ON BOUNDEDLY-CONVEX FUNCTIONS ON PSEUDO-TOPOLOGICAL VECTOR SPACES

Notions of a boundedly convex function and of a Lipschitz-continuous function are extended to the case of functions on pseudo-topological vector spaces. It is proved that for “good” pseudo-topologizers Ψ , any continuous Ψ -boundedly convex function is Ψ -differentiable and its derivative is Ψ -Lipschitz-continuous. As a corollary, it is shown that any boundedly convex function is Hyers-Lang differentiable.


Introduction.
In [5] Joachim Focke, proved that for any continuous boundedlyconvex function f on a Banach space B, its Fréchet derivative f : B → B * (which always exists for such functions) is Lipschitz-continuous, that is, there exists M > 0, such that for every x 1 ,x 2 from B, f (x 1 ) − f (x 2 ) ≤ M x 1 − x 2 . (1.1) Recall that a convex function f : B → R is called boundedly-convex if there exists M > 0 such that, for any x 1 ,x 2 from B and any λ 1 ,λ 2 ≥ 0 with λ 1 + λ 2 = 1, This means that the "deviation from linearity" for f is not greater than for (1/2)M • 2 , in the case of a norm generated by a scalar product.See Remark 3.2.Notice that we can take M from (1.2) as M for (1.1).We extend this result to the case of functions f defined on arbitrary pseudo-topological (in particular topological) vector spaces (One often uses the term "convergence space" as a synonym to "pseudo-topological space".Below, we recall necessary definitions concerning pseudo-topological vector spaces.For more details, see [6].)For this end, we have to do three things: (1) extend the notion of bounded convexity, (2) extend the notion of Lipschitz-continuity, (3) specify the definition of differentiability we use since, as it is well-known, the notion of Fréchet differentiability can be extended to the case of nonnormable spaces by many different ways.We do these things by appealing the notion of a pseudo-topologizer, which was introduced in [4] and was thoroughly investigated in [1].In those papers, it was attached to each pseudo-topologizer Ψ a corresponding notion of differentiability (called Ψdifferentiability).Here, we attach to each pseudo-topologizer Ψ notions of Ψ -bounded convexity and of Ψ -Lipschitz-continuity.Besides, we introduce an extension of the notion of bounded convexity (that does not append pseudo-topologizers).We prove that, roughly speaking, for "good" pseudo-topologizers Ψ , any continuous Ψ -boundedly convex function on a pseudo-topological vector space is Ψ -differentiable and that its derivative is Ψ -Lipschitz-continuous.Furthermore, we see that both the pseudotopologizer F s of convergence on a linear filter system S and their onion modification F # s are "good".As a consequence, we derive that any continuous boundedly convex function on a topological vector space is Hyers-Lang differentiable.
The Focke's result corresponds to the case where Ψ is the pseudo-topologizer F b of convergence on a system of bounded sets.

Notation. Throughout, we use the following notation:
R the reals, R + := [0, +∞) r the filter of the neighborhoods of zero in R r + the trace of the filter r on R + X # the onion modification of X (see below) [A] the filter in a set X, generated by a subset A ⊂ X, i.e., the filter of all the subsets of X that contain A [x] the (trivial ultra-) filter generated by a singleton {x} F(X,Y ) the set of all mappings from X into Y L(X, Y ) the set of all continuous linear mappings from X into Y If there enter filters in an expression, then this expression is to be interpreted as the image of the product of these filters by the corresponding mapping.
For example, if x is a filter in a vector space X, then r x denotes the image of the filter r×x (that is, of the filter with the basis consisting of all the products I ×U, where I ∈ r, U ∈ x) under the multiplication mapping R×X → X, (t, x) tx, that is the filter with basis consisting of sets of the form I t U, where t > 0, U ∈ x (I t U := {τx|τ ∈ I t , x ∈ U}).
As another example, if f is a filter in F(X,Y ) and x is a filter in X, then f(x) denotes the image of the filter x×f under the evaluation mapping X ×F(X,Y ) → Y , (x, f ) f (x), that is, the filter with basis consisting of all sets of the form Preliminary notions.A pseudo-topology (or a convergence structure) ψ in a set X is a mapping from X into the power set of the set of all filters in X, that satisfies the following conditions (where we write x ↓ x ψ instead of "x ∈ ψ(x)"; one reads this relation as "the filter x convergent to the point x in the pseudo-topology ψ"): (a) ∀x ∈ X : A pseudo-topological space (or a convergence space) is the pair (X, ψ), where X is a set, and ψ is a pseudo-topology in X. Usually, we simply write X instead of (X, ψ), The pseudo-topology generated in a natural sense by a topology, is being identified with this topology.
A net {x ı } ı∈I in a pseudo-topological space X is said to converge to a point where x is the filter of "tails" of the net, that is A ∈ x : Y , and is called continuous if it is continuous at each point.The pseudo-topology ψ induced on a subset X ⊂ X by a pseudo-topology ψ in X is defined as follows: x ↓ A pseudo-topological vector space (p.v.s.) is a vector space X equipped with a pseudotopology that is compatible with the vector structure in X (in the sense that the operations of addition and multiplication by a scalar are continuous, the real line being equipped with its natural topology).
If we weaken the compatibility conditions by replacing the requirement of continuity of the multiplication mapping by the following two conditions: we obtain the notion of a pseudo-topological vector group (p.v.g.).In p.v.s.'s it holds, besides, the condition For p.v.g.'s, we simply write x ↓ instead of x ↓ 0 .
A filter x in a p.v.g. is called bounded if r x ↓.A set B in a p.v.g. is called bounded if the filter [B] is bounded (that is, if r B ↓).A point x in a p.v.g. is called bounded if the set {x} is bounded (that is if r x ↓).Thus, a p.v.s. is a p.v.g.such that all its points are bounded.
For every p.v.g.X, the associated onion (or equable) p.v.g.X # is defined by the following conditions: as a vector space X # coincides with X, and x ↓ X # : ∃y ↓ X : x ⊃ y = ry .
It is clear that the pseudo-topology of X # is stronger than the pseudo-topology of X, that is the identity mapping X # → X is continuous.
Any topological vector space (t.v.s.) X is a p.v.s. and satisfies the condition X # = X.

Pseudo-topologizers
Definition 2.1 [1].Let A and B be subcategories of the category P V G of all p.v.g.'s with the continuous linear mappings as morphisms.A pseudo-topologizer Ψ on A × B is a covariant functor Ψ : A • ×B → P V G (where A • denotes the dual category to A) that satisfies the following conditions: (a) for any two objects X and Y from A and B, respectively, Ψ (X, Y ) is (as a vector space) the vector subspace in F(X,Y ) that contains L(X, Y ); (b) for any two morphisms u ∈ L(X 2 ,X 1 ) and v ∈ L(Y 1 ,Y 2 ) of the categories A and B, respectively, and for any mapping For any pseudo-topologizer Ψ , the formula Ψ # (X, Y ) = Ψ (X, Y ) # defines a pseudotopologizer Ψ # which is called the onion modification of Ψ .Definition 2.2 [1].Let A be a subcategory of P V G.We say that a linear filter system S in A is given if, for any p.v.g.X from A, a nonempty set S(X) of filters in X is given such that the following conditions are fulfilled: (a) for each X and Y from A, if x ∈ S(X) and l ∈ L(X, Y ), then l(x) ∈ S(Y ); (b) for each X from A, if x, y ∈ S(X), then x + y ∈ S(X).In the case where, for every X, all filters form S(X) are filters of the from [A], where A is a subset of X, we say about a set system.
Important examples of linear filter systems are: is the set of all bounded sets in X.
Definition 2.3 [1].Let S be a filter system in A. We define the pseudo-topologizer F s (of convergence on S) on A × P V G by the conditions: (a) Lemma 2.4.Let X be a p.v.g., let Y be a p.v.s., and let f : Definition 2.5 (See [1]).We say that a pseudo-topologizer Ψ on A × B possesses the property (EXP) if the well-known algebraical isomorphism ) is a P V G isomorphism for any object Y from B and for any objects X 1 and X 2 from A such that X 1 ×X 2 is also an object from A. We say that Ψ possesses the property (IMB) if, for any X from A and for any Here, "imbedding" X 1 ⊂ X 2 means that X 1 is a vector subspace in X 2 and that the pseudo-topology of X 1 coincides with the pseudo-topology induced from X 2 .
Definition 2.6.We say that a pseudo-topologizer Ψ on A × B possesses the property (SAT) if the following condition is fulfilled: for any X from A and any Y from B, where the "saturation" f of a filter f in F(X,Y ) is defined as the filter generated by the filter basis { F | F ∈ f}, the set F being for any F ⊂ F(X,Y ) defined by the formula (2.4) (The fact that the sets F,F ∈ f, are really a filter basis follows from the relation F1 Theorem 2.7.For any linear filter system S, the pseudo-topologizers F S and F #

S possess the properties (EXP), (IMB), and (SAT).
Proof.The assertion on (EXP) and (IMB) was proved in [1,Thm. 1.41].Let us prove the assertion on (SAT).It is clear from (2.4) that (2.5) It follows at once from (2.5) that for any filter f in F(X,Y ), any filter x in X, and any set U in X, we have This completes the proof.

3.
Ψ -bounded convexity and Ψ -Lipschitz-continuity.Here, we introduce the notions of bounded convexity, Ψ -bounded convexity, and Ψ -Lipschitz-continuity and recall the notion of Ψ -differentiability.Definition 3.1.Let Ψ be a pseudo-topologizer on a category A of p.v.g.'s containing R as an object, and let X be a p.v.g.We say that a convex function f : X → R is boundedly convex if there exists a continuous homogeneous function of degree 2 q : X → R such that for any x 1 ,x 2 from X and any nonnegative numbers λ 1 ,λ 2 with If, in addition, q is a bounded point in Ψ (X, R), we say that f is Ψ -boundedly convex.
Remark 3.2.For any nonnegatively definite quadratic form (f (x) = b(x, x), where b is a symmetric bilinear form, such that b(x, x) ≥ 0 for all x), so that such forms satisfy condition (3.1) with the last "≤" changed by "=" and with q = f .Remark 3.3.It is evident that the addition, to f , of a constant or a linear function does not disturb the validity of (3.1), and that a translation of f by any vector h (that is, the pass from f to the function x f (x − h)) also does not disturb (3.1).Remark 3.4.It is easy to see that for normed spaces X, the definition of a boundedly convex function reduces to the usual one given in the introduction.Definition 3.5.Let Ψ be a pseudo-topologizer on A×B, and let X and Y be p.v.g.'s from A and B, respectively.We say that a mapping f : where f t,x being (for t ∈ R + and x ∈ X) a mapping from X into Y defined as follows: f t,x = 0 if t = 0, and This definition is indeed an extension of the usual one that is seen from the following lemma.
Lemma 3.6.Let X and Y be normed spaces.Then a mapping f :

continuous if and only if f is Lipschitz continuous in the usual sense, that is, if and only if there exists
(3.6) Proof.First of all, notice that equation (3.3) for Ψ = Ψ b means that for every bounded set B in X, the set is bounded in Y. Now, let f satisfy (3.6).Then for each t ∈ R + and each x ∈ X, we have (3.9)Indeed, any h can be written in the form h = h e, where e = 1.Without loss of generality, we can assume that h =: α > 0. So, we have (3.10) Definition 3.7 [4].Let Ψ be a pseudo-topologizer on A × B and let X and Y be p.v.g.'s from A and B, respectively.We say that a mapping f : where f (x) ∈ L(X, Y ) (the derivative of f at x) and r satisfies the condition that is, where r t (for t ∈ R + ) is a mapping from X into Y defined as follows: r t = 0 if t = 0, and Notice that F B -differentiability (respectively, F c -differentiability) is the so-called Frölicher-Bucher (respectively, Michal-Bastiani) differentiability and that, for normed spaces, F b -differentiability is just Fréchet differentiability.Remark 3.8.As shown in [2], for the case of topological vector spaces F # C -differentiability coincides with F # B -differentiability.This is the so-called Hyers-Lang differentiability.

The main results.
Here is the exact formulation of our results.Comparing with the above roughly speaking formulation in the introduction, a condition of continuity appears now twice.Theorem 4.1.Let Ψ be a pseudo-topologizer on P LG×P LG that possesses the properties (EXP), (IMB), and (SAT).Let X be an arbitrary p.v.s.If a continuous convex function f : X → R is Ψ -boundedly convex, the corresponding function q (see Definition 3.1) being also continuous, then f is everywhere Ψ -differentiable and its derivative f : X → L(X, R) is Ψ -Lipschitz continuous, L(X, R) being supplied with the pseudo-topology induced from Ψ (X, R).
Proof.We have the following steps.
Step 1.Here we show that f is everywhere Gateaux differentiable.Since f is convex, the restriction of f onto each straight line is a convex continuous function.As is wellknown (see, e.g., [7]), for this restriction, there exist both one-sided derivatives at each point of the straight line.This means that our function f is differentiable at each point x in any direction h.Denote the corresponding mapping by f (x).We need to verify that this mapping is linear and continuous.It is evident that it is positively homogeneous.By Remark 3.3, we may assume that x = 0 and f (0) = 0.
Step 3. Here, we verify that f (0) is continuous.Take whence it follows that As t ↓ 0, we obtain If h → 0, then f (h) and q(h) tend to zero by the supposed continuity of f and q.Hence, f (0)h → 0 if h → 0, that is, f (0) is continuous at 0 and, thereby, everywhere.Thus, we have proved that f is everywhere Gateaux differentiable.
Step 4. Now, we prove that f is Ψ -differentiable at each point x.By Remark 3.3 and the fact that the addition of continuous affine functions and translations do not disturb Ψ -differentiability, we may assume that x = 0,f (0) = 0,f (0) = 0 (where f (0) is the Gateaux derivative at 0 which was proved to exist).We need to verify that where Since the function f is convex and f (0) = 0 and f (0) = 0, both the values f (−th) and f (th) are nonnegative.Hence, that is, If t ↓ 0, then tq → 0 in Ψ (X, Y ) since q is Ψ -bounded.Therefore, also r t → 0 in Ψ (X, Y ).
Step 5. Now, we show that, for any x 1 ,x 2 from X, Again, we may assume, without loss of generality, that x = 0,f (0) = 0, and f (0) = 0 (since equation (4.8) does not disturb by the addition to f of constants and linear functions).The relation to be proved takes then the form (if we put for any h ∈ X.But this follows at once from (4.8).
Step 6.Now, we go to the proof of the main assertion on Ψ -Lipschitz continuity of our derivative.We need to show that By the properties (EXP) and (IMB) which are fulfilled for Ψ by the assumption, equation (4.16) is equivalent to the relation where f R + ,X is the set in F(X ×X, R), that corresponds to the set f R + ,X by the canonical isomorphism In the next step, we show that where But r q ↓ Ψ (X, R) by the fact that q is Ψ -bounded.Hence, both the terms in the righthand side of (4.20) converge to 0 in F(X × X, R) by condition (b) of Definition 2.1.Thereby, (4.17) is proved.
Step 7. It remains to show equation (4.19).We have, for t ∈ R + and x, h 1 ,h 2 ∈ X, (Here, the bar is to be understood in the same sense as in equation (4.17) above.)Put )th 1 , so that x 0 is the center of the parallelogram with the vertices x 1 ,...,x 4 .By (3.1), If we take the sum of the four inequalities from (4.22) to (4.25), the first two being multiplied by 2 and −2, respectively, then we obtain whence it follows that If we substitute here h 2 by −h 2 , then we find that the left-hand side of (4.27) times −1 also does not exceed the right-hand side of (4.27).So, the left-hand side belongs to I 1 (1/2q(h 1 ) + 2q(h 2 )).Therefore, (see (4.21)) So, if we put, for short, whence it follows that The theorem is proved.
Corollary 4.2.Let S be a linear filter system in a category A of p.v.g.'s, let X be a p.v.g. from A, and let f : X → R be a continuous convex function.If f is F s -boundedly convex, then f is everywhere F # s -differentiable and its derivative f : X → L(X, R) is F s -Lipschitz-continuous, L(X, R) being supplied with the pseudo-topology induced from F # s (X, R).
Proof.If the function q that appears in Definition 3.1 is a bounded point in F s (X, R), then q is also a bounded point in F # S (X, R) since r q ↓ F S (X, R) ⇒ r q ↓ F # S (X, R) by the definition of the onion modification.So, the assertion of Corollary 4.2 follows from Theorem 4.1.
Corollary 4.3.Let X be a p.v.g., and let f : X → R be a continuous convex function.If f is boundedly convex, then f is everywhere Hyers-Lang differentiable and its derivative f : Proof.This follows from Lemma 2.4 and Remark 3.8.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning .8) so that, any bounded B, the set (3.7) is bounded in Y .Conversely, let the set (3.7) be bounded for any bounded B. Take the unit ball as B. Let the norms of all elements of the corresponding set (3.7) do not exceed M. Then for any x and h, f (x + h) − f (x) ≤ M h .
1) and divide by t: and π 1 and π 2 are the canonical projections of the product X × X onto the factors.It follows from (4.19) that

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation