The Closed Neighborhood and Filter Conditions in Solid Sequence Spaces

Let E be a topological vector space of scalar sequences, with topology T; (E,T) satisfies the closed neighborhood condlt[on Iff there is a basis of nelghborhoods at the origin, for , consisting of sets whlch are closed with respect to the topology 7 of coordlnate-wlse convergence on E; (E,) satisfies the filter condition Iff every filter, Cauchy wlth respect to , convergent with respect to 7, converges with respect to. Examples are given of solid (deflnltion below) normed spaces of sequences which (a) fall to satisfy the filter condition, or (b) satisfy the filter condition, but not the closed neighborhood condition. (Robertson and others have given examples fulfilling (a), and examples fulfilllng (b), but these examples were not solid, normed sequence spaces.) However, it is shown that among separated, separable solid pairs (E,T), the filter and closed neighborhood conditions are equivalent, and equivalent to the usual coordlnate sequences constituting an unconditlonal Schauder basis for (E,T). Consequently, the usual coordinate sequences do constitute an unconditional Schauder basis in every complete, separable, separated, solid pair (E,).


BACKGROUND AND DEFINITIONS.
Suppose that E is a topological vector space under two topologles, and .
Following Robertson [I] and Garllng [2], we shall say that (E,) satisfies the closed neighborhood condition (with respect to 7) if and only if there is a base of r-nelghborhoods of the origin which are -closed, and that (E,T) satisfies the filter condition (with respect to ) if and only if each filter in E, Cauchy with respect to T, and convergent with respect to , converges with respect to T. (More exactly, Robertson might say that (E,) satisfies the filter condition with respect to and the identity injection of (E,T) into (E,).) Observe that if (E,z) is complete, then (E,z) trivially satisfies the filter condition, with respect to any .Although the result will not be used much here, a theorem of Robertson [I], Theorem I) is worth noting: if (E,) is separated (Hausdorff), and is finer than , then the completion of (E,T) is naturally embedded in the completion of (E,) if and only if (E,T) satisfies the filter condition, with respect to .
The importance of the closed neighborhood condition arises from a result of Bourbaki ([3], Proposition 8, Chap.i., I), which may also be found in Treves [4] (Lemma 34.2); this result is approximately Proposition I0 of [I], which we restate here.
l_f (E,) is separated, T is finer than , and (E,) satisfies the closed neighborhood condition, then (E,z) satisfies the filter condition.
In this paper E will be a subspace of , the vector space of all scalar sequences (the scalars may be either the real or complex numbers); will be the topology of coordlnate-wlse convergence on E. This is the relative topology on E induced by the product topology on m, thought of as a countable product of copies of the scalar field.
If x E the solid hull of x is A subset of is solid if and only if it contains the solid hull of each of its elements.
Throughout, E will be solid.
A topology T on E (with which E becomes a t.v.s.) will be called solid if there is a neighborhood base at the origin for consisting of solid sets.When both E and are solid, we will refer to (E,z) as a solid pair.
Note that if a norm p on E satlslfes y E S(x)E implies that p(y) p(x), (I.I) and E is solid, then the norm topology on E associated with p is solid.Conversely, if E is solid, any solid norm topology on E is associable with such a norm (since the convex hull of a solid set is solid); when satisfies (I.I) will be called a solid norm.
LEMMA 1.2.Suppose that E is solid, and p is a solid norm on E. Let denote the topology on E defined by p, and let U {x e E; p(x) }.Then (E, ) satisfies the closed neighborhood condition if and only if the closure of U i__n_n (E,w)is bounded in (E, ).
PROOF.If the closure W of U in (E,) is bounded in (E,), then {rW; r > 0} Is a neighborhood base at the origin in (E,) consisting of sets closed in (E,).If, on the other hand, (E,)satslfes the closed neighborhood condtion, then the closure in (E,) of some set rU, r > 0, is contained in U; thus Wc_r-Iu.
The coordinate projections on E are the functlonals f de[ined by n f (x) =x n n The following lemma is well known..5).LEMMA 1.3.Suppose that (E,) is a solid pair.The following are equivalent.(c) (E,) is separated.
The coordinate sequences are the sequences en e defined by en (m) m, e (m) 0 otherwise.
We shall assume throughout that E contains the coordinate n sequences.The finite sections are the functions P E E defined by n P (x) Y. xkek.Following the terminology in Kothe [6] if for each x e E the kgn sequence (Pn(X)) converges to x in (E, ), we will say that (E, ) is AK (for Abschnitt-Konvergenz).
When (E, ) is a separated solid pair, (E,)is AK If and only if the coordinate sequences form a Schauder basis for (E,); the solidity of E and then guarantees that the basis is unconditional.
The following is Proposition 2.9 of [5].LEMMA 1.4.Suppose that (E,) is an AK solid pair.Then (E,) satisfies the closed neighborhood condition.
In the next section we give two examples of solid, solidly normed spaces E the first fails to satisfy the filter condition, and the second satisfies the filter condition, but not the closed neighborhood condition.The object is to augment the supply of such examples; see Lindenstrauss and Tzafrlfi [7], Garllng [2], and Gaposhkin and Kadets ([8]).A far as we know, these are the first known (or, in the case of Example 2.1, noticed) such examples in which (E,x) is a solid, solidly normed sequence space, and is the topology of coordlnate-wlse convergence on E.
The norm topology in Example 2.1 is not consistent in the sense of Ruckle [9]; by Theorem of [2] It could not possibly be.The norm in Example 2.2 is consistent.
In section 3 it is shown that among separable, separated solid pairs (E,z), the filter and closed neighborhood conditions are equivalent, and are equivalent to (E,) being AK.
As an easy consequence, it follows that in every complete separable separated solid pair (E,), the coordinate sequences form an unconditional Schauder basis of (E,z).(Thus, the famous separable Banach spaces with no Schauder basis, while possibly realizable as sequence spaces, are not realizable as solid, solidly normed sequence spaces.) In Section 4 we discuss a problem mentioned in [5], still unresolved, related to the results of this paper, and one other problem that arises from these results.
Then (x(k)) k is Cauchy with respect to p, converges to zero in (E,), and O(x (k)) > for all k; thus, by Lemma 1.3, (x (k)) cannot converge tn (E,'), the norm topology associated with p. Consequently, (E,) does not satisfy the filter condition. [.Pn(X)" Clearly E is solid and p is a solid norm on E; E is an "I direct sum" of the spaces Jn ), each isomorphic to , so E, with , is complete.Consequently, (E,) satisfies the filter condition, where, as usual, T is the topology determined by p.
For each n, let gn denote the characteristic sequence of Jn; i.e. gn(m) if m Jn' and gn(m) 0 otherwise.Note that (gn n + for each n.Also, O(Pk(gn )) for each k, and gn is the limit, in (E,), of (Pk(gn))k By Lemma 1.2, (E,) does not satisfy the closed neighborhood condition.Suppose that (E,) is a separable, separated solid pair.Suppose that (E,x) is not AK.Then there is some x E E such that the sequence (P (x)) does not converge to x in (E x).But (P (x)) does converge to x In (E,); n n therefore, if (a) holds, it must be that (P (x)) is not Cauchy in (E,x).Therefore, n there is a solid T-nelghborhood U of the origin in E and two sequences (nt), (m t) of positive integers satisfying and n t < m t nt+ P (x)-P (x) If t < r then n t < m t nr, so Pint(x) Pnt(X) e S(P (x) P (x)); n n t r consequently, Pn (x) Pnt(X) U for r > t, because P (x) P (x) U and U is r mt nt solid.
For  (E,x) is not separable.We have arlved at this conclusion assuming (a) and not (c).
Consequently, (a) implies (c), when (E,x) satisfies the hypothesis of the theorem.
If (E,x) is a complete separable, separated solid pair, then (E,x) is AK.
PROOF.If (E,x) is complete, then (E,x) satisfies the filter condition.
It has been pointed out to the author that the hard part of the theorem above, (a) implies (c), resembles a corollary of a well known result about Banach lattices ([7], Proposition l.a.7).Since the scalars may be the complex numbers here, it would -I -I (nor-A (E), nor-A ()) necessarily satisfy the filter condition?Problems 4.1 and 4.2 are equivalent in the following sense.Suppose that C is a class of separated solid pairs, with the property that if (E,T) c C and (E,) satisfies the fiter condition, then the completion (E,)of (E,)also is an element of C. (By Robertson's Theorem ([I]), (E,) will be a sequence space; it is straightforward to see that (,) will be a solid pair).If the answer to the question in 4.1 or 4.2 is yes, whenever (E,T) is confined to C, then the answer to the question in the other is yes, whenever (E,) is in C (with A fixed, or allowed to range over some selection of matrices).Thus, for instance, the result of Haydon and Levy mentioned above implies that when (E,) Is locally convex, metrlzable, and satlfles the filter condition, and A satisfies the hypothesis of Problem 4.1 with respect to(E,), then (nor-A (E), nor-A ()) sat[sfles the filter condition.
For another instance, the answer to 4.2 is yes whenever the rows of A are finite sequences.
We omit the proof of the equivalence.
In one direction the proof uses the theorem of Robertson mentioned above, and the fact that the completion of a solid pair s a solid pair, when it is a sequence space.In the other direction the proof resembles that of Theorem 2.13 of [5].
Finally, we note a problem left unresolved by the results of sections 2 and 3.
PROBLEM 4.3.Does there exist a separable, separated solid pair (E,), preferabl normed, which does not satisfy the filter condition?
The space in Example 2.1 is not separable.
To see this, let Dr; r c c} be a collection of infinite sets of positive integers indexed by the continuum c with the property that r r 2 implies that K 0 K is finite.
(Such a collection may be r r 2 obtained by putting the positive integers in one-to-one correspondence with the rational numbers, and for each real number r, taking K to be the elements of a r x(r) sequence of distinct rationals convergent to r.) For each r c, let e be x(r) The theorem of section 3 bears on Problem 4.3; it suffices to find a separable, separated solld pair (E,z) which is not AK.The proof of that theorem says more; for each x E E, the sequence (Pn(X)) will have to be Cauchy.

ACKNOWLEDGEMENT.
Besides Richard Raydon and Mireille Levy, whose invaluable (to me) contribution is mentioned earlier in this section, I would like to thank R.N. Mohapatra, Stan Rajnak, and M.S. Ramanujan for various helpful and illuminating remarks on the subject of this paper.

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

( a )
The coordinate p.roJectlons are continuous on E. (b) is finer than7.

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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