 0 è' + çè ræè', çè ; ' 2 ë' ;' 2 ë ; ' 2 ë' 2, ç; ' ë ; ' 2 ë', ç; ' 2, çë ; ' 2 ë' 2 ;' + çë: è4.2è 4.4 Proposition. The function rè'è is the representation in polar coordinates of a ç-periodic convex closed curve, i.e., it deænes a norm p r in R 2 ècf. è3.3èè. Proof. Periodicity and closedness follow directly from the construction ècf. Lemma 4.3è. Convexity: The four arcs constituting the curve rè'è are convex èlemma 4.2 aèè. Now, the composed curve isconvex, because the lines t and t 2 are support lines of rè'è in' and ' 2 respectively èlemma 4.2 bèè. 4.5 Theorem. rè'è =r æ è,'è 8' 2 S ; i.e., p r = p 0 r, p r is selfpolar è 0 : Polarization with respect to M 2, cf. Lemma 4.0 bèè. Proof. Since r and r æ are ç-periodic, it is enough to consider ' 2 ë' 2, ç; ' 2 ë. Case : ' 2 ë' 2, ç; ' ë In this case,' 2 ëè ;è 2 ë. Consequently there is a è 2 ë' ;' 2 ë such that,' èlemma 4.4 bèè. Thus, Lemma 4.7 bè implies: r æ è,'è rèèè cosèè+'è =. On the other hand,' 0 èèè. Lemma 4.2 cè yields: = ræè'è r 0 èèè cosèè + 'è =rè'è rèèè cosèè + 'è. Thus, r æ è,'è =rè'è. Case 2: ' 2 ë' ;' 2 ë Here,' 2 ëè 2, ç; è ë. So, there must be a è 2 ë' 2, ç; ' ë such that,' èlemma 4.4 bèè. Lemma 4.7 bè implies again: r æ è,'è rèèè cosèè+'è =. On the other hand ' æ è,èè. Indeed, if è 2 è' 2, ç; ' è, æ è,èè =,@rèèè due to case. If è = ', then,' è ë ëè 2, ç; è ë=ë@ èlè rè' è;è ë. So, ' 2 ë' ;,@ èlè rè' èë = ë' ;@ èrè r æ è,èèë ècf. case è. Now, the inequality r æ èèè cosèè, ' è ç =r = r æ è,èè cosè,è,' è 8è èlemma 4.7 aèè shows, using Lemma 4.2, ' æ è,èè, or: ' 2 ë' ;@ èrè r æ è,èèë æ è,èè. The ænal case è = ' 2, ç can be treated analogously. Finally apply Lemma 4.7 bè to r æ and use Proposition 4.8: rè'è r æ è,èè cosèè + 'è =. An application of case yields rè'è rèèè cosèè + 'è =, and thus r æ è,'è =rè'è. 4.3 Uniqueness and completeness of the construction The aim of the present section is to show è that diæerent choices of the ingredients I, II for the construction lead to diæerent selfpolar norms and 2è that every selfpolar norm can be obtained by the above construction èthm. 4.7è. 26

27 4.6 Lemma. Let p denote any selfpolar norm on M 2. Then the curve r p è'è from eq. è3.3è touches each of the four hyperbolae èx 0 è 2, èx è 2 = æ in exactly one point. The tangents of the hyperbolae in these points are support lines of r p è'è. Proof. Uniqueness: The hypothesis means: r æ pè,'è =r p è'è for every '. By Lemma 4.7 aè it follows that ër p è'èë 2 çjcosè2'èj, for all '. Thus, the entire curve r p è'è must lie between the four hyperbolae. On the other hand it has to be convex, so there cannot be more than one point of intersection with each of the hyperbolae. Existence of points of intersection: Consider the hyperbola h := h in the quadrant, ç é'é ç. Set, similarly to 4 4 Deænition 4.3: h 0 è:=æ, ç =,' 2 0 ; ' 0 2 ë, ç ; ç ë: 4 4 d d', hè'è cos ' æ hè'è sin ' '=' 0 Now deæne æ : ë, ç 4 ; ç ë!fintervals of 4 S g by æè'è p S By Lemma 4.4 bè: æè'è '2è,ç=4 ;ç=4è =è@èrè r p è, ç ç è 4 4 ;@èlè r p è ç è, ç 4 èè=è@èrè r p è, ç è, ç 4 4 ;@èlè r p è ç è+ ç è. This latter interval contains the 4 4 zero angle, èrè r p è, ç è 2 è, 3ç ; ç è and 4 4 r p è ç è 2 è, ç ; 3ç è. Hence, there must be some ' 0 2 è, ç 4 ; ç è, for which 02 4 æè' 0è, 0 è p è' 0 è. The hyperbola considered here has the 0 è=,' 0. Using this, Lemma 4.7 bè implies: ër p è' 0 èë 2 cosè2' 0 è=rpè,' æ 0 è r p è' 0 è cosè2' 0 è =. Since the equation of the hyperbola is ëhè'èë 2 cosè2'è =, the above identity shows that r p è'è intersects hè'è in the direction ' 0. The 0 è p è' 0 è means that the tangent ofh in this point of intersection is a support line of r p. The remaining three hyperbolae can be treated in the same way because of symmetry. At this stage the main theorem of the present section can be proven: 4.7 Theorem. Set P g := f è' ;' 2 è 2 S æ S j, ç 4 é' é ç 4 ; ç 4 é' 2 é 3ç 4 ; cos 2 è' + ' 2 è é cosè2' è jcosè2' 2 èjg æfç :ë0; ç 2 ë! Rj ç describes a convex arc from è; 0è to è; ç 2 è, completely lying inside the triangle è; 0è; è p 2; ç 4 è; è; ç 2 è g; P e := f è' ;' 2 è 2 S æ S j, ç 4 é' é ç 4 ; ç 4 é' 2 é 3ç 4 ; cos 2 è' + ' 2 è = cosè2' è jcosè2' 2 èjg: There is a bijection from P := P g ëp e èdisjoint unionè into the set of selfpolar norms on M 2 èessentially given by the construction described insection 4.2è. 27