Applied Categorical Structures 7: 279–295, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands.
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p-Banach Spaces and p-Totally Convex Spaces RALF KEMPER Fachbereich Mathematik, Fernuniversität, 58084 Hagen, Germany (Received: 11 December 1996; accepted: 12 September 1997) Abstract. We introduce the categories Vecp of p-normed vector spaces, Banp of p-Banach spaces, ACp of p-absolutely and TCp of p-totally convex spaces (0 < p ≤ 1). It will be shown that TCp (ACp ) is the Eilenberg–Moore category of Banp (Vecp ). Then congruence relations on TCp (ACp )-spaces are studied. There are many differences between TCp (ACp )-spaces and totally (absolutely) convex spaces (i.e. p = 1) (Pumplün and Röhrl, 1984, 1985), which will become apparent in Section 4. Mathematics Subject Classifications (1991): 18A40, 18C05, 18C20, 46A05. Key words: p-Banach space, p-normed vector space, (p-) totally (resp. (p-) absolutely) convex space, congruence relation, Eilenberg–Moore category, monad.
Introduction In [8] Pumplün and Röhrl introduced the category TC (resp. TCfin ) of (finitely) totally convex (t.c.) spaces, which are the Eilenberg–Moore algebras of the monad induced by the unit ball functor from the category of Banach spaces (resp. normed vector spaces) with linear contractions to the category of sets. In accordance with [2] we use the term “absolutely convex” for the spaces Pumplün and Röhrl call “finitely totally convex”. AC denotes the category of absolutely convex spaces. In the present paper the categories TCp of p-totally convex and ACp of pabsolutely convex spaces are introduced and investigated (0 < p ≤ 1). For p = 1 they coincide with the categories TC and AC and the results proved by Pumplün and Röhrl ([8, 9]) are contained as a special case in this paper, resp. in a continuation of this. The notion of p-absolutely convex spaces is a generalization of p-absolutely convex subsets of K-vector spaces and p-totally convex spaces are a generalization of p-totally convex subsets of topological K-vector spaces, K = R, C. In Section 1 we introduce the categories Vecp and Banp of p-normed vector spaces and p-Banach spaces [4] over the field K = R, C. p : Banp → Set (resp. p,fin : Vecp → Set) denotes the canonical unit ball functor. p and p,fin have left adjoints lp (resp. lp,fin ). Here, for a set X, lp (X) is the usual lp -spaces on X ([4]), while lp,fin (X) is the subspace of lp (X) consisting of all functions of finite support. p and p,fin are premonadic ([10]), i.e. the comparison functor from
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Banp (resp. Vecp ) to the corresponding category of Eilenberg–Moore algebras is full and faithful. After the definition of p-totally and p-absolutely convex spaces and the proof of some computational rules in Section 2 it is shown in Section 3 that the categories TCp and ACp are the Eilenberg–Moore algebras for p (resp. p,fin ). Both categories possess an internal hom-functor and a tensor product, they are closed categories in the sense of [7] and even autonomous categories in the sense of Linton ([6]). In Section 4 a semi p-norm on p-absolutely (resp. p-totally) convex spaces is introduced and properties of congruence relations on these spaces are studied. There are many differences between totally (resp. absolutely) and ptotally (resp. p-absolutely) convex spaces; this is especially true for the central Theorem 4.10, for which R. Börger gave a very important hint. 1. p-Normed Vector Spaces and p-Banach Spaces p will always denote a real number with 0 < p ≤ 1, and one puts ωp := ( 12 )1/(p−1); N is the set of natural numbers and K is the field R or C of real or complex numbers. For a set M we put |M| := card(M). DEFINITION 1.1. Let (Vi , k ki ) (i = 0, 1) be p-normed vector spaces ([4, p. 115]). A K-linear mapping f : V0 → V1 is called a contraction if and only if for every x ∈ V0 kf (x)k1 ≤ kxk0 holds. VecK,p or briefly Vecp is the category of all p-normed vector spaces together with these contractions. A p-Banach space is a complete p-normed vector space ([4, p. 116]). The full subcategory of Vecp which is determined by all p-Banach spaces is denoted by Banp . It is obvious that for all p, s with 0 < s ≤ p Vecp (resp. Banp ) is a subcategory of Vecs (resp. Bans ). All topological statements will refer to the topology induced by the p-norm. For a bounded linear mapping f : V → W between p-normed vector spaces kf (x)k | x ∈ V \{0} = sup kf (x)k | kxk ≤ 1 kf k := sup kxk = inf M ≥ 0 | kf (x)k ≤ Mkxk (x ∈ V ) denotes the usual p-norm. Every (V , k k) ∈ Vecp with respect to the supremum pnorm on V ×V is a locally p-convex space ([4, 6.5.1]) and the p-norm k k: V → R is continuous. PROPOSITION 1.2. For a p-normed vector space (V , k k) and a closed subspace U the canonical projection π : V → V /U satisfies: (i) π is a coequalizer in Vecp . (ii) kπ k = 0 or kπ k = 1 holds.
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Proof. (i) Obviously, W := {(x, y) ∈ V × V | x − y ∈ U } is a p-normed subspace of V ×V with respect to the supremums p-norm on V ×V . Apparently, π is the coequalizer of the restrictions of the canonical projections π1 , π2 : V × V → V to W in Vecp . (ii) Let kπ k 6= 0 and k k0 denote the p-norm on V /U . Obviously, kπ k−1 k k0 is a p-norm on V /U and the canonical projection πˆ : (V , k k) → (V /U, kπ k−1 k k0 ) is a Vecp -morphism. The identical mapping ϕ: (V /U, k k0 ) → (V /U, kπ k−1 k k0 ) is K-linear and ϕ ◦ π = πˆ holds. Obviously, ϕ is a Vecp -morphism. Because of kπ k = 6 0 there exists a v ∈ V \U and kπ k−1 kv + U k0 = kπ k−1 kϕ(v + U )k0 ≤ 2 kv + U k0 implies kπ k = 1. For a set mapping g: X → KP put supp g := {x ∈ X | g(x) 6= 0}. For a set I and real numbers αi ≥ 0 (i ∈ I ), i∈I αi < ∞ always implies that supp {αi | i ∈ countable. For a set X define lp (X) := {g ∈ Set(X, K) | I } := {i ∈ I | αi 6= 0} is P p supp g is countable and x∈X |g(x)| < ∞}. The p-norm k k: lp (X) → R is P p 1/p defined by kgk := ( x∈X |g(x)| ) (g ∈ lp (X)). For x ∈ X, δ x : X → K is the Dirac function at x, that is δ x (x) = 1 and δ x (y) = 0 for y 6= x (y ∈ X). PROPOSITION 1.3 ([4, p. 118]). For every set X lp (X) is a p-Banach space and (δ x )x∈X is a Schauder-basis ([4, p. 292]) of lp (X). The unit ball functor p,fin : Vecp → Set is induced by the mapping assigning to each p-normed vector space V its closed unit ball p,fin (V ) := {x ∈ V | kxk ≤ 1}. Its restriction to Banp is denoted by p . Especially one has for K ∈ Banp p (K) = 1 (K) = {α ∈ K | |α| ≤ 1}. PROPOSITION 1.4 (cf. [8, 1.1]). The functor p : Banp → Set has lp as left adjoint, is premonadic, but fails to be monadic ([10, 2.10]). Proof. Let X be a set. The mapping ηX : X → p ◦ lp (X) is defined by ηX (x) := δ x (x ∈ X). For a set mapping ψ: X → p (B) obviously a Banp ˆ lp (X) → B with p (ψ) ˆ ◦ ηX = ψ is uniquely determined. ψ: ˆ morphism ψ: P ˆ lp (X) → B defined by ψ(g) := x∈X g(x)ψ(x) is a (well defined) Banp ˆ ◦ ηX = ψ, rendering the left adjoint in question. The morphism with p (ψ) counit ε of this adjunction is uniquely determined by the equation εB (δ x ) = x (x ∈ p (B), B ∈ Banp ). Since for B ∈ Banp and any x ∈ B\{0}, x = kxkεB (δ x/kxk) = εB (kxkδ x/kxk ) holds, εB : lp ◦ p (B) → B is a surjection. Ker εB := εB−1 ({0B }) and lp ◦ p (B)/Ker εB are complete. πB : lp ◦ p (B) → lp ◦ p (B)/Ker εB denotes the canonical projection. Thus there exist a K-linear isomorphism ϕ: lp ◦ p (B)/Ker εB → B with ϕ ◦ πB = εB . By an elementary computation, ϕ is a Banp -isomorphism. By 1.2(i), πB is a coequalizer in Vecp , and thus in Banp . ϕ ◦ πB = εB implies that εB is a coequalizer in Banp (B ∈ Banp ). Consequently, p is premonadic ([10, 3.6]). That p fails to be monadic follows by [8, 4.6]. 2
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For any set := K(X) and a p-norm k k: lp,fin (X) → R is defined P X put lp,finp(X) 1/p by kgk := ( x∈X |g(x)| ) (g ∈ lp,fin (X)). Obviously we have PROPOSITION 1.5. For every set X (lp,fin (X), k k) is a p-normed vector space and the family (δ x )x∈X is a basis of lp,fin (X). Similar to 1.4, resp. to the case p = 1 one proves the following two propositions: PROPOSITION 1.6. The functor p,fin : Vecp → Set possesses lp,fin as left adjoint. p,fin is premonadic and fails to be monadic. PROPOSITION 1.7. Banp is a reflective subcategory of Vecp . The reflections morphism is a dense embedding. LEMMA 1.8. The canonical forgetful functor V : Vecp → Set (resp. V : Banp → Set) has no left adjoint. Proof. Assume, V : Banp → Set has a left adjoint F : Set → Banp with unit η: Set → V ◦ F . Define the mapping f : N → V (K) by f (n) := nkηN (n)k + 1 (n ∈ N); here k k means the p-norm on F (N). Then a simple computation leads to a contradiction. 2 The category Vecp has products, namely V = {v = (vi )i∈I ∈ Xi∈I Vi | sup{kvi k | i ∈ I } < ∞} for Vi ∈ Vecp (i ∈ I ) with kvk := sup{kvi k | i ∈ I } for v ∈ V , and equalizers hence it is complete. Banp as a full, reflective subcategory is also complete and products and equalizers in Banp are the same as in Vecp . For categorical reasons the cocompleteness of Vecp and Banp results from the completeness but for later applications the coproduct and the coequalizers in Vecp and Banp will be explicitly constructed now. Let (Vi )i∈I be a family of p-normed vector spaces P and put V := {v = (vi )i∈I ∈ Xi∈I Vi | |supp v| < ∞}. For v ∈ V put kvk := ( i ∈ supp v kvi kp )1/p . The mapping µi : Vi → V is defined by (µi (x))i := x and (µi (x))j := 0 for j 6= i (i, j ∈ I ). Obviously we have PROPOSITION 1.9. (V , µi : Vi → V )i∈I is a coproduct of (Vi )i∈I in Vecp . For Vecp -morphisms f, g: V → W define U := (f − g)(V ), i.e. the closure of the subspace (f − g)(V ) of W and let π : W → W/U be the canonical projection. Then we have the following PROPOSITION 1.10. π : W → W/U is a coequalizer of f and g in Vecp . Let (BP i )i∈I be a family of p-Banach spaces and define P B := {b = (bi )i∈I ∈ Xi∈I Bi | i∈I kbi kp < ∞}. For b ∈ B put kbk := ( i∈I kbi kp )1/p . Define the mapping µi : Bi → B by (µi (x))i := x and (µi (x))j := 0 for j 6= i (i, j ∈ I ). Then we have
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PROPOSITION 1.11. (B, µi : Bi → B)i∈I is a coproduct of (Bi )i∈I in Banp . Let f, g: B → B 0 be Banp -morphisms and U := (f − g)(B). Then B 0 /U is complete and one gets for the canonical projection π : B 0 → B 0 /U the following PROPOSITION 1.12. π : B 0 → B 0 /U is a coequalizer of f and g in Banp . DEFINITION 1.13. For V , W ∈ Vecp (V , W ∈ Banp ) put Hom(V , W ) := {f : V → W | f is K-linear and bounded}. For V , W ∈ Vecp , f ∈ Hom(V , W ) by definition kf k = inf{M ≥ 0 | kf (x)k ≤ Mkxk for all x ∈ V } holds. Obviously, (Hom(V , W ), k k) is canonically a p-normed vector space. For B, B 0 ∈ Banp by a straightforward computation ([4, 3.2.3]) (Hom(B, B 0 ), k k) is a p-Banach space. Here we have the following PROPOSITION 1.14. The functors Hom(−, −): Vecp × Vecp → Vecp (resp. Hom(−, −): Banp × Banp → Banp ) are internal ([6, p. 637]) with respect to
p,fin : Vecp → Set (resp. p : Banp → Set). A K-bilinear mapping ψ: V × W → Z (V , W, Z ∈ Vecp ) is called bounded if and only if there exists a λ ≥ 0 with kψ((v, w))k ≤ λkvkkwk for all v ∈ V , w ∈ | v ∈ V , w ∈ W, kvkkwk 6= 0}, W . In this case one puts kψk := sup{ kψ((v,w))k kvkkwk and in case kψk ≤ 1 ψ is called a bilinear contraction. The following definition is the natural generalization of the usual notation of the projective tensor product to the case p < 1. DEFINITION 1.15. Let V , W ∈ Vecp (Banp ). A triple (E, ⊗: V × W → E) is called a projective tensor product of V and W in Vecp (Banp ), if E ∈ Vecp (Banp ), ⊗: V × W → E is a bilinear contraction and for every bounded bilinear mapping ψ: V × W → Z (Z ∈ Vecp (Banp )) there exists a uniquely determined bounded K-linear mapping ϕ: E → Z with ϕ ◦ ⊗ = ψ. Next we obtain the following PROPOSITION 1.16. Let V , W ∈ Vecp . Then there exists a projective tensor product of V and W in Vecp uniquely determined up to isomorphism, denoted by V ⊗ W . The functor − ⊗ − is left adjoint to Hom(−, −). bC := Q(B ⊗ C) (where Q is the For p-Banach spaces B and C put B ⊗ bC be the isometric inclusion and put completion functor). Let in: B ⊗ C ,→ B ⊗ b b ⊗ := in ◦ ⊗: B × C ,→ B ⊗C. Obviously we have bC, ⊗ b) is a projective tensor product PROPOSITION 1.17. For B, C ∈ Banp , (B ⊗ b− is left adjoint to Hom(−, −). of B and C in Banp . The induced functor −⊗ It is well known, that K is a (single object) cogenerator in the categories Vec1 and Ban1 . For p < 1, K is not a cogenerator in Vecp (Banp ), which is shown by the following example.
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Let p < 1 and Lp ([0, 1]) := {f : [0, 1] → K | f is Borel-measurable and p 0 |f (x)| dx < ∞}. Obviously, under pointwise addition and scalar-multiplication, Lp ([0, 1]) is a K-vector subspace of K[0,1] and one defines Mp := {f ∈ Lp ([0, 1]) | f = 0 almost everywhere}. Mp is a K-vector subspace of Lp ([0, 1]), and one puts Lp ([0, 1]) := Lp ([0, 1])/Mp . Let π : Lp ([0, 1]) → Lp ([0, 1]) be the R1 canonical projection and define kπ(f )k := ( 0 |f (x)|p dx)1/p (f ∈ Lp ([0, 1])). Then one has
R1
PROPOSITION 1.18 ([5, p. 158]). Lp ([0, 1]) is a p-Banach space and the-zero morphism is the only K-linear continuous mapping Lp ([0, 1]) → K.
2. p-Totally and p-Absolutely Convex Spaces P Put p := p ◦ lp (N) = {α = (αi )i∈N ∈ KN | i |αi |p ≤ 1}. For α ∈ p define supp α := {i ∈ N | αi 6= 0}. Finally, put p,fin := p,fin ◦ lp,fin (N) = {α ∈ p | |supp α| < ∞}. By a routine computation one gets PROPOSITION 2.1. Let α = (αi )i∈N ∈ p (p,fin ), β i = (βji )j ∈N ∈ p (p,fin ) P (i ∈ N). Then ( i αi βji )j ∈N ∈ p (p,fin ) holds. DEFINITION 2.2. An p -algebra is a set X together with a mapping p → Set(XN , X), α 7→ αX (α ∈ p ). A morphism from the p -algebra X to the p algebra Y is a set mapping f : X → Y satisfying for any α ∈ p f ◦αX = αY ◦f N , where f N : XN → Y N is defined componentwise. Obviously, the p -algebras together with their morphisms form a category, the composition of the morphisms being the set-theoretical one. DEFINITION 2.3. (i) For an p -algebra X, α ∈ p , x := (xi )i∈N ∈ XN one P defines i αi xi := αX (x). j j (ii) Put δ j := (δi )i∈N ∈ KN , where δi is the Kronecker-symbol, i.e. δii = 1 and j δi = 0 for j 6= i (i, j ∈ N). DEFINITION 2.4 (cf. [8, 2.2]). (i) An p -algebra (X, (αX , α ∈ p )) is called a p-totally convex space if and only if X 6= ∅ and the following two axioms are satisfied: X j δi xi = xj holds. (TCp 1) For all (xi )i∈N ∈ XN , j ∈ N, i
(TCp 2) For all α, β i ∈ p (i ∈ N), (xi )i∈N ∈ XN , XX X X i i αi βj xj = αi βj xj holds. i
j
j
i
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(ii) The full subcategory of the category of p -algebras which is determined by all p-totally convex spaces is denoted by TCp . If D := (X, (αX , α ∈ p )) is a p-totally convex space, then we write x ∈ D instead of x ∈ X and D ∈ TCp instead of (X, (αX , α ∈ p )) ∈ TCp . PROPOSITION 2.5. In a p-totally convex space D, the following computational rules are valid: (i) Let α ∈ p and let T be a set with supp Pα ⊂ T ; then P for all x = (xi )i∈N , N y = (yi )i∈N ∈ D with xi = yi (i ∈ T ) i αi xi = i αi yi . N (ii) Let P N be a permutation, α ∈ p and x = (xi )i∈N ∈ D ; then P ϕ: N → . i αi xi = i αϕ(i) xϕ(i) P (iii) For (0)i∈N ∈ pP0D := i 0xi does not depend on (xi )i∈N ∈ D N . (iv) For all α ∈ p i αi 0D = 0D holds. (v) For x := (xi )i∈N ∈ D N put supp x := {i ∈ N | xi 6= 0} and let T be a set with supp P x ⊂ T . If for α, β ∈ p , αi = βi for all i ∈ T , then P α x = i i i i βi xi . (vi) Let ϕ: N → N be an injective mapping, α, β ∈ p with αk = βϕ(k) (k ∈ N) / ϕ(N)). Then for all x, y ∈ D N with xϕ(k) = yk (k ∈ N) and βi = 0P(i ∈ P i βi xi = i αi yi holds. (vii) Let ϕ: N → N×N be a bijective mapping with the components ϕ1 , ϕ2 : N → N, i.e. ϕ(n) = (ϕ1 (n), ϕ2 (n)) (n ∈ N); then for all α, β i ∈ p , xji ∈ D (i, j ∈X N) X X (n) (n) αi βji xji = . αϕ1 (n) βϕϕ21(n) xϕϕ21(n) i
n
j
(viii) For all α, β ∈ p , xji ∈ D (i, j ∈ N) X X X X αi βj xji = βj αi xji i
j
j
i
holds. Proof. The proof of (i)–(vii) is similar to [8, 2.4], resp. [11, 7.1]. For the proof of (viii) put β i := β (i ∈ N) in (vii). Then we get X X X (n) αi βj xji = αϕ1 (n) βϕ2 (n) xϕϕ21(n) i
j
n
X X X (n) = = βj αi xji . βϕ2 (n) αϕ1 (n) xϕϕ21(n) n
j
2
i
DEFINITION 2.6. Put p,fin := {α ∈ p | |supp α| < ∞}. Replacing p by p,fin in 2.2 one gets the category of p,fin -algebras. An p,fin -algebra X is called a p-absolutely convex space if and only if X 6= ∅ holds and in 2.4 (TCp 1) and the restriction of (TCp 2) to α, β i ∈ p,fin (i ∈ N) is valid. ACp denotes the full subcategory of the category of p,fin -algebras which is determined by all p-absolutely convex spaces.
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REMARK 2.7. The statements in 2.5 by restriction of p to p,fin are also valid for p-absolutely convex spaces. D N and β = (βi )i∈N ∈ DEFINITION 2.8. Let D ∈ TCp (ACp ), x = (xi )i∈N ∈P p (p,fin ) with supp β = {1}. Then one defines β1 x1 := i βi xi . By 2.8, on D ∈ TCp (ACp ) a (K)-scalar-multiplication is defined. By 2.5 we get easily the following PROPOSITION 2.9. Let D ∈ TCp (ACp ), α, β ∈ (K), x ∈ D, (γi )i∈N ∈ p p,fin and (xi )i∈N ∈ D N . Then the following statements are valid: (i) (ii) (iii) (iv) (v)
1x = x. α0D = 0D . α(βx) = (αβ)x. P P γ x = ( )x. i i i γiP P i (αγi )xi = α( i γi xi ).
As a category of equationally defined universal algebras over Set, TCp (ACp ) is complete and cocomplete. Let Di ∈ TCp (ACp ) (i ∈ I ). On the cartesian product Xi∈I Di a TCpQ(ACp )-structure is defined componentwise. This TCp (AC Q p )-space with the canonical projections π : is denoted by i∈I Di and togetherQ j i∈I Di → Q Dj (j ∈ I ) this yields the product ( i∈I Di , πj : i∈I Di → Dj )j ∈I of the family (Di )i∈I in TCp (ACp ). Q In particular, for D ∈ TCp (ACp ) the cartesian power D N := i∈N D is a p-totally (p-absolutely) convex space and by 2.5(viii) we have the following PROPOSITION 2.10. For D ∈ TCp (ACp ) and α ∈ p (p,fin ) the mapping αD : D N → D is a TCp (ACp )-morphism. LEMMA 2.11. For 0 < s ≤ p, TCp (ACp ) is a subcategory of TCs (ACs ). Proof. This is a consequence of s ⊂ p (s,fin ⊂ p,fin ).
2
3. The Comparison Functor and the Tensor Product For a p-normed vector space V and αP ∈ p,fin one defines α p,fin (V ) : p,fin (V )N →
p,fin (V ) by α p,fin (V ) ((xi )i∈N ) := i αi xi (xi ∈ p,fin (V ), i ∈ N). This equips b p,fin (V ).
p,fin (V ) with the structure of a p-absolutely convex space denoted by Obviously, the restrictions of Vecp -morphisms are ACp -morphisms of the associb p,fin : Vecp → ACp . ated p-absolutely convex spaces, hence we have a functor Similarly, if B is a p-Banach space and α ∈P p , the mapping α p (B) : p (B)N →
p (B) is defined by α p (B) ((xi )i∈N ) := i αi xi (xi ∈ p (B), i ∈ N). This b p : Banp → TCp . b p (B) and hence a functor yields a p-totally convex space
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In a category K of commutative equationally defined universal algebras one knows ([7, p. 179]) that the forgetful functor U : K → Set possesses a left adjoint. This left adjoint of the forgetful functor Up : TCp → Set (resp. Up,fin : ACp → Set) will now be constructed explicitly. PROPOSITION 3.1. The forgetful functor Up : TCp → Set (resp. Up,fin : ACp → b p,fin ◦ lp,fin ) as a left adjoint. b p ◦ lp (resp. Set) has b p ◦ lp and define the mapping ηX : X → Up ◦ Proof. Let X be a set, F := x → Up (D) (D ∈ TCp ) be a set F (X) by ηX (x) := δ (x ∈ P X). Let f : X x g(x)δ (g ∈ F (X)) (supp g is countable mapping. Because of g = x ∈ supp g and can therefore be imbedded into N) a TCp -morphism ϕ: F (X) → D with X). Taking this as a Up (ϕ)◦ηX = f is uniquely determined by ϕ(δ x ) = f (x) (x ∈P definition one gets the mapping ϕ: F (X) → D with ϕ(g) := x ∈ supp g g(x)f (x) (g ∈ F (X)). ϕ(g) is well defined, since the definition of ϕ(g) does not depend on the chosen imbedding of supp g into S N (cf. 2.5). Let α ∈ p , gi ∈ F (X) (i ∈ N) and let S be a countable set with i∈N supp gi ⊂ S and S ,→ N be an imbedding of S into N. Then one has XX X X X =ϕ αi gi = ϕ αi gi (x)δ x αi gi (x) δ x ϕ i
i
=
XX x∈S
=
X
x∈S
x∈S
i
X X αi gi (x) f (x) = αi gi (x)f (x)
i
i
x∈S
αi ϕ(gi ).
i
Obviously, the equation Up (ϕ) ◦ ηX = f is fulfilled and F is a left adjoint. The proof in the finitary case is analogous. 2 b p,fin ◦ b p ◦ lp ◦ Up → TCp (resp. ε: Similar to 1.4, one gets for the counit ε: b p ◦ lp , ε, η) (resp. (Up,fin , b p,fin ◦ lp,fin ◦ Up,fin → ACp ) of the adjunction (Up , x lp,fin , ε, η)) εD (δ ) = x (x ∈ D), D ∈ TCp (resp. D ∈ ACp ) and εD (D ∈ TCp ) (resp. D ∈ ACp ) is uniquely determined by this equations. A well-known result about categories of equationally defined universal algebras over Set ([7, p. 179]) implies that the forgetful functor Up : TCp → Set (resp. Up,fin : ACp → Set) is monadic. COROLLARY 3.2. The forgetful functor Gp : TCp → ACp has a left adjoint. Proof. By 3.1, the forgetful functors Up : TCp → Set and Up,fin : ACp → Set possess a left adjoint. Obviously, Up = Up,fin ◦ Gp holds. The category TCp b p,fin ◦lp,fin , ε, η) possesses coequalizers and the counit ε of the adjunction (Up,fin , is pointwise a regular epimorphism and our assertion follows. 2 THEOREM 3.3. (i) Up : TCp → Set is the Eilenberg–Moore category of p : b p : Banp → TCp the comparison functor. b p is full and Banp → Set with faithful.
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(ii) Up,fin : ACp → Set is the Eilenberg–Moore category of p,fin : Vecp → b p,fin : Vecp → ACp the comparison functor. b p,fin is full and faithful. ACp with Proof. (i) By 1.4, lp : Set → Banp is left adjoint to p with unit ηX : X → p ◦ lp (X), ηX (x) = δ x (x ∈ X, X ∈ Set), and counit εB : lp ◦ p (B) → B, uniquely b p ◦lp : Set → TCp determined by εB (δ x ) = x (x ∈ p (B), B ∈ Banp ). By 3.1, 0 b p ◦ lp ◦ Up (D) → D, is left adjoint to Up with unit ηX (X ∈ Set) and counit εD : 0 x bp = uniquely determined by εD (δ ) = x (x ∈ D, D ∈ TCp ). Because of Up ◦
p a routine calculation shows that the monads of these adjunctions are equal ([10, 2.4]). By 1.4 we are finished. The proof of (ii) is similar. 2 PROPOSITION 3.4. For the forgetful functors Up : TCp → Set (resp. Up,fin : b p (K), −) (resp. Up,fin ' ACp ( b p,fin (K), −) holds, in ACp → Set) Up ' TCp ( b b particular p (K) (resp. p,fin (K)) is a generator in TCp (ACp ). b p (K), D) by Proof. For D ∈ TCp define the mapping ρD : Up (D) → TCp ( b p (K)). Obviously, this defines a bijective ρD (x)(α) := αx (x ∈ Up (D), α ∈ 2 natural transformation. The proof for Up,fin is similar. Let C, D ∈ TCp (ACp ), ϕP i ∈ TCp (C, D) (ACp (C, P D)) (i ∈ N) and P α ∈ p (p,fin ). Define the mapping i αi ϕi : C → D by ( i αi ϕi )(x) := i αi ϕi (x) (x ∈ C). Then (cf. 2.10, [6, p. 640]) we have the PROPOSITION 3.5. With the above definition TCp (C, D) (ACp (C, D)) is a ptotally (p-absolutely) convex subspace of D Up (C) (D Up,fin (C) ). DEFINITION 3.6. (i) For C, D ∈ TCp the above p-totally convex space with underlying set TCp (C, D) is denoted by Homp (C, D). (ii) For TCp -morphisms g: C 0 → C, h: D → D 0 one defines the mapping Homp (g, h): Homp (C, D) → Homp (C 0 , D 0 ) by Homp (g, h)(f ) := h ◦ f ◦ g (f ∈ Homp (C, D)). Obviously, Homp (g, h) in 3.6(ii) is a TCp -morphism. Homp,fin (C, D) for pabsolutely convex spaces C, D and Homp,fin (g, h) for ACp -morphisms g: C 0 → C, h: D → D 0 is defined as in the infinitary case. Now we have PROPOSITION 3.7. Homp (−, −): TCp × TCp → TCp and Homp,fin (−, −): ACp × ACp → ACp are internal Hom-functors (in the sense of [6, p. 637]) for TCp and ACp . From 2.10 and [6, p. 240] one gets the following PROPOSITION 3.8. TCp (resp. ACp ) is an autonomous category in the sense of Linton, i.e. possesses a tensor product, which, together with coherence morphisms turns it into a symmetric monoidal closed category. The induced functor − ⊗ −: TCp × TCp → TCp (resp. − ⊗ −: ACp × ACp → ACp ) is a left adjoint of the internal Hom-functor Homp (−, −): TCp × TCp → TCp (resp. Homp,fin (−, −): ACp × ACp → ACp ).
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289
4. Congruence Relations DEFINITION 4.1. Let D ∈ TCp and “∼” an equivalence relation on D. Then “∼” is called a congruence relation only if for all ai , bi ∈ D (i ∈ N), P if and P α ∈ p , ai ∼ bi (i ∈ N) implies i αi ai ∼ i αi bi . Congruence relations in ACp are defined analogously. Obviously, for D ∈ TCp (ACp ) and a congruence relation “∼” on D the quotient D/∼ possesses a uniquely structure as p-totally (p-absolutely) convex space such that the canonical projection π : D → D/∼ is a TCp (ACp )-morphism. DEFINITION 4.2. Let D ∈ ACp . The mapping k kD : D → R is defined as a Minkowski-functional by kxkD := inf{|λ| | λ ∈ (K) and x = λy} (x ∈ D). Often one writes simply kxk instead of kxkD . PROPOSITION 4.3. Let D ∈ ACp . Then one has (i) For every x ∈ D kxk = inf{λ ∈ [0, 1] | there exists y ∈ D with x = λy}. (ii) For each ACp -morphism f : C → D, kf (x)kD ≤ kxkC , x ∈ C, holds. (iii) If “∼” is a congruence relation on D and π : D → D/∼ the canonical projection, then kπ(y)kD/∼ = inf{kxkD | π(x) = π(y)} (y ∈ D). λ Proof. (i) follows from λy = |λ|( |λ| y), y ∈ D, λ ∈ (K)\{0} and (ii) is trivial. (iii) Let y ∈ D and define γ := inf{kxkD | π(x) = π(y)}. From (ii) we get kπ(y)k ≤ y. Let λ ∈ [0, 1], x 0 ∈ D with π(y) = λπ(x 0 ) = π(λx 0 ). This implies γ ≤ λ, thus γ ≤ kπ(y)k, and we are finished. 2
LEMMA 4.4. Let D ∈ TCp (ACp ) and x ∈ D. P Then kxk = inf{kαk | α ∈ p (p,fin ) and there are xi ∈ D (i ∈ N) with x = i αi xi } holds. ∈ D (i ∈ Proof. Obviously, the assertion P holds for x = 0. In case x 6= 0 let xi P N), α ∈ p (p,fin ) with x = i αi xi . This implies kαk > 0 and y := i kααii k xi is well defined. From x = kαky the assertion follows. 2 PROPOSITION P 4.5 (cf. [8,P6.2]). For all D ∈ TCp (ACp ), α ∈ p (p,fin ), xi ∈ D (i ∈ N) k i αi xi kp ≤ i |αi |p kxi kp holds. Proof. Let ε > 0. Then there exist λi ∈ [0, 1], yP i ∈ D with xi = Pλi yi and kxipk ≤ 1/p p λiP≤ min{kxi k + εP , 1} (i ∈ N). 4.4 yields k α x k = k i i i i αi (λi yi )k = P P kP i (αi λi )yi kp ≤ i |αi λi |p ≤ i |αi |p (kxi k + ε 1/p )p ≤ i |αi |p (kxi kp + ε) ≤ 2 ( i |αi |p kxi kp ) + ε. PROPOSITION 4.6 (cf. [8, 6.5]). QFor a family (Di )i∈I of p-totally (p-absolutely) convex spaces and x = (xi )i∈I ∈ i∈I Di , k(xi )i∈I k = sup{kxi k | i ∈ I } holds. Proof. Define σ := sup{kxi k | i ∈ I }. It follows immediately from 4.3(ii) that σ ≤ k(xi )i∈I k. On the other hand let us assume that there exists a λ with σ < λ < kxk. Then there are yi ∈ Di with xi = λyi (i ∈ I ), implying x = λ(yi )i∈I . This 2 contradicts λ < kxk and we get kxk = sup{kxi k | i ∈ I }.
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RALF KEMPER
DEFINITION 4.7. For an ACp -morphism f : C → D one puts kf ks := inf{λ ∈ [0, 1] | kf (x)k ≤ λkxk for all x ∈ C}. PROPOSITION 4.8. If f : C → D is an ACp -morphism then the following statements are hold: (i) kf (x)k ≤ kf ks kxk for every x ∈ C. (ii) kf ks = sup{kf (x)k | x ∈ C}. (x)k | x ∈ (iii) If there exists x0 ∈ C with kx0 k 6= 0, then kf ks = sup{ kfkxk C with kxk 6= 0}. Proof. (i) This follows from 4.7. (ii) Put σ := sup{kf (x)k | x ∈ C}. Because of kxk ≤ 1 (x ∈ C) (i) implies σ ≤ kf ks . For all x, y ∈ C, λ ∈ [0, 1], x = λy implies kf (x)k = kf (λy)k = kλf (y)k ≤ λkf (y)k ≤ λσ . This leads to kf (x)k ≤ σ kxk (x ∈ C), thus kf ks ≤ σ and finally kf ks = σ = sup{kf (x)k | x ∈ C}. (x)k | x ∈ C with kxk 6= 0} is well (iii) Because of the assumption, σ := sup{ kfkxk defined. From (i) we get σ ≤ kf ks , and from the definition of σ kf (x)k ≤ σ kxk (x ∈ C with kxk 6= 0). This is also true for x ∈ C with kxk = 0 (4.3(ii)) and (x)k | x ∈ C with kxk 6= 0}. 2 implies kf ks ≤ σ , thus kf ks = sup{ kfkxk DEFINITION 4.9. For 0 < γ ≤ 1 one defines γ (K) := {β ∈ (K) | |β| ≤ γ } ◦ ◦ and γ (K) := {β ∈ K | |β| < γ }. For γ = 1 one simply writes (K) ( (K)) ◦ instead of 1 (K) ( 1 (K)). In the category AC (and TC) the following central theorem is valid: Let D ∈ AC and “∼” a congruence relation on D. Then, for x, y ∈ D, S := {α ∈ (K) | ◦ αx ∼ αy} equals {0}, (K) or (K) ([8, 4.1] and [3, 1.1]). Furthermore, for x, y, z ∈ D, M := {α ∈ [0, 1] | αx + (1 − α)z ∼ αy + (1 − α)z} equals {0}, [0, 1[ or [0, 1] ([2, 1.5]). In the following we generalizes this result as far as possible to the case p < 1. THEOREM 4.10. Let p < 1, D ∈ ACp and “∼” a congruence relation on D. Then for all x, y, z ∈ D, σ, τ, γ ∈ ]0, 1], x0 , y0 , z0 ∈ D with x = σ x0 , y = τy0 and z = γ z0, and for every α ∈ ]0, 1[ with αx ∼ αy, one puts S := β ∈ [0, 1] | β 1/p x + (1 − β)1/p z ∼ β 1/p y + (1 − β)1/p z and has: (i) For γ < 1 ≤ σ p/(1−p) + τ p/(1−p) 1 − γp 0, p/(1−p) ⊂ S. (σ + τ p/(1−p))1−p − γ p
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(ii) For σ p/(1−p) + τ p/(1−p) < 1 S = [0, 1] holds, in particular x ∼ y. Proof. Obviously, 0 ∈ S. For 0 < β ≤ 1 and all ε with 0 < ε < min{1, α −1 β 1/p } a mapping fε,β : [εα, β 1/p ] → R is defined by fε,β (t) := ((t − εα)σ )p + ((β 1/p − t)τ )p + ε p + (1 − β)γ p (t ∈ [εα, β 1/p ]). 0 fε,β is differentiable in ]εα, β 1/p [ with fε,β (t) = p((t − εα)p−1 σ p − (β 1/p − t)p−1 τ p ). For t0ε := (β 1/p σ p/(1−p)+εατ p/(1−p))·(σ p/(1−p)+τ p/(1−p) )−1 , εα < t0ε < 0 (t) > 0 if and only if t < β 1/p follows and an elementary computation shows fε,β ε ε 0 1/p t0 , and fε,β (t) < 0 if and only if t > t0 (εα < t < β ). Since fε,β is continuous in εα and β 1/p , fε,β has an absolute maximum in t0ε . A simple computation leads to
lim fε,β (t0ε ) = β(σ p/(1−p) + τ p/(1−p) )1−p + (1 − β)γ p .
ε→0
Hence limε→0 fε,β (t0ε ) < 1 is equivalent with β (σ p/(1−p) + τ p/(1−p))1−p − γ p < 1 − γ p . Consequently, in (i) limε→0 fε,β (t0ε ) < 1 is equivalent with β < (1 − γ p ) (σ p/(1−p) + τ p/(1−p))1−p − γ p
−1
,
and using the assumption in (ii) limε→0 fε,β (t0ε ) < 1 is fulfilled for all β ∈ ]0, 1]. Hence, for every β ∈ ]0, 1] with limε→0 fε,β (t0ε ) < 1 there exists an ε0 with 0 < ε0 < min{1, α −1 β 1/p }, such that for all ε ∈ ]0, ε0 ], t ∈ [εα, β 1/p ], fε,β (t) ≤ fε,β (t0ε ) < 1 holds. Put M := λ ∈ [0, β 1/p ] | (β 1/p σ )x0 + (1 − β)1/p γ z0 ∼ (λσ )x0 + + (β 1/p − λ)τ y0 + (1 − β)1/p γ z0 . M is well-defined, since (β 1/p σ )p + ((1 − β)1/p γ )p = βσ p + (1 − β)γ p ≤ 1 (and (β 1/p τ )p + ((1 − β)1/p γ )p ≤ 1) holds, and for all λ ∈ [0, β 1/p ], (λσ )p + ((β 1/p − λ)τ )p + (1 − β)γ p ≤ 1. This can be seen as follows: For λ = β 1/p it was shown above. For λ ∈ M with λ < β 1/p there exists an ε ∈ ]0, ε0 ] with t := λ + εα ≤ β 1/p . For λ = t − εα, t ∈ [εα, β 1/p ], we get p (λσ )p + (β 1/p − λ)τ + (1 − β)γ p p p = (t − εα)σ + (β 1/p − t)τ + εατ + (1 − β)γ p p p ≤ (t − εα)σ + (β 1/p − t)τ + ε p + (1 − β)γ p = fε,β (t) ≤ fε,β (t0ε ) < 1.
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RALF KEMPER
For every λ ∈ M\{0} there exists an ε1 ∈ ]0, ε0 ] with ε1 α ≤ λ. For all ε ∈ ]0, ε1 ] µ := λ − εα ∈ [0, λ[ holds. If λ ∈ M\{0} and ε1 , ε, µ are as above, the following terms are well-defined because M is well-defined. Then we have (µσ )x0 + (β 1/p − µ)τ y0 + (1 − β)1/p γ z0 = (µσ )x0 + (β 1/p − λ)τ + εατ y0 + (1 − β)1/p γ z0 = (µσ )x0 + (β 1/p − λ)τ y0 + ε α(τy0 ) + (1 − β)1/p γ z0 ∼ (µσ )x0 + (β 1/p − λ)τ y0 + ε α(σ x0 ) + (1 − β)1/p γ z0 = (µσ )x0 + (β 1/p − λ)τ y0 + (εασ )x0 + (1 − β)1/p γ z0 = (λσ )x0 + (β 1/p − λ)τ y0 + (1 − β)1/p γ z0 ∼ (β 1/p σ )x0 + ((1 − β)1/p γ )z0. This implies µ ∈ M, thus [λ − ε1 α, λ] ⊂ M. Obviously, β 1/p ∈ M\{0} holds. Thus there exists an ε1 ∈ ]0, ε0 ] with β 1/p − ε1 α > 0. By the above equations, this implies β 1/p − n0 ε1 α ∈ M, where n0 := max{n ∈ N | β 1/p − nε1 α > 0}. Because of β 1/p − (n0 + 1)ε1 α ≤ 0 there exists an ε2 ∈ ]0, ε1 ] with (β 1/p − n0 ε1 α) − ε2 α = 0. Again the above equations yield 0 = (β 1/p − n0 ε1 α) − ε2 α ∈ M because of β 1/p − n0 ε1 α ∈ M\{0}. One has β 1/p x + (1 − β)1/p z = (β 1/p σ )x0 + (1 − β)1/p γ z0 ∼ (β 1/p τ )y0 + (1 − β)1/p γ z0 = β 1/p y + (1 − β)1/p z, i.e. β ∈ S, and we are finished.
2
COROLLARY 4.11. If p < 1, D ∈ ACp and “∼” is a congruence relation on D, then for x, y, z ∈ D and 0 < α < 1 with αx ∼ αy one defines S := β ∈ [0, 1] | β 1/p x + (1 − β)1/p z ∼ β 1/p y + (1 − β)1/p z . Then: (i) For kzk < 1 ≤ kxkp/(1−p) + kykp/(1−p) 1 − kzkp 0, ⊂ S holds. (kxkp/(1−p) + kykp/(1−p) )1−p − kzkp (ii) From kxkp/(1−p) + kykp/(1−p) < 1, S = [0, 1] follows, in particular x ∼ y. Proof. (i) There exist σn , τn , γn ∈ ]0, 1], xn , yn , zn ∈ D with x = σn xn , y = p/(1−p) p/(1−p) + τn , γn < 1 (n ∈ N) and limn→∞ σn = τn yn , z = γn zn , 1 ≤ σn kxk, limn→∞ τn = kyk and limn→∞ γn = kzk. Because of 4.10(i), for all n ∈ N p p/(1−p) p/(1−p) 1−p p +τn ) −γn )−1 ] ⊂ S holds and the assertion follows. [0, (1−γn )((σn (ii) There exist σ, τ, γ ∈ ]0, 1], x0 , y0 , z0 ∈ D with x = σ x0 , y = τy0 , z = γ z0 and σ p/(1−p) + τ p/(1−p) < 1, and the assertion is implied by 4.10(ii). 2
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In the continuation of this paper it will be shown that the result obtained in 4.11(i) for the set S is the best possible one; furthermore it will be proved that in the remaining cases, i.e. kzk = 1 ≤ kxkp/(1−p) + kykp/(1−p) “no further statement” about the set S can be made. COROLLARY 4.12. If “∼” is a congruence relation on D, D ∈ ACp , p < 1, and if one defines S := {α ∈ (K) | αx ∼ αy} for x, y ∈ D, then S = {0} or the following statements are fulfilled: ◦
(i) 1 ≤ kxkp/(1−p) + kykp/(1−p) implies βp (K) ⊂ S for βp := (kxkp/(1−p) + kykp/(1−p) )(p−1)/p . (ii) kxkp/(1−p) + kykp/(1−p) < 1 implies S = (K), in particular x ∼ y. Proof. (i) Put z := 0 in 4.11 and notice that αx ∼ αy (α ∈ (K)) implies βx ∼ βy for all β ∈ (K) with |β| = α. 2 COROLLARY 4.13. If D ∈ ACp , p < 1 and S are as in 4.12, then S = {0} or ◦
ωp (K) ⊂ S (where ωp = (1/2)1/p−1 ). Proof. Put σ := 1, τ := 1, γ := 0, x0 := x, y0 := y, z := 0, z0 := 0 in 4.10. In the same way as in 4.12 one gets the assertion. 2 COROLLARY 4.14. If “∼” is a congruence relation on D ∈ ACp , then for x, y ∈ D with kxk, kyk < ωp , αx ∼ αy (α ∈ (K)\{0}) implies x ∼ y. Proof. There exist σ ∈ ]0, ωp [, x0 , y0 ∈ D with x = σ x0 and y = σy0 . Put S := {β ∈ (K) | βx0 ∼ βy0 }. ασ ∈ S implies S 6= {0}. By 4.13, resp. [8, 4.1], 2 σ ∈ S holds, thus x = σ x0 ∼ σy0 = y. THEOREM 4.15 (cf. [8, 6.9]). For x ∈ D, D ∈ TCp , kxk = 0 implies x = 0. Proof. For p = 1 the assertion is proved in [8, 1 and define P 6.9].p Let pP< p ∞ ( 12 )n/p (n ∈ N0 ). Then εn ∈ ]0, 1] (n ∈ N0 ), ∞ ε ≤ εn :=P n=1 n+1 n=1 εn = 1, and p p ∞ ε2 + n=1 εn+1 ≤ 1 follow. Because of kxk = 0 there exist yn ∈ D with x = εn yn , ω in particular εn yn = εn+1 yn+1 = εn (( 12 )1/p yn+1 ) (n ∈ N0 ). For α := 2p we get 1/p 1/p 1 1 αyn = α yn+1 = (αyn+1 ) (n ∈ N0 ). 2 2 P Then for z := ∞ n=1 εn+1 yn−1 , one has X ∞ ∞ X α = α(ε2 y0 ) + α εn+2 yn = ε2 (αx) + εn+2 (αyn ) n=1
n=1
1/p ∞ X 1 = ε2 (αx) + εn+1 (αyn ) = ε2 (αx) + εn+1 (αyn−1 ) 2 n=1 n=1 X ∞ εn+1 yn−1 = α(ε2 x) + αz. = α(ε2 x) + α ∞ X
n=1
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This implies 0 = α α(ε2 x) + αz − α(αz) = α 2 ε2 x = αε2 (αy0 ) 1/p 1/p 1 1 y1 . = αε2 α y1 = α 2 ε2 2 2 Because of ε1 = ( 12 )1/p < ωp 4.13 yields x = ε1 y1 = 0.
2
REMARK 4.16. In order to see that 4.15 is false for D ∈ ACp , take an arbitrary K-vector space V 6= {0}. V has a canonical structure of a p-absolutely convex space and one has kxk = 0 for any x ∈ V . THEOREM 4.17. For D ∈ ACp and x ∈ D the following statements hold: ◦
(i) S := {α ∈ (K) | kαxk = 0} equals {0}, (K) or (K). In case kxk < 1 ◦ S 6= (K). ◦
(ii) M := {β ∈ (K) | βx = 0} equals {0}, (K) or (K). In case kxk < 1 ◦ M 6= (K). Proof. For p = 1 the assertions in (i) and (ii) are proved in [8, 4.1, 6.10] resp. are a direct consequence of this. ◦ (i) For p < 1 assume S 6= {0}. For α ∈ S\{0} and β ∈ (K) there exists an ε0 ∈ ]0, 1[ with p/(1−p) (p−1)/p
|β| < 1 + ε0
.
Since kαxk = 0, for ε ∈ ]0, ε0 ] there exists a yε ∈ D with αx = (αε)yε = α(εyε ), hence |α|x = |α|(εyε ). Putting z := 0, z0 := 0, γ := 0, σ := 1, x0 := x, y := εyε , y0 := yε , τ := ε, in 4.10, one gets |β|x = |β|(εyε ) because of |α| ∈ S\{0} and p/(1−p) (p−1)/p
|β| < 1 + ε0
≤ 1 + ε p/(1−p)
(p−1)/p
from 4.10(i) or (ii). This leads to βx = β(εyε ) = ε(βyε ) (ε ∈ ]0, ε0 ]), thus kβxk = ◦ 0, i.e. (K) ⊂ S. If there exists a γ ∈ S with |γ | = 1, one gets easily S = (K). ◦ Thus S is (K) or (K). In case kxk < 1 there exists a σ ∈ ]0, 1[, x0 ∈ D with x = σ x0 . Put S 0 := {τ ∈ (K) | kτ x0 k = 0}. Because of k(ασ )x0 k = kαxk = 0 ◦ and ασ 6= 0 we have S 0 6= {0}. This implies (K) ⊂ S 0 , i.e. σ ∈ S 0 and leads to kxk = kσ x0 k = 0 resp. 1 ∈ S, implying S = (K). (ii) For p < 1 assume M 6= {0}. Put y := 0 in 4.12. In case kxk = 1 4.12(i) ◦ yields (K) ⊂ M, and M = (K) otherwise (4.12(ii)). If there exists a γ ∈ M ◦ with |γ | = 1, in case kxk = 1 M = (K) follows immediately. Thus M = (K) or (K) holds. 2
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