Characterizations of ultrabarrelledness and barrelledness involving the singularities of

families of convex mappings

By Wolfgang W. Breckner at Cluj-Napoca, Alfred Göpfert at Halle and Tiberiu Trif at Cluj-Napoca

Abstract: The paper reveals that ultrabarrelled spaces (respectively barrelled spaces) can be characterized by means of the density of the so-called weak singularities of families consisting of continuous convex mappings that are defined on an open absolutely convex set and take values in a locally full ordered topological linear space (respectively locally full ordered locally convex space). The idea to establish such characterizations arose from the observation that, in virtue of well-known results, the density of the singularities of families of continuous linear mappings allows to characterize both the ultrabarrelled spaces and the barrelled spaces.

Key Words: Ultrabarrelled spaces, barrelled spaces, ordered topological linear spaces, convex mappings

1991 Mathematics Subject Classification: 46A08, 26B25, 52A41

Running Head: Breckner et al., Characterizations of ultrabarrelledness and barrelledness

1. Introduction

In the present paper we are concerned with the characterization of two important types of topological linear spaces, namely ultrabarrelled spaces and barrelled spaces. For a general information on these spaces the reader is referred to the books [1], [9]-[11], [14] and [16].

The main concept which we use in our investigations is that of a singularity of a family of mappings. In order to recall here the definition of this concept let us suppose that M is a nonempty subset of a topological linear space X and that F is a family of mappings from M into a topological linear space Y. Let o_{Y} denote the zero-element of Y.

If x_{0} is a point in M, then F is said to be:

(i) equicontinuous at x_{0} if for every neighbourhood V of o_{Y} there exists a neighbourhood U of x_{0} such that

{f(x) − f(x_{0}) | f ∈ F} ⊆ V for every x ∈ U ∩ M;

(ii) bounded at x_{0} if the set {f(x_{0}) | f ∈ F} is bounded, i.e. for every neighbourhood V of o_{Y} there exists a positive integer n such that

{f(x_{0}) | f ∈ F} ⊆ nV.

If F is equicontinuous (respectively bounded) at each point of M, then F is called equicontinuous (respectively pointwise bounded) on M.

Any point in M at which F is not bounded is said to be a singularity of F. The set of all singularities of F is denoted by S_{F}. Obviously, F is pointwise bounded on M if and only if S_{F} is empty.

The study of the singularities of families of mappings goes back mainly to S. Banach and H. Steinhaus [2] who stated, for families of continuous linear mappings from a Banach space into another normed linear space, the so-called principle of condensation of the singularities. This principle as well its generalizations given in [3], [4], [6]-[8] and [17] are very useful tools in approximation theory and numerical analysis (see, for instance, [12]).

In the present paper we reveal a new application of the singularities. The roots of this application lie in the well-known fact that both the ultrabarrelled spaces and the barrelled spaces can be characterized by using the equicontinuity of pointwise bounded families consisting either of continuous linear mappings or of continuous convex mappings (see, for instance, [1], [9] and [13]). Our goal is to add to these results characterizations of the ultrabarrelled spaces and the barrelled spaces in terms of the sets of singularities of such families of mappings. First we show that the density of the sets S_{F} associated with families F of continuous linear mappings allows to characterize both types of topological linear spaces under consideration. Next it is noticed that such characterizations do not remain true when the linear mappings are replaced by convex mappings. However, the main results of the paper reveal that ultrabarrelled spaces and barrelled spaces can be characterized by means of the singularities of families F of continuous convex mappings defined on open absolutely convex sets when the density of the set S_{F} ∪ (−S_{F}) is used. The elements belonging to this set S_{F} ∪ (−S_{F}) are called weak singularities of F. They have already been considered in [6] to generalize the principle of the condensation of the singularities of families of continuous linear mappings for certain families of continuous nonlinear mappings.

Throughout the paper all linear spaces and all topological linear spaces that occur are over the field R of real numbers. The set of all positive integers is denoted by N. Given any subset M of a topological space, we denote by cl M and int M the closure and the interior of this set, respectively. The terms topological linear space and locally convex space are abbreviated to TLS and LCS, respectively.

2. Characterizations involving the singularities of families of linear mappings

The ultrabarrelled spaces have been introduced by W. Robertson [15]. In our exposition we shall apply the definition given in [1] which is based on the concept of a string.

Let X be a TLS, and let o_{X} be its zero-element. A sequence (U_{n}) of subsets of X is said to be a:

(i) string in X if all its terms U_{n} (n ∈ N) are absorbing and balanced, on the one hand, and satisfy

U_{n+1} + U_{n+1} ⊆ U_{n} for all n ∈ N,

on the other hand; (ii) closed string in X if it is a string in X whose terms U_{n} (n ∈ N) are closed; (iii) topological string in X if it is a string in X whose terms U_{n} (n ∈ N) are neigh-

bourhoods of o_{X}.

A TLS is called an ultrabarrelled space if all closed strings in that space are topological strings.

Results given in [7] and [16] furnish easily the following theorem collecting some characterizations of the ultrabarrelled spaces by means of families of continuous linear mappings.

Theorem 2.1. Let X be a TLS. Then the following assertions are equivalent:

(A) X is an ultrabarrelled space.

(AL_{1}) Given any TLS Y and any family F of continuous linear mappings from X into Y which is not equicontinuous at o_{X}, the set S_{F} contains a G_{δ}-set which is dense in X.

(AL_{2}) Given any TLS Y and any family F of continuous linear mappings from X into Y which is not equicontinuous at o_{X}, the set S_{F} is dense in X.

(AL_{3}) Given any TLS Y and any family F of continuous linear mappings from X into Y which is pointwise bounded on X, F is equicontinuous at o_{X}.

(AL_{4}) Given any TLS Y and any family F of continuous linear mappings from X into Y which is pointwise bounded on X, F is equicontinuous on X.

Proof. (A) ⇒ (AL_{1}) It is proved in [7], Theorem 3.1 (ii).

(AL_{1}) ⇒ (AL_{2}) Obvious.

(AL_{2}) ⇒ (AL_{3}) Suppose that F is not equicontinuous at o_{X}. Then, in virtue of assertion (AL_{2}), the set S_{F} must be dense in X, and hence it cannot be empty. Consequently, F is not pointwise bounded on X, which is impossible.

(AL_{3}) ⇒ (AL_{4}) Let x_{0} be any point in X. Given any neighbourhood V of o_{Y}, the equicontinuity of F at o_{X} implies the existence of a neighbourhood U_{0} of o_{X} such that

(2.1) {f(x) | f ∈ F} ⊆ V for every x ∈ U_{0}.

Put U := x_{0} + U_{0}. Then U is a neighbourhood of x_{0}. On the other hand, (2.1) implies that

{f(x) − f(x_{0}) | f ∈ F} = {f(x − x_{0}) | f ∈ F} ⊆ V

for every x ∈ U. Consequently, F is equicontinuous at x_{0}.

(AL_{4}) ⇒ (A) It is proved in [16], pp. 10-12.

The barrelled spaces have been introduced by N. Bourbaki. Usually they are defined as follows. An LCS X is called a barrelled space if each absorbing, absolutely convex and closed subset of X is a neighbourhood of o_{X}. It can be easily seen that each LCS which is an ultrabarrelled space is also a barrelled space. But there exist barrelled spaces that are not ultrabarrelled spaces (see, for instance, the example given on p. 256 in [15]).

The following theorem is quite similar to Theorem 2.1. Based on results given in [7] and [9], it collects some characterizations of the barrelled spaces by means of families of continuous linear mappings.

Theorem 2.2. Let X be an LCS. Then the following assertions are equivalent:

(B) X is a barrelled space.

(BL_{1}) Given any LCS Y and any family F of continuous linear mappings from X into Y which is not equicontinuous at o_{X}, the set S_{F} contains a G_{δ}-set which is dense in X.

(BL_{2}) Given any LCS Y and any family F of continuous linear mappings from X into Y which is not equicontinuous at o_{X}, the set S_{F} is dense in X.

(BL_{3}) Given any LCS Y and any family F of continuous linear mappings from X into Y which is pointwise bounded on X, F is equicontinuous at o_{X}.

(BL_{4}) Given any LCS Y and any family F of continuous linear mappings from X into Y which is pointwise bounded on X, F is equicontinuous on X.

Proof. (B) ⇒ (BL_{1}) It is proved in [7], Theorem 3.1 (ii).

(BL_{1}) ⇒ (BL_{2}), (BL_{2}) ⇒ (BL_{3}), (BL_{3}) ⇒ (BL_{4}) These implications are obtained exactly like the corresponding implications in Theorem 2.1.

(BL_{4}) ⇒ (B) It follows from [9], Theorem 7.1.1 (1).

It seems that the equivalences (A) ⇔ (AL_{1}) ⇔ (AL_{2}) and (B) ⇔ (BL_{1}) ⇔ (BL_{2})

enclosed in Theorem 2.1 and Theorem 2.2, respectively, have not been explicitly mentioned up to now, though they are simple consequences of well-known results as the proofs of the theorems 2.1 and 2.2 reveal. We take these new equivalences as a starting-point for the investigations in this paper. More precisely, having them in mind, we ask the following question: Do the above-mentioned equivalences remain true when in (AL_{1}) and (AL_{2}) (respectively in (BL_{1}) and (BL_{2})) the family F of continuous linear mappings from X into Y is replaced by a family of continuous convex mappings from X into an ordered TLS (respectively ordered LCS) Y ? The answer to this question will be given in the next section and will start our investigations.

3. Characterizations involving the singularities of families of convex mappings

Before we deal with the question asked in the preceding section, we recall some fundamental definitions from the theory of ordered topological linear spaces.

By an ordered linear space we mean a linear space Y on which there is defined a binary relation ≤ such that for all x, y, z ∈ Y and all numbers a > 0 the following conditions are satisfied:

(i) x ≤ x; (ii) x ≤ y and y ≤ z imply x ≤ z;

(iii) x ≤ y implies x + z ≤ y + z and ax ≤ ay.

When Y is an ordered linear space, then the set

Y_{+} := {x ∈ Y | o_{Y} ≤ x}

is called the positive wedge of Y.

Given an ordered linear space Y, a subset M of Y is said to be full if

(x_{1} + Y_{+}) ∩ (x_{2} − Y_{+}) ⊆ M for all x_{1}, x_{2} ∈ M. (3.1)

The inclusion (3.1) expresses that each x ∈ Y, for which there exist elements x_{1}, x_{2} ∈ M satisfying x_{1} ≤ x and x ≤ x_{2}, must belong to M.

An ordered TLS is defined to be an ordered linear space endowed with a linear topology. An ordered TLS Y is called locally full if it admits a neighbourhood-base at o_{Y} consisting of full sets.

Given a nonempty convex subset M of a linear space and an ordered linear space Y, any mapping f : M → Y satisfying

f((1 − a)x_{1} + ax_{2}) ≤ (1 − a)f(x_{1}) + af(x_{2})

for all x_{1}, x_{2} ∈ M and all a ∈ [0, 1] is said to be convex.

After these preliminaries we return to the question of characterizing ultrabarrelled spaces and barrelled spaces by using the singularities of families of continuous convex mappings. Unexpectedly, the answer to this question is negative. This is shown by the following simple example taken from [6]. Set X := R and Y := R. Next define for each n ∈ N the mapping f_{n} : X → Y by f_{n}(x) := exp(nx). Obviously, all the mappings f_{n} (n ∈ N) are continuous and convex, while the family F := {f_{n} | n ∈ N} is not equicontinuous at o_{X}. Nevertheless the set S_{F} is not dense in X, because S_{F} = ]0, ∞[. Consequently, none of the equivalences

(A) ⇔ (AL_{1}), (A) ⇔ (AL_{2}), (B) ⇔ (BL_{1}) and (B) ⇔ (BL_{2})

remains valid when families of continuous convex mappings are used.

However, in the sequel we show that ultrabarrelled spaces and barrelled spaces can be characterized by means of the singularities of families F of continuous convex mappings when the density of the set S_{F} is replaced by the density of the larger set S_{F} ∪(−S_{F}). The theory can be developed assuming either that all convex mappings involved are defined on the whole TLS X or that they are defined only on an open absolutely convex subset M of X. In what follows we merely deal with the second of these cases.

Given a TLS X, we denote by O_{ac}(X) (respectively O_{c}(X)) the class consisting of all nonempty, open and absolutely convex (respectively convex) subsets of X. Further, we assign to any family F of mappings from a set M ∈ O_{ac}(X) into a linear space Y the mapping F : M → 2^{Y} defined by

F (x) := {f(x) | f ∈ F} ∪ {f(−x) | f ∈ F}. (3.2)

By means of this set-valued mapping F we associate with any subset V of Y the following sets:

D(V) := {x ∈ M | ∀n ∈ N : F (x) ∩ (Y \ nV) ≠ ∅},
E(V) := {x ∈ M | ∀f ∈ F : f(x) − f(o_{X}) ∈ V and f(−x) − f(o_{X}) ∈ V }.

By using these notations we now derive the main results of our paper. These results are based on the following two lemmas.

Lemma 3.1. Let X be a TLS, and let F be a nonempty family of convex mappings from a set M ∈ O_{ac}(X) into an ordered TLS Y. Then the following assertions are true:

1^{∘} If V is a full balanced subset of Y, then E(V) is balanced.

2^{∘} If V is a full absolutely convex subset of Y, then E(V) is absolutely convex.

3^{∘} If F is bounded at o_{X} and if V is a full balanced subset of Y for which there exists a balanced neighbourhood W of o_{Y} such that W + W + W ⊆ V, on the one hand, and such that D(W) is not dense in M, on the other hand, then E(V) is absorbing.

Proof. 1^{∘} Let f be any mapping in F. Then the set

E_{f}(V) := {x ∈ M | f(x) − f(o_{X}) ∈ V and f(−x) − f(o_{X}) ∈ V } (3.3)

is balanced. To see this it suffices to show that

(3.4) aE_{f}(V) ⊆ E_{f}(V) for every a ∈ [0, 1],

because of the equality −E_{f}(V) = E_{f}(V). Fix any a ∈ [0, 1] and any x ∈ E_{f}(V). Then the elements

u := f(x) − f(o_{X}) and v := f(−x) − f(o_{X})

lie in V. On the other hand, the convexity of f yields the following inequalities:

−av ≤ f(ax) − f(o_{X}) ≤ au, −au ≤ f(−ax) − f(o_{X}) ≤ av. (3.5)

Since the set V is balanced and full and contains both the elements u and v, from (3.5) it results that

f(ax) − f(o_{X}) ∈ V and f(−ax) − f(o_{X}) ∈ V.

Consequently ax lies in E_{f}(V). Thus (3.4) holds, and hence E_{f}(V) is balanced as claimed.

Since f was arbitrarily chosen in F and E(V) is the intersection of all the sets E_{f}(V) (f ∈ F), it follows that E(V) is balanced.

2^{∘} Again let f be any mapping in F. We claim that the set E_{f}(V), defined by (3.3), is convex. To prove this let x_{1} and x_{2} be in E_{f}(V), and let a be taken from [0, 1]. Next define

u_{i} := f(x_{i}) − f(o_{X}) and v_{i} := f(−x_{i}) − f(o_{X}) for i = 1, 2.

Taking into account that V is convex, it follows that

u := (1 − a)u_{1} + au_{2} and v := (1 − a)v_{1} + av_{2}

lie in V. On the other hand, by the convexity of f, the element x := (1 − a)x_{1} + ax_{2} satisfies the following inequalities:

−v ≤ f(x) − f(o_{X}) ≤ u, −u ≤ f(−x) − f(o_{X}) ≤ v. (3.6)

Since the set V is balanced and full and contains both the elements u and v, from (3.6) it results that

f(x) − f(o_{X}) ∈ V and f(−x) − f(o_{X}) ∈ V.

Consequently, x lies in E_{f}(V), and hence E_{f}(V) is convex.

Since f was arbitrarily chosen in F and E(V) is the intersection of all the sets E_{f}(V) (f ∈ F), it follows that E(V) is convex. But, according to assertion 1^{∘}, E(V) is also balanced. Thus E(V) must be absolutely convex.

3^{∘} Since D(W) is not dense in M, we can find an x_{0} ∈ M and a neighbourhood U of x_{0} such that U ∩ D(W) = ∅. (3.7)

Let x be any element of X. Since U ∩ M is a neighbourhood of x_{0}, there exists a number r > 0 such that (3.8) x_{0} + ax ∈ U ∩ M for every a ∈ [−r, r].

From (3.8) it follows that x_{0} + rx and x_{0} − rx lie in the absolutely convex set M, and hence rx also lies in M. On the other hand, we conclude from (3.8) and (3.7) that neither x_{0} + rx nor x_{0} − rx lies in D(W). Hence there must exist positive integers n_{1} and n_{2} such that

F (x_{0} + rx) ⊆ n_{1}W and F (x_{0} − rx) ⊆ n_{2}W,

respectively. Put n := max {n_{1}, n_{2}}. Since W is balanced, we then have

F (x_{0} + rx) ∪ F (x_{0} − rx) ⊆ nW. (3.9)

Next we take into consideration that the set {f(o_{X}) | f ∈ F} is bounded and choose a positive integer m such that

{f(o_{X}) | f ∈ F} ⊆ mW. (3.10)

Now we associate with the numbers n and m a positive integer p such that n/2p ≤ 1 and m/p ≤ 1. After that we fix any f ∈ F and set

u := \frac{1}{2p}[f(x_{0} + rx) + f(−x_{0} + rx)] − \frac{1}{p}f(o_{X}),

v := \frac{1}{2p}[f(x_{0} − rx) + f(−x_{0} − rx)] − \frac{1}{p}f(o_{X}).

According to (3.9) and (3.10) as well as to the choice of p and W, we then have

u ∈ \frac{1}{2p}(nW + nW) − \frac{m}{p}W ⊆ W + W + W ⊆ V,

v ∈ \frac{1}{2p}(nW + nW) − \frac{m}{p}W ⊆ W + W + W ⊆ V.

On the other hand, in virtue of Proposition 2.2 stated in [6], the convexity of f yields the following inequalities for α := r/p:

−v ≤ f(αx) − f(o_{X}) ≤ u, −u ≤ f(−αx) − f(o_{X}) ≤ v. (3.11)

Since the set V is balanced and full and contains both the elements u and v, from (3.11) it results that

f(αx) − f(o_{X}) ∈ V and f(−αx) − f(o_{X}) ∈ V.

Since f was arbitrarily chosen in F, this result shows that αx ∈ E(V). Consequently E(V) is absorbing.

Lemma 3.2. Let X be a TLS, let F be a nonempty family of continuous mappings from a set M ∈ O_{ac}(X) into a TLS Y, and let V be a closed subset of Y. Then D(V) is a G_{δ}-set in X.

Proof. The set-valued function F : M → 2^{Y}, assigned to F by (3.2), is lower semicontinuous on M, i.e. for every x_{0} ∈ M and every open subset T of Y with F (x_{0}) ∩ T ≠ ∅, there exists a neighbourhood U of x_{0} such that

F (x) ∩ T ≠ ∅ for all x ∈ U ∩ M. (3.12)

Indeed, let x_{0} be any point in M, and let T be any open subset of Y with F (x_{0}) ∩ T ≠ ∅. Then we can select a function f_{0} ∈ F such that at least one of the points f_{0}(x_{0}) or f_{0}(−x_{0}) lies in T. If f_{0}(x_{0}) ∈ T, then the continuity of f_{0} at x_{0} implies the existence of a neighbourhood U of x_{0} so that f_{0}(x) ∈ T for all x ∈ U ∩ M. If f_{0}(−x_{0}) ∈ T, then the continuity of f_{0} at −x_{0} implies the existence of a neighbourhood U of x_{0} so that f_{0}(−x) ∈ T for all x ∈ U ∩ M. Consequently in both cases (3.12) is valid.

Next we put

D_{n}(V) := {x ∈ M | F (x) ∩ (Y \ nV) ≠ ∅} for all n ∈ N.

We claim that all the sets D_{n}(V) (n ∈ N) are open. To see this, we fix any positive integer n. Let x_{0} be any point in D_{n}(V). Then we have F (x_{0}) ∩ (Y \ nV) ≠ ∅. Now, taking into consideration that F is lower semicontinuous on M and that Y \ nV is an open set, it follows that there is a neighbourhood U of x_{0} such that

F (x) ∩ (Y \ nV) ≠ ∅ for all x ∈ U ∩ M.

Hence we have U ∩ M ⊆ D_{n}(V). But U ∩ M is a neighbourhood of x_{0}. Consequently, x_{0} is an interior point of D_{n}(V). Since x_{0} was arbitrarily chosen in D_{n}(V), the set D_{n}(V) must be open.

Finally, since D(V) is the intersection of all the sets D_{n}(V) (n ∈ N), it results that D(V) is a G_{δ}-set in X.

The next theorem is our first main result. Just like Theorem 2.1 it refers to the characterization of the ultrabarrelled spaces. But, unlike Theorem 2.1, now families of continuous convex mappings and the weak singularities of these families are used.

Theorem 3.3. Let X be a TLS. Then the following assertions are equivalent:

(A) X is an ultrabarrelled space.

(AC_{1}) Given any M ∈ O_{ac}(X), any locally full ordered TLS Y and any family F of continuous convex mappings from M into Y which is bounded at o_{X}, but which is not equicontinuous at o_{X}, the set S_{F} ∪ (−S_{F}) contains a G_{δ}-set which is dense in M.

(AC_{2}) Given any M ∈ O_{ac}(X), any locally full ordered TLS Y and any family F of continuous convex mappings from M into Y which is bounded at o_{X}, but which is not equicontinuous at o_{X}, the set S_{F} ∪ (−S_{F}) is dense in M.

(AC_{3}) Given any M ∈ O_{ac}(X), any locally full ordered TLS Y and any family F of continuous convex mappings from M into Y which is pointwise bounded on M, F is equicontinuous at o_{X}.

(AC_{4}) Given any M ∈ O_{c}(X), any locally full ordered TLS Y and any family F of continuous convex mappings from M into Y which is pointwise bounded on M, F is equicontinuous on M.

Proof. (A) ⇒ (AC_{1}) Let V be a neighbourhood-base at o_{Y} consisting of full balanced sets. Since Y is locally full, such a neighbourhood-base exists (see, for instance, Theorem 2.4.1 in [5]). We claim that there is a V ∈ V such that D(V) is dense in M.

To prove this we assume that none of the sets D(V) (V ∈ V) is dense in M. This assumption yields a contradiction as the following two steps reveal.

Step 1. We prove that each of the sets cl E(V) (V ∈ V) is a neighbourhood of o_{X}. To this end we consider any V ∈ V. After that we construct a sequence (V_{n}) in V such that V_{1} = V and V_{n+1} + V_{n+1} + V_{n+1} ⊆ V_{n} for all n ∈ N. Then, according to the assertions 1^{∘} and 3^{∘} of Lemma 3.1, all the sets E(V_{n}) (n ∈ N) are balanced and absorbing. Consequently, the sets U_{n} (n ∈ N) defined by U_{n} := 2^{1−n}E(V_{n}) are also balanced and absorbing. But, in addition, these sets satisfy

U_{n+1} + U_{n+1} ⊆ U_{n} for all n ∈ N. (3.13)

To prove this assertion we consider any elements x_{1} and x_{2} in U_{n+1}. Then we have 2^{n}x_{i} ∈ E(V_{n+1}) for i = 1, 2. Now let f be any mapping in F. Then the elements u_{i} (i = 1, 2) and v_{i} (i = 1, 2) defined by

u_{i} := f(2^{n}x_{i}) − f(o_{X}) and v_{i} := f(−2^{n}x_{i}) − f(o_{X}),

respectively, belong to V_{n+1}. This fact implies that

u := \frac{1}{2}u_{1} + \frac{1}{2}u_{2} ∈ V_{n+1} + V_{n+1} ⊆ V_{n}, v := \frac{1}{2}v_{1} + \frac{1}{2}v_{2} ∈ V_{n+1} + V_{n+1} ⊆ V_{n}.

But, on the other hand, the convexity of f yields

−v ≤ f(2^{n−1}(x_{1} + x_{2})) − f(o_{X}) ≤ u, −u ≤ f(−2^{n−1}(x_{1} + x_{2})) − f(o_{X}) ≤ v.

Since the set V_{n} is balanced and full and contains both the elements u and v, it results that

f(2^{n−1}(x_{1} + x_{2})) − f(o_{X}) ∈ V_{n} and f(−2^{n−1}(x_{1} + x_{2})) − f(o_{X}) ∈ V_{n}.

This being true for any f ∈ F, we must have 2^{n−1}(x_{1} + x_{2}) ∈ E(V_{n}), and hence x_{1} + x_{2} ∈ U_{n}. This completes the proof of (3.13).

Summing up, all the sets U_{n} (n ∈ N) are balanced, absorbing and satisfy (3.13). Consequently, the sequence (cl U_{n}) is a closed string in X. Taking into account that X is ultrabarrelled, we conclude that cl U_{1} is a neighbourhood of o_{X}. In other words, cl E(V) is a neighbourhood of o_{X}.

Step 2. Let V_{0} be any neighbourhood of o_{Y}. Then we can select a V ∈ V such that cl V ⊆ V_{0}. Put U := cl E(V). Since all the mappings in F are continuous, we have

{f(x) − f(o_{X}) | f ∈ F} ⊆ cl V for all x ∈ U ∩ M,

and hence

{f(x) − f(o_{X}) | f ∈ F} ⊆ V_{0} for all x ∈ U ∩ M.

But, according to Step 1, the set U is a neighbourhood of o_{X}. Consequently, F must be equicontinuous at o_{X}, which contradicts the hypothesis of assertion (AC_{1}).

We have arrived at this contradiction because of our assumption that none of the sets D(V) (V ∈ V) is dense in M. Consequently, there exists a V ∈ V for which D(V) is dense in M. Next let W be any closed neighbourhood of o_{Y} such that W ⊆ V. Then we have D(V) ⊆ D(W), and hence D(W) is dense in M. Furthermore, D(W) satisfies D(W) ⊆ S_{F} ∪ (−S_{F}) and, according to Lemma 3.2, it is a G_{δ}-set. This completes the proof of the implication (A) ⇒ (AC_{1}).

(AC_{1}) ⇒ (AC_{2}) Obvious.

(AC_{2}) ⇒ (AC_{3}) Assume that F is not equicontinuous at o_{X}. Then, in virtue of assertion (AC_{2}), the set S_{F} ∪ (−S_{F}) must be dense in M. Consequently, S_{F} has to be nonempty, which contradicts the supposed pointwise boundedness of F on M.

(AC_{3}) ⇒ (AC_{4}) Let x_{0} be any point in M. We choose a balanced neighbourhood U of o_{X} such that x_{0} + U ⊆ M. Let conv U be the convex hull of U. Then T := int (conv U) belongs to O_{ac}(X) and satisfies x_{0} + T ⊆ M. Next we assign to each f ∈ F the mapping g_{f} : T → Y defined by g_{f}(x) := f(x_{0} + x). Obviously, all the mappings g_{f} (f ∈ F) are continuous and convex. Moreover, the family {g_{f} | f ∈ F} is pointwise bounded on T. Therefore, in view of (AC_{3}), this family must be equicontinuous at o_{X}. In other words, F must be equicontinuous at x_{0}.

(AC_{4}) ⇒ (A) This implication follows from the implication (AL_{4}) ⇒ (A) stated in Theorem 2.1, because (AC_{4}) implies (AL_{4}).

Corollary 3.4. Let X be an ultrabarrelled space, let Y be a locally full ordered TLS, and let F be a family of continuous convex mappings from a set M ∈ O_{ac}(X) into Y which is bounded at o_{X}, but which is not equicontinuous at o_{X}. Then S_{F} ∪ (−S_{F}) is residual in M.

Proof. Apply the implication (A) ⇒ (AC_{1}) of Theorem 3.3 and a well-known characterization of the residual sets (see, for instance, Theorem 2.1 in [3]).

Corollary 3.5. Let X be a TLS of the second category, let Y be a locally full ordered TLS, and let F be a family of continuous convex mappings from a set M ∈ O_{ac}(X) into Y which is bounded at o_{X}, but which is not equicontinuous at o_{X}. Then S_{F} ∪ (−S_{F}) is residual in M.

Proof. According to Theorem 6.2.2 in [9], X is an ultrabarrelled space. By applying Corollary 3.4 we conclude that S_{F}∪(−S_{F}) is residual in M.

Next we are going to prove our second main result which relates to the characterization of barrelled spaces. Unlike Theorem 2.2, this time families of continuous convex mappings and the weak singularities of these families are used

Theorem 3.6. Let X be an LCS. Then the following assertions are equivalent:

(B) X is a barrelled space.

(BC_{1}) Given any M ∈ O_{ac}(X), any locally full ordered LCS Y and any family F of continuous convex mappings from M into Y which is bounded at o_{X}, but which is not equicontinuous at o_{X}, the set S_{F} ∪ (−S_{F}) contains a G_{δ}-set which is dense in M.

(BC_{2}) Given any M ∈ O_{ac}(X), any locally full ordered LCS Y and any family F of continuous convex mappings from M into Y which is bounded at o_{X}, but which is not equicontinuous at o_{X}, the set S_{F} ∪ (−S_{F}) is dense in M.

(BC_{3}) Given any M ∈ O_{ac}(X), any locally full ordered LCS Y and any family F of continuous convex mappings from M into Y which is pointwise bounded on M, F is equicontinuous at o_{X}.

(BC_{4}) Given any M ∈ O_{c}(X), any locally full ordered LCS Y and any family F of continuous convex mappings from M into Y which is pointwise bounded on M, F is equicontinuous on M.

Proof. (B) ⇒ (BC_{1}) We argue similarly with the proof of the implication (A) ⇒ (AC_{1}) in Theorem 3.3. Let V be a neighbourhood-base at o_{Y} consisting of full and absolutely convex sets. Since Y is a locally full ordered LCS, such a neighbourhood-base exists (see, for instance, Theorem 2.5.1 in [5]). We claim that there is a V ∈ V for which D(V) is dense in M.

To prove this we assume that none of the sets D(V) (V ∈ V) is dense in M. Then each of the sets cl E(V) (V ∈ V) is a neighbourhood of o_{X}. Indeed, if V ∈ V, then according to the assertions 2^{∘} and 3^{∘} of Lemma 3.1, the set E(V) is absolutely convex and absorbing. Consequently, cl E(V) is absolutely convex, absorbing and closed. Thus, by the barrelledness of X, cl E(V) must be a neighbourhood of o_{X}.

Taking into consideration that all the sets cl E(V) (V ∈ V) are neighbourhoods of o_{X}, it results, like in Step 2 of the proof of the implication (A) ⇒ (AC_{1}) in Theorem 3.3, that F is equicontinuous at o_{X}, which contradicts the hypothesis of assertion (BC_{1}). Therefore there must exist a V ∈ V for which D(V) is dense in M. Selecting any closed neighbourhood W of o_{Y} that satisfies W ⊆ V, we then conclude, like in the final part of the proof of the implication (A) ⇒ (AC_{1}) in Theorem 3.3, that D(W) is a G_{δ}-set enclosed in S_{F} ∪ (−S_{F}) and dense in M.

(BC_{1}) ⇒ (BC_{2}) Obvious.

(BC_{2}) ⇒ (BC_{3}), (BC_{3}) ⇒ (BC_{4}) These implications are obtained exactly like the corresponding implications in Theorem 3.3.

(BC_{4}) ⇒ (B) This implication follows from the implication (BL_{4}) ⇒ (B) stated in Theorem 2.2, because (BL_{4}) is a consequence of (BC_{4}).

Corollary 3.7. Let X be a barrelled space, let Y be a locally full ordered LCS, and let F be a family of continuous convex mappings from a set M ∈ O_{ac}(X) into Y which is bounded at o_{X}, but which is not equicontinuous at o_{X}. Then S_{F} ∪ (−S_{F}) is residual in M.

Proof. Apply the implication (B) ⇒ (BC_{1}) of Theorem 3.6 and the characterization of residual sets used in the proof of Corollary 3.4.

Corollary 3.8. Let at least one of the following conditions be satisfied:

(i) X is an ultrabarrelled space and Y is a locally full ordered TLS which is locally bounded;

(ii) X is a barrelled space and Y is a locally full ordered LCS which is locally bounded. Further, let F be a family of continuous convex mappings from a set M ∈ O_{ac}(X) into Y which is bounded at o_{X}, but which is not equicontinuous at o_{X}. Then S_{F} ∪ (−S_{F}) is a G_{δ}-set and dense in M.

Proof. Since Y is locally bounded, we can choose a closed bounded neighbourhood V of o_{Y}. Then we have D(V) = S_{F} ∪ (−S_{F}). Indeed, the inclusion D(V) ⊆ S_{F} ∪ (−S_{F}) is obvious, while the inclusion S_{F} ∪ (−S_{F}) ⊆ D(V) is proved as follows. Let x be any element in S_{F} ∪ (−S_{F}). Then there is a neighbourhood W of o_{Y} such that

F (x) ∩ (Y \ nW) ≠ ∅ for all n ∈ N. (3.14)

Further, let m be a positive integer such that V ⊆ mW. Then we have Y \nmW ⊆ Y \nV for all n ∈ N. In virtue of (3.14) it follows that

F (x) ∩ (Y \ nV) ≠ ∅ for all n ∈ N.

Consequently, x lies in D(V), and thus S_{F} ∪ (−S_{F}) ⊆ D(V) holds.

According to Lemma 3.2 the equality D(V) = S_{F} ∪(−S_{F}) implies that S_{F} ∪(−S_{F}) is a G_{δ}-set. In addition, in virtue of the implications (A) ⇒ (AC_{2}) in Theorem 3.3 and (B) ⇒ (BC_{2}) in Theorem 3.6, the set S_{F} ∪ (−S_{F}) is dense in M.

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Universitatea Babeş-Bolyai, Fac. de Matematică şi Informatică,

Str. Kogălniceanu Nr. 1, RO-3400 Cluj-Napoca, Romania e-mail: breckner@math.ubbcluj.ro Martin-Luther-Universität, FB Mathematik und Informatik, Institut für

Optimierung und Stochastik, D-06099 Halle, Germany e-mail: goepfert@mathematik.uni-halle.d400.de

Universitatea Babeş-Bolyai, Fac. de Matematică şi Informatică, Str. Kogălniceanu Nr. 1, RO-3400 Cluj-Napoca, Romania

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