Decision Support for Intermodal Transport

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S. COBZAS¸

Received 14 February 2005 and in revised form 6 June 2005

The aim of the present paper is to introduce the asymmetric locally convex spaces and to prove some basic properties. Among these I do mention the analogs of the Eidelheit- Tuckey separation theorems, of the Alaoglu-Bourbaki theorem on the weak compactness of the polar of a neighborhood of 0, and a Krein-Milman-type theorem. These results extend those obtained by Garc´ıa-Raﬃet al. (2003) and Cobzas¸ (2004).

1. Introduction

LetXbe a real vector space. Anasymmetric seminormonX is a positive sublinear functionalp:X→[0;∞), that is,psatisfies the conditions

(AN1) p(x)≥0,

(AN2) p(tx)=t p(x),t≥0, (AN3) p(x+y)≤p(x) +p(y),

for allx,y∈X. The function ¯p:X→[0,∞), defined by ¯p(x)=p(−x),x∈X, is another positive sublinear functional onX, called theconjugateofp, and

p^s(x)=maxp(x),p(−x), x∈X, (1.1) is a seminorm onX. The inequalities

p(x)−p(y)≤p^s(x−y), p(x)¯ −p(y)¯ ≤p^s(x−y) (1.2) hold for allx,y∈X. If the seminormp^sis a norm onX, then we say that pis anasym- metric normonX. This means that, beside (AN1)–(AN3), it satisfies also the condition

(AN4) p(x)=0 andp(−x)=0 imply thatx=0.

The pair (X,p), whereXis a linear space and pis an asymmetric seminorm onX is called aspace with asymmetric seminorm, respectively, aspace with asymmetric norm, ifp is an asymmetric norm.

In the last years, the properties of spaces with asymmetric norms were investigated in a series of papers, emphasizing similarities as well as diﬀerences with respect to the theory of (symmetric) normed spaces, see [3,5,6,7,12,13,16,17]. This study was stimulated

International Journal of Mathematics and Mathematical Sciences 2005:16 (2005) 2585–2608 DOI:10.1155/IJMMS.2005.2585

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also by their applications in the complexity of algorithms, see [11,14,18]. The aim of the present paper is to develop the basic results in the theory of asymmetric locally convex spaces, a natural extension of asymmetric normed spaces and of (symmetric) locally convex spaces as well.

The functionρ:X×X→[0;∞) defined byρ(x,y)=p(y−x),x,y∈X, is an asym- metric semimetric onX. Denote by

B_p(x,r)=

x∈X:p(x−x)< r, B_p(x,r)=

x∈X:p(x−x)≤r, (1.3) the open, respectively closed, ball inXof centerxand radiusr >0. Denoting by

B_p=B_p(0, 1), B_p=B_p(0, 1) (1.4) the corresponding unit balls, then

B_p(x,r)=x+rB_p, B_p(x,r)=x+rB_p. (1.5) The unit ballsB_pandBpare convex absorbing subsets of the spaceXandpagrees with the Minkowski functional associated to any of them.

An asymmetric seminorm p onX generates a topologyτ_p onX, having as basis of neighborhoods of a pointx∈Xthe family{B_p(x,r) :r >0}of open p-balls. The family {B_p(x,r) :r >0}of closedp-balls is also a neighborhood basis atxforτ_p.

The ballB_p(x,r) isτ_p-open but the ballB_p(x,r) need not to beτ_p-closed, as can be seen from the following typical example.

Example 1.1. Consider onRthe asymmetric seminormu(α)=max{α, 0},α∈R, and denote byRuthe spaceRequipped with the topologyτugenerated byu. The conjugate seminorm is ¯u(α)= −min{α, 0}, andu^s(α)=max{u(α), ¯u(α)} = |α|.The topologyτ_u, called theupper topologyofR, is generated by the intervals of the form (−∞;a),a∈R, and the family{(−∞;α+) :>0}is a neighborhood basis of a pointα∈R. The set (−∞; 1)=B_u(0, 1) is τ_u-open, and the ballB_u(0, 1)=(−∞; 1] is notτ_u-closed because R\Bu(0, 1)=(1;∞) is notτu-open.

Remark 1.2. As can be easily seen, the continuity of a mapping f from a topological space (T,τ) to (R,τu) is equivalent to its upper semicontinuity as a mapping from (T,τ) to (R,| · |).

The topologyτ_p is translation invariant, that is, the addition + :X×X→X is continuous, but the multiplication by scalars·:R×X→Xneed not be continuous, as it is shown by some examples, as, for example, that given in [5]. We will present another one in the context ofExample 1.1.

Example 1.3. In the space (R,u) from Example 1.1, the interval (−∞; 1/2) is a τ_u- neighborhood of 0=(−1)0 but for anyα,β >0, the neighborhood (−∞;−1 +α)×(−∞, β) of (−1, 0) contains the point (−1,−1) and (−1)(−1)=1∈/ (−∞; 1/2).

The discontinuity of the multiplication by scalars, (α,x)→αx, forα= −1 follows also from the fact that the interval (−∞;a) isτ_u-open but−(−∞;a)=(−a;∞) is notτ_u-open.

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The following proposition will be useful in the study of the continuity of linear mappings between asymmetric locally convex spaces.

Proposition1.4. IfX is a real vector space, f,g:X→Rare sublinear functionals, and α,β >0, then the following conditions are equivalent:

∀x∈X, g(x)≤β=⇒f(x)≤α, (1.6)

∀x∈X, uf(x)≤α

βug(x). (1.7)

Ifg(x)≥0for allx∈X, then these two conditions are also equivalent to the following one:

∀x∈X, f(x)≤α

βg(x). (1.8)

Proof. (1.6)⇒(1.7). Let x∈X. If g(x)≤0, theng(nx)=ng(x)≤0< α,n∈N, so that n f(x)=f(nx)≤β,n∈N, implying that f(x)≤0 and

uf(x)=0=α

βug(x). (1.9)

Ifg(x)>0, theng((β/g(x))x)=β, so that f

β g(x)x

≤α ⇐⇒ f(x)≤α

βg(x) ⇐⇒ uf(x)≤α

βu(g(x)). (1.10) (1.7)⇒(1.6). Letx∈X. Ifg(x)≤0< β, thenu(g(x))=0, so that

f(x)≤uf(x)≤α

βug(x)=α

βg(x)≤α. (1.11)

Ifg(x)>0, then by hypothesis,

f(x)≤uf(x)≤α

βug(x)=α

βg(x)≤α. (1.12)

Sinceg(x)≥0,x∈X, implies thatu(g(x))=g(x),x∈X, the equivalence (1.7)⇔(1.8) is

obvious.

Let nowPbe a family of asymmetric seminorms on a real vector spaceX. Denote by Ᏺ(P) the family of all nonempty finite subsets ofP, and forF∈Ᏺ(P),x∈X, andr >0, let

BF(x,r)=

y∈X:p(y−x)≤r, p∈F=

Bp(x,r) :p∈F, B_F(x,r)=

y∈X:p(y−x)< r, p∈F=

B_p(x,r) :p∈F (1.13) denote the closed, respectively, open multiball of centerxand radiusr. It is immediate that these multiballs are convex absorbing subsets ofX.

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Putting

p_F(x)=maxp(x) :p∈F, x∈X, (1.14) thenp_Fis an asymmetric seminorm onXand

B_F(x,r)=B_p_F(x,r), B_F(x,r)=B_p_F(x,r). (1.15) Theasymmetric locally convextopology associated to the familyPof asymmetric seminorms on a real vector spaceX is the topologyτP having as basis of neighborhoods of any pointx∈Xthe familyᏺ(x)= {B_F(x,r) :r >0,F∈Ᏺ(P)}of open multiballs. The familyᏺ(x)= {B_F(x,r) :r >0, F∈Ᏺ(P)}of closed multiballs is also a neighborhood basis atxforτ_P.

It is easy to check that the familyᏺ(x) fulfills the requirements of a neighborhood basis, that is,

(BN1)x∈B_F(x,r),

(BN2) for B_F₁(x,r1) andB_F₂(x,r2) inᏺ(x), there exists B_F(x,r)∈ᏺ(x) such that B_F(x,r)⊂B_F₁(x,r1)∩B_F₂(x,r2).

For (BN2), one can takeF=F1∪F2andr=min{r1,r2}.

Obviously, for P= {p}, we obtain the topologyτp of an asymmetric seminormed space (X,p) considered above, that is,τ_{p}=τp.

The topologyτ_Pis derived from a quasiuniformityᐃPonXhaving as vicinities the sets

WF()=

(x,y)∈X×X:p(y−x)<, p∈F, (1.16) forF∈Ᏺ(P) and>0. Replacing the sign< by≤in the above definition, the corresponding sets will form a basis for the same quasiuniformityᐃP. A good source for the properties of quasiuniform spaces is the book [10] (see also [4]). Quasiuniform structures related to asymmetric normed spaces were investigated in [1,2,9].

We say that the familyPisdirectedif for anyp1,p2∈P, there existsp∈Psuch thatp≥ p_i,i=1, 2, wherep≥qstands for the pointwise ordering:p(x)≥q(x) for allx∈X. If the familyPis directed, then for anyτP-neighborhood of a pointx∈X, there existp∈Pand r >0 such thatB_p(x,r)⊂V(resp.,Bp(x,r)⊂V). Indeed, ifB_F(x,r)⊂V, then there exists p∈Psuch thatp≥qfor allq∈Fso thatB_p(x,r)⊂B_F(x,r)⊂V. Similarly, the vicinities defined by (1.16) withF= {p},p∈P, and>0 form a basis for the quasiuniformity ᐃP.

The family

Pd=

pF:F∈Ᏺ(P), (1.17)

wherepFis defined by (1.14), is a directed family of asymmetric seminorms generating the same topology asP, that is,τ_P_d=τ_P. Therefore, without restricting the generality, we can always suppose that the familyPof asymmetric seminorms is directed.

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BecauseB_F(x,r)=x+B_F(0,r) andB_F(x,r)=x+B_F(0,r), the topologyτ_P is translation invariant,

ᐂ(x)=

x+V:V∈ᐂ(0), (1.18)

where byᐂ(x) we have denoted the family of all neighborhoods with respect toτP of a pointx∈X.

The addition + :X×X→Xis continuous. Indeed, forx,y∈Xand the neighborhood B_F(x+y,r) ofx+y, we haveB_F(x,r/2) +B_F(y,r/2)⊂B_F(x+y,r).

As we have seen inExample 1.3, the multiplication by scalars need not be continuous, even in asymmetric seminormed spaces.

The topologyτp generated by an asymmetric norm is not always Hausdorff. A nec- essary and sufficient condition in order thatτ_p be Hausdorffis given in the following proposition.

Proposition1.5 (see [13]). For an asymmetric seminormpon a real vector spaceX, put p(x)˜ =infp(x) +p(x−x) :x∈X, x∈X. (1.19) (1)The functionalp˜is a (symmetric) seminorm onX,p˜≤p, andp˜is the greatest of the seminorms onXmajorized byp.

(2)The topologyτpgenerated bypis Hausdorﬀif and only ifp(x)˜ >0for everyx=0.

Proof. We will give a proof of the first assertion, diﬀerent from that given in [13]. The second assertion will be proved in the more general context of asymmetric locally convex spaces.

First, observe that replacingxbyx−xin (1.19), we get p(˜ −x)=infp(x) +p(x+x) :x∈X

=infp(x−x) +p(x−x) +x:x∈X=p(x),˜ (1.20) so that ˜pis symmetric. The positive homogeneity of ˜p, ˜p(αx)=αp(x),˜ x∈X,α≥0, is obvious. Forx,y∈Xand arbitraryx,y∈X, we have

˜

p(x+y)_≤p(x+y) +p(x+y₋x₋y)_≤p(x) +p(x₋x) +p(y) +p(y₋y), (1.21) so that passing to infimum with respect tox,y∈X, we obtain the subadditivity of ˜p,

p(x˜ +y)≤p(x) + ˜˜ p(y). (1.22) Suppose now that there exists a seminormqonXsuch thatq≤p, that is, for allz∈X, q(z)≤p(z), and ˜p(x)< q(x)≤p(x), for somex∈X. Then, by the definition of ˜p, there existsx∈Xsuch that ˜p(x)< p(x) +p(x−x)< q(x), leading to the contradiction

q(x)≤q(x) +q(x−x)=q(x) +q(x−x)≤p(x) +p(x−x)< q(x). (1.23)

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The following characterization of the Hausdorﬀseparation property for locally convex spaces is well known, see, for example, [19, Lemma VIII.1.4].

Proposition1.6. Let(X,Q)be a locally convex space, whereQis a family of seminorms generating the topologyτQofX. The topologyτQis Hausdorﬀseparated if and only if for everyx∈X,x=0, there existsq∈Qsuch thatq(x)>0.

In the case of asymmetric locally convex spaces, we have the following characterization.

Proposition1.7. Let P be a family of asymmetric seminorms on a real vector spaceX.

The asymmetric locally convex topologyτP is Hausdorﬀseparated if and only if for every x∈X,x=0, there exists p∈P such thatp(x)˜ >0, wherep˜is the seminorm associated to the asymmetric seminormpthrough the formula (1.19).

Proof. Suppose thatPis directed and let

P˜= {p˜:p∈P}. (1.24)

Denote byτP˜the locally convex topology onXgenerated by the family ˜Pof seminorms.

The topologyτ_P is finer thanτP˜. Indeed, ˜G∈τP˜ is equivalent to the fact that for every x∈G, there exist˜ p∈P andr >0 such thatBp˜(x,r)⊂G. Because˜ p(y−x)< r implies that ˜p(y−x)≤p(y−x)< r, we haveB_p(x,r)⊂Bp˜(x,r)⊂G, so that ˜˜ G∈τP. If for every x∈X,x=0, there existsp∈Psuch that ˜p(x)>0, then the locally convex topologyτP˜is separated Hausdorﬀ, and so will be the finer topologyτ_P.

Conversely, suppose that the topologyτPis Hausdorﬀand show that ˜p(x)=0 for all p∈Pimplies thatx=0.

Letx∈Pbe such that ˜p(x)=0 for allp∈P. By the definition (1.19) of the seminorm p, for every˜ p∈Pandn∈N, there exists an elementx_(p,n)∈Xsuch that

px(p,n)

+px(p,n)−x<1

n. (1.25)

Define the order onP×Nby (p,n)≤(q,m) if and only ifp≤qandn≤m. Since the familyPof asymmetric seminorms is directed, the setP×Nis also directed with respect to the order we just defined. Therefore,{x(p,n): (p,n)∈P×N}is a net inXand by (1.25), we have

px(p,n)

<1

n, px(p,n)−x< 1

n, (1.26)

for all (p,n)∈P×N.

We will prove that the net{x(p,n)}converges to both 0 andx. Since the topologyτPis Hausdorﬀ, this will imply thatx=0.

To prove that the net{x_(p,n)}converges to 0, we have to show that for everyp∈P, the net{p(x(p,n))}tends to 0, that is,

∀p∈P,∀>0,∃ p0,n0

∈P×N,∀(q,n)∈P×N, such that (q,n)≥

p0,n0

=⇒px(q,n)

<. (1.27)

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Let p∈Pand>0. Put p0=pand letn0∈Nbe such that 1/n0<. Then for every (q,n)∈P×Nsuch thatq≥pandn≥n0, we have

px(q,n)

≤qx(q,n)

<1 n^≤

1

n0<. (1.28)

The convergence of{p(x(p,n)−x)}to 0, which is equivalent to theτP-convergence of {x_(p,n)}tox, can be proved similarly, using the second inequality in (1.26).

Corollary 1.8. Let (X,P)be an asymmetric locally convex space. If the topology τ_P is Hausdorﬀ, then for everyx∈X,x=0, there existsp∈Psuch thatp(x)>0.

Proof. If the topologyτP is Hausdorﬀ, then for everyx∈X,x=0, there exists p∈P such that ˜p(x)>0. Replacingxby−x and takingx=0 in the definition (1.19) of the seminorm ˜p, we get

p(x)=p(0) +p(0 +x)≥p(x)˜ >0. (1.29) As in the symmetric case, asymmetric locally convex topologies can be defined through some basic families of convex absorbing sets.

A nonempty familyᏯof subsets of a real vector spaceXis called anasymmetric locally convex basisprovided that

(BC1) eachC∈Ꮿis convex and absorbing;

(BC2) for everyC1,C2∈Ꮿ, there existsC∈Ꮿsuch thatC⊂C1∩C2; (BC3) for everyC∈Ꮿandα >0, there existsD∈Ꮿsuch thatD⊂αC.

Define a mappingᐁ:X→2^Xby

ᐁ(x)= {U⊂X:∃C∈Ꮿsuch thatx+C⊂U}. (1.30) Recall that for an absorbing subsetCofX, the Minkowski functionalp_C:X→[0;∞) is defined by

pC(x)=inf{t >0 :x∈tC}. (1.31) IfCis absorbing and convex, thenpCis a positive sublinear functional, and

x∈X:p_C(x)<1⊂C⊂

x∈X:p_C(x)≤1. (1.32) Conversely, ifpis a positive sublinear functional onX, thenC= {x∈X:p(x)<1} andC= {x∈X:p(x)≤1}are convex absorbing subsets ofX, andpC=pC=p.

Denoting by

P= {pC:C∈Ꮿ} (1.33)

the family of all Minkowski functionals associated to the sets inᏯ, thenPis a family of asymmetric seminorms onX. By (BC1) and the fact thatp_C≤p_DifD⊂C, it follows that the familyPis directed.

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Proposition1.9. The familyᐁ(x)of subsets ofX given by (1.30) satisfies the axioms of a neighborhood system, so that it defines a topologyτ_ᏯonX. This topology agrees with the asymmetric locally convex topology generated by the family (1.33) of asymmetric seminorms.

Proof. One can easily check that the familyᐁof subsets of X satisfies the axioms of a neighborhood system.

Since both of the topologiesτ_Ꮿandτ_Pare translation invariant, in order to prove their coincidence, it suﬃces to show that they have the same 0-neighborhoods. Denote byᐂ the neighborhood mapping associated toτP. IfU∈ᐁ(0), then there existsC∈Ꮿsuch thatC⊂U. The inclusions

x∈X:pC(x)<1⊂C⊂U (1.34) show thatU∈ᐂ(0).

Conversely, ifV∈ᐂ(0), then there existC∈Ꮿandr >0 such that{x∈X:p_C(x)≤ r} ⊂V. By (BC3), there existsD∈Ꮿsuch thatD⊂rC. But then

D⊂rC⊂

x∈X:pC(x)≤r⊂V, (1.35)

so thatV∈ᐁ(0).

2. Bounded linear mappings between asymmetric locally convex spaces and the dual space

Let (X,P), (Y,Q) be two asymmetric locally convex spaces with the topologiesτPandτQ

generated by the familiesPandQof asymmetric seminorms onXandY, respectively. In the following, when we say that (X,P) is an asymmetric locally convex space, we under- stand thatXis a real vector space,Pis a family of asymmetric seminorms onX, andτPis the asymmetric locally convex topology associated toP.

A linear mappingA:X→Y is called (P,Q)-bounded if for everyq∈Q, there exist F∈Ᏺ(P) andL≥0 such that

∀x∈X, q(Ax)≤Lmaxp(x) :p∈F. (2.1) If the familyPis directed, then the (P,Q)-boundedness ofAis equivalent to the condition that for everyq∈Q, there existp∈PandL≥0 such that

∀x∈X, q(Ax)≤Lp(x). (2.2)

The continuity of the mappingAfrom (X,τP) to (Y,τQ) is called (τP,τQ)-continuity.

We will use also the term (P,Q)-continuity for this property, and (P,u)-continuity in the case of (τ_P,τ_u)-continuous linear functionals.

Because both of the topologiesτPandτQare translation invariant, we have the following result. Recall that a mappingFbetween two quasiuniform spaces (X,ᐁ) and (Y,ᐃ) is called quasiuniformly continuous if for everyW∈ᐃ, there existsU∈ᐁsuch that (F(x),F(y))∈Wfor every (x,y)∈U.

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Proposition2.1. Let(X,P)and(Y,Q)be asymmetric locally convex spaces andA:X→Y a linear mapping. The following conditions are equivalent.

(1)The mappingAis(P,Q)-continuous onX.

(2)The mappingAis continuous at0∈X.

(3)The mappingAis continuous at some pointx0∈X.

The following proposition emphasizes the equivalence of continuity and boundedness for linear mappings.

Proposition2.2. Let(X,P)and(Y,Q)be two asymmetric locally convex spaces andA: X→Y a linear mapping. The following assertions are equivalent.

(1)The mappingAis(P,Q)-continuous onX.

(2)The mappingAis continuous at0∈X.

(3)The mappingAis(P,Q)-bounded.

(4)The mappingAis quasiuniformly continuous with respect to the quasiuniformities ᐃPandᐃQ.

Proof. The equivalence (1)⇔(2) follows from the preceding proposition.

Suppose that the familiesPandQare directed.

(2)⇔(3). Forq∈Q, consider theτ_Q-neighborhoodV =B_q(0, 1) ofA0=0∈Y, and letU be a neighborhood of 0∈Xsuch thatA(U)⊂V. If p∈Pandr >0 are such that Bp(0,r)⊂U, then

∀x∈X, p(x)≤r=⇒q(Ax)≤1. (2.3) ByProposition 1.4applied to f(x)=q(Ax) andg(x)=p(x), this relation implies that

∀x∈X, q(Ax)≤1

rp(x). (2.4)

Conversely, ifAis (P,Q)-bounded, then for everyτQ-neighborhoodV of 0∈Y, there existq∈QandR >0 such thatBq(0,R)⊂V. Letp∈PandL≥0 be such that the condition (2.2) is fulfilled. Takingr:=R/(L+ 1), we have

∀x∈Bp(0,r), q(Ax)≤Lp(x)≤ L

L+ 1R≤R, (2.5)

which shows thatA(B_p(0,r))⊂B_q(0,R)⊂V, that is,Ais continuous at 0∈X.

The implication (3)⇒(4) follows from the (P,Q)-boundedness of the mappingAand the definition (1.16) of the vicinities.

To prove (4)⇒(3), suppose thatAis (ᐃP,ᐃQ)-quasiuniformly continuous. Forq∈Q, let W= {(y,y)∈Y×Y :q(y−y)≤1} ∈ᐃQ, and let U= {(x,x)∈X×X:p(x− x)≤r} ∈ᐃPbe such that (x,x)∈Uimplies that (Ax,Ax)∈W. Takingx=0, it follows that

∀x∈X, p(x)≤r=⇒q(Ax)≤1, (2.6)

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so that, byProposition 1.4,

∀x∈X, q(Ax)≤1

rp(x). (2.7)

In the case of linear functionals on an asymmetric locally convex space, we have the following characterization of continuity, whereuis as inExample 1.1.

Proposition2.3. Let(X,P)be an asymmetric locally convex space andϕ:X_→_Ra linear functional. The following assertions are equivalent.

(1)ϕis(P,u)-continuous at0∈X.

(2)ϕis(P,u)-continuous onX.

(3)There existp∈PandL≥0such that

∀x∈X, ϕ(x)≤Lp(x). (2.8)

(4)ϕis upper semicontinuous from(X,τ_P)to(R,| · |).

Remark 2.4. If the familyPis not directed, then the (P,u)-continuity of the functionalϕ is equivalent to the condition that there existF∈Ᏺ(P) andL≥0 such that

∀x∈X, ϕ(x)≤Lmaxp(x) :p∈F=Lp_F(x). (2.9) The dual of an asymmetric locally convex space. For an asymmetric locally convex space (X,P), denote byX=X_Pthe set of all linear (P,u)-continuous functionals. IfP= {p}, then we obtain the dual spaceX_pof an asymmetric normed space (X,p) considered in [13].

LetX^#be the algebraic dual space toX, that is, the space of all linear functionals onX.

In contrast to the symmetric case,X=X_Pis not a subspace ofX^#, but merely a convex cone, that is,

(i)ϕ,ψ∈X⇒ϕ+ψ∈X, (ii)ϕ∈Xandα≥0⇒αϕ∈X.

There are examples in the caseP= {p}of p-bounded linear functionalsϕon a space with asymmetric norm (X,p) such that−ϕis notp-bounded, see [5]. A simpler example can be exhibited in the space (R,u) fromExample 1.1.

Example 2.5. The identity mappingϕ(t)=t,t∈R, is (τu,τu)-continuous because

∀t∈R, ϕ(t)=t≤max{t, 0} =u(t), (2.10) but −ϕ is not (τu,τu)-continuous, because it is impossible to find L≥0 such that (−ϕ)(t)≤Lu(t) for allt∈R. Indeed, takingt= −1, we obtain the contradiction

1=(−ϕ)(−1)≤L·u(−1)=0. (2.11) Remark 2.6. It is easy to check that a linear functional ϕ(t)=at, t∈R, is (τu,τu)- continuous if and only ifa≥0. Indeed ifa≥0, thenϕ(t)=at≤u(at)=au(t),t∈R. Ifa <0, then, reasoning as above, one concludes thatϕfails to be continuous.

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Suppose that the familyP of asymmetric seminorms is directed, and for p∈P, let p^s(x)=max{p(x),p(−x)}be the symmetric seminorm attached top, and let

P^s=

p^s:p∈P. (2.12)

Denote byX^∗=(X,P^s)^∗the dual space of the locally convex space (X,P^s). Since for a seminormqand a linear functionalϕwe have

∀x∈X, ϕ(x)≤Lq(x) ⇐⇒ ∀x∈X, ϕ(x)≤Lq(x), (2.13) we haveX=X_P⊂X^∗=(X,P^s)^∗. Indeed, ifϕ∈X,p∈P, andL≥0 are such that for allx∈X,ϕ(x)≤Lp(x), then, the inequalityp≤p^sand the above equivalence imply that

|ϕ(x)| ≤Lp^s(x),x∈X, showing thatϕ∈X^∗.

Let p be an asymmetric seminorm on a real vector space X and letϕ:X→Rbe a linear functional. Put

ϕ|p=supϕBp

. (2.14)

We say that the functionalϕisp-bounded if there existsL≥0 such that

∀x∈X, ϕ(x)≤Lp(x). (2.15)

A numberL≥0 satisfying (2.15) is called ap-Lipschitz constant forϕ. The functionalϕ isp-bounded if and only ifϕ|p<∞andϕ|pis the smallestp-Lipschitz constant forϕ.

Thep-boundedness ofϕis also equivalent to its (τp,τu)-continuity. The functional · |p

defined by (2.14) is an asymmetric norm on the asymmetric dualX_pof (X,p), that is, ϕ+ψ|p≤ ϕ|p+ψ|p,αϕ|p=αϕ|p, for allϕ,ψ∈X_p andα≥0. Also,ϕ|p>0 for ϕ∈X_p\ {0}.

Similar considerations can be done with respect to the conjugate asymmetric seminorm ¯p(x)=p(−x) ofpand

ϕ|p¯=supϕBp¯

. (2.16)

Some properties of the norm · |pare collected in the following proposition.

Proposition2.7. Letpbe an asymmetric seminorm on a real vector spaceXandϕ:X→R a linear functional.

(1)The following equalities hold:

ϕ|p=supϕB_p, ϕ|p¯=supϕBp¯

. (2.17)

Moreover, if the functionalϕ=0isp-bounded, thenϕ|p>0andϕ(x0)= ϕ|p, for some x0∈Bp, implies thatp(x0)=1.

(2)Ifϕ=0is(p, ¯p)-bounded, then ϕB_p=

− ϕ|p^¯;ϕ|p

. (2.18)

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Ifϕisp-bounded but notp-bounded, then¯ ϕB_p=

− ∞;ϕ|p

. (2.19)

Proof. (1) If (x_n) is a sequence inB_p such thatϕ(x_n)→ ϕ|p forn→ ∞, thenx_n=(1− 1/n)xn∈B_pandϕ(x_n)→ ϕ|p. Because

supϕB_p≤supϕBp

= ϕ|p, (2.20)

it follows that supϕ(B_p)= ϕ|p.

Ifϕ=0, andz∈Xis such thatϕ(z)>0, then the inequalityϕ(z)≤ ϕ|pp(z) implies thatϕ|p>0.

Ifx0∈B_pis such thatϕ(x0)= ϕ|p, thenϕ|pp(x0)≥ϕ(x0)= ϕ|p>0, so thatp(x0)>

0. Ifp(x0)<1, thenx1=(1/ p(x0))x0∈Bpand ϕx1

=ϕx0

px0

= ϕ|p

px0

>ϕ|p=supϕB_p, (2.21) a contradiction.

(2) Suppose thatϕis (p, ¯p)-bounded. We have

ϕ|p¯=supϕ(x) : ¯p(x)<1=supϕ(−x) :p(x)<1

= −infϕ(x) :p(x)<1. (2.22)

Similar calculations show that

ϕ|p¯= −infBp

. (2.23)

BecauseB_pis convex, it follows thatϕ(B_p) is an interval inRand − ϕ|p¯;ϕ|p

⊂ϕB_p⊂ − ϕ|p¯;ϕ|p

. (2.24)

Ifϕ|p∈ϕ(B_p), thenϕ|p=ϕ(x0), for somex0∈Xwithp(x0)<1, in contradiction to the assertion (1) of the proposition. Similarly, if−ϕ|p¯∈ϕ(B_p), then−ϕ|p¯=ϕ(x1), for somex1∈Xwithp(x1)<1. But then, forx₁=(1/ p(x1))x1∈B_p, we obtain the contradiction

ϕx₁=ϕx1

px1

=−ϕ|p^¯

px1

<−ϕ|p^¯=infϕBp

. (2.25)

Ifϕisp-bounded but not ¯p-bounded, then

ϕ|p¯=supϕB_p= ∞, (2.26)

so that, by (2.22), infϕ(B_p)= −ϕ|p¯= −∞. Since ϕ(B_p) is an interval in R,ϕ|p= supϕ(B_p), andϕ|p∈/ ϕ(B_p), it follows thatϕ(B_p)=(−∞;ϕ|p).

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Extension of bounded linear functionals. As in the symmetric case, an extension result for continuous linear functionals defined on subspaces of an asymmetric locally convex space will be particularly useful in developing a duality theory for such spaces.

Proposition2.8. Let(X,P)be an asymmetric locally convex space andYa subspace ofX.

Ifϕ:Y→Ris a(P,u)-continuous linear functional, then there exists a(P,u)-continuous linear functionalΦ:X→Rsuch thatΦ|Y=ϕ.

Proof. Suppose that the familyP is directed. ByProposition 2.3, there exist p∈P and L≥0 such that

∀y∈Y, ϕ(y)≤Lp(y). (2.27)

By the Hahn-Banach dominated extension theorem, there exists a linear functional Φ:X→Rsuch thatΦ|Y=ϕand

∀x∈X, Φ(x)≤Lp(x), (2.28)

which, by the sameProposition 2.3, is equivalent to the (P,u)-continuity ofΦ.

The following existence result is well known in the symmetric case.

Proposition2.9. (1)Ifp is an asymmetric norm on a real vector spaceX andx0∈Xis such thatp(x0)>0, then there exists a p-bounded linear functionalϕ:X→Rsuch that

(i)ϕx0

=px0

, (ii)ϕ|p=1.

(2)Let(X,P)be an asymmetric locally convex space. If the topologyτ_Pis Hausdorﬀ, then for everyx0∈X,x0=0, there existsψ∈Xsuch thatψ(x0)=1.

Proof. (1) LetZ=Rx0 andϕ0:Z→Rbe defined byϕ0(tx0)=t p(x0), t∈R. Thenϕ0

is linear andϕ0(tx0)=t p(x0)=p(tx0) fort≥0. Sinceϕ0(tx0)=t p(x0)<0≤p(tx0) for t <0, it follows thatϕ0(z)_≤p(z) for allz_∈Z. By the Hahn-Banach extension theorem, there exists a linear functionalϕ:X→Rsuch thatϕ|Z=ϕ0andϕ(x)≤p(x) for allx∈X, implying thatϕ|p≤1. Since

ϕ|p=supϕ(x) :x∈Bp

≥supϕ0(z) :z∈Z∩Bp

≥ϕ0

1 px0x0

=1, (2.29) it follows thatϕ|p=1.

(2) Ifx0=0 andτPis Hausdorﬀ, then byCorollary 1.8, there exists p∈P such that p(x0)>0. Ifϕ:X→Ris ap-bounded linear functional satisfying the conditions (i) and (ii) of the first assertion, then we can takeψ=(1/ p(x0))ϕ.

Thew-topology of the dualX. This is the analog of the weak^∗-topology (w^∗-topology) on the dual of a locally convex space. In the case of an asymmetric normed space (X,p), it was considered in [13].

Let (X,P) be an asymmetric locally convex space andX=X_P the asymmetric dual cone. Aw-neighborhood of an elementϕ∈Xis a subsetWofXfor which there exist