# A study of the strong topologies on finite dimensional probabilistic normed spaces.

1. Introduction

The notion of the probabilistic normed space (briefly, the PN space) was first introduced by Sertnev (1962)  in order to study the best approximation. Alsina et al.,1993  generalized this definition. The generalized notion includes the former one as a special case. There is a natural topology, called the strong topology, on each PN space. In studying the strong topologies on the generalized PN spaces, Alsina et al., 1997  investigated the continuity of the probabilistic norm; they pointed out that each PN space is a topological group but may not be a topological vector space. Saadati and Amini (2005)  researched a special class of finite dimensional topological vector PN spaces with Archimedean triangle function [[tau].sup.*]; they showed that each PN space in this class is strongly complete. The normability of the PN spaces was studied by Zhang (2008)  and Lafuerza-Guillen et al. (2009) . However, the strong topological structure on a non-topological vector PN space has been studied scarcely in previous literature. In this paper, the detailed strong topological structure on the finite dimensional PN space is presented.

Preliminaries of this work are given in Section 2. In Section 3, we introduce the notion of the absorbed exponent of a set. The strong topological characteristics can be reflected by the absorbed exponents of the base vectors. Another notion called the absorbed function is introduced to express the absorbed exponent of a singleton. Some properties which will be used in Section 4 are obtained. In Section 4, we obtain all strong topological structures on finite dimensional PN spaces and show that each finite dimensional PN space is strongly complete.

2. Preliminaries

We adopt the usual notations and terminologies which were used in previous literature [1-11, 13, 15, 16].

[[DELTA].sup.+] is the set of probability distribution functions. [d.sub.L] is the modified Levy metric (or called Sibley metric) . ([[DELTA].sup.+], [d.sub.L]) is a compact metric space. The convergence in ([[DELTA].sup.+], [d.sub.L]) is equivalent to the weak convergence of distribution functions.

A triangle function [tau] is a binary operation on [[DELTA].sup.+], namely a function [tau] : [[DELTA].sup.+] x [[DELTA].sup.+] [right arrow] [[DELTA].sup.+] that is associative, commutative, nondecreasing and has e0 as an identity. The continuity of [tau] means uniform continuity with respect to the product topology on [[DELTA].sup.+] x [[DELTA].sup.+].

Definition 2.1.  A PN space is a quadruple (V, v, [tau], [[tau].sup.*]), where V is a real vector space, [tau] and [[tau].sup.*] are continuous triangle functions with [tau] [less than or equal to] [[tau].sup.*] and v is a mapping from V into [[DELTA].sup.+] such that for all p, q in V, the following conditions hold:

(PN1) [v.sub.p] = [[epsilon].sub.0] iff (if and only if) p = [theta], here [theta] is the null vector of V;

(PN2) [v.sub.-p] = [v.sub.p] ;

(PN3) [v.sub.p+q] [greater than or equal to] [tau]([v.sub.p], [v.sub.q]);

(PN4) [v.sub.p] [less than or equal to] [[tau].sup.*]([v.sub.ap], [v.sub.(1- a)p]) for all a in [0,1].

The Menger PN space under T, denoted by (V, v, T), is a PN space (V, v, [tau], [[tau].sup.*]) in which [tau] = [[tau].sub.t] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some continuous t-norm T and it's T-conorm [T.sup.*], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A space (V, v, [tau]) is called the Serstnev space if it satisfies (PN1), (PN3) and the following condition:

For each p [member of] V and 0 [not equal to] [lambda] [member of] R, the identity [v.sub.[lambda]p](x) = [v.sub.p](x/[absolute value of [lambda]]) holds.

The strong topology on a PN space (V, v, [tau], [[tau].sup.*]) is determined by the system of neighborhoods

[N.sub.p]([lambda]) = {q : [d.sub.L]([v.sub.p-q], [[epsilon].sub.0]) < [lambda]} = {q : [v.sub.p-q]([lambda]) > 1 - [lambda]},

for each vector p [member of] V. The PN space under the strong topology is a Hausdorff space and satisfies the first countability axiom. Briefly, we call it a [T.sub.2] and [A.sub.1] space. {[N.sub.q] (1/n): n [member of] [Z.sub.+]} is a countable neighborhood base for each q [member of] V, where [Z.sub.+] is the set of all positive integers.

Definition 2.2. [4,10] Let (V, v, [tau], [[tau].sup.*]) be a PN space. A sequence {[p.sub.n]} [[subset].bar] V is said to be strongly convergent to p [member of] V, if for each [lambda] > 0, there exists a positive integer N such that [p.sub.n] [member of] [N.sub.p]([lambda]) for n [greater than or equal to] N. Also the sequence {[p.sub.n]} is called a strong Cauchy sequence, if for every [lambda] > 0, there is a positive integer N such that [v.sub.pn-pm]([lambda]) > 1 - [lambda] whenever m, n > N. (V, v, [tau], [[tau].sup.*]) is said to be strongly complete iff every strong Cauchy sequence in V is strongly convergent to a point in V.

Theorem 2.3.  Let (V, v, [tau], [[tau].sup.*])bea PN space. If 0 [less than or equal to] [absolute value of [alpha]] [less than or equal to] [absolute value of [beta]], then [v.sub.[beta]p [less than or equal to] [v.sub.[alpha]p] for an arbitrary vector p [member of] V .

3. Absorbed Exponent and Absorbed Function

Definition 3.1. Let (V, v, [tau], [[tau].sup.*]) be a PN space and A [[subset].bar] V. We say that A can be absorbed by a neighborhood [N.sub.[theta]](t) of the null vector [theta], if there is a [[delta].sub.0] > 0 such that [lambda]A [[subset].bar] [N.sub.[theta]](t) when [absolute value of [lambda]] < [[delta].sub.0]([lambda] [member of] R), where [lambda]A = {[lambda]p : p [member of] A}.

Definition 3.2. Let (V, v, [tau], [[tau].sup.*]) be a PN space and A [[subset].bar] V. The absorbed exponent of A, denoted [[alpha].sub.A], is defined by:

[[alpha].sub.A] = inf {t > 0 : for arbitrary [t.sub.0] > t, A can be absorbed by [N.sub.[theta]]([t.sub.0])}.

In each PN space, we have [N.sub.[theta]](t) = V for arbitrary t > 1. An arbitrary set A can be absorbed by V naturally, so [[alpha].sub.A] exists and 0 [less than or equal to] [[alpha].sub.A] [less than or equal to] 1. This means that the definition of the absorbed exponent is valid. When A is a singleton {p}, we write [[alpha].sub.p] instead of [[alpha].sub.{p}]. The following theorem gives some properties about the absorbed exponent.

Theorem 3.3. Let (V, v, [tau], [[tau].sup.*]) be a PN space.

(1) [[alpha].sub.[theta]] = 0;

(2) If the real number k [not equal to] 0, then [[alpha].sub.kA] = [[alpha].sub.A] for each A [[subset].bar] V;

(3) For a vector p [member of] V, [[alpha].sub.p] = 0 iff [a.sub.n]p [right arrow] [theta] (n [right arrow] [infinity]) for arbitrary sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [a.sub.n] [right arrow] 0(n [right arrow] [infinity]);

(4) A, B [[subset].bar] V .If [[alpha].sub.A] = [[alpha].sub.B] = 0, then [[alpha].sub.A+B] = 0, where A + B = {p + q : p [member of] A, q [member of] B};

(5) If A [[subset].bar] B [[subset].bar] V, then [[alpha].sub.A] [less than or equal to] [[alpha].sub.B];

(6) [alpha].sub.A[union]B] = max([[alpha].sub.A], [[alpha].sub.B]) for A, B [[subset].bar] V.

Proof.

(1) For each t > 0, one has [theta] [member of] [N.sub.[theta]](t), so [[alpha].sub.[theta]] = 0.

(2) For a fixed real number k [not equal to] 0 and each t > [[alpha].sub.A], there is a real number [delta] > 0 such that [lambda]A [[subset].bar] [N.sub.[theta]](t) when [absolue value of [lambda]] < [delta]. Taking [[delta].sub.k] = [delta]/[absolute value of k], then [rho](kA) [[subset].bar] [N.sub.[theta]](t) when [absolute value of [rho]] < [[delta].sub.k]. This implies that [[alpha].sub.kA] [less than or equal to] [[alpha].sub.A].

Conversely, [[alpha].sub.A] = [alpha](1/k)(kA),so [[alpha].sub.A] [less than or equal to] [[alpha].sub.kA]. Hence [[alpha].sub.A] = [[alpha].sub.kA] for each real number k [not equal to] 0.

(3) If [[alpha].sub.p] = 0, then {p} can be absorbed by [N.sub.[theta]](t) for every t > 0. So for each t > 0, there is a real number [delta] > 0 such that [lambda]p [member of] [N.sub.[theta]](t) when [absolute value [lambda]] < [delta]. For a sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [a.sub.n] [right arrow] 0(n [right arrow] [infinity]), there is a positive integer N such that [absolute value of [a.sub.n]] < [delta] when n > N. So [a.sub.n]p [member of] [N.sub.[theta]](t) when n > N. This implies that [a.sub.n]p [right arrow] [theta] (n [right arrow] [infinity]).

Conversely, if [a.sub.n]p [right arrow] [theta](n [right arrow] [infinity]) for each sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [a.sub.n] [right arrow] 0(n [right arrow] [infinity]). Letting [a.sub.n] = 1/n, we get p/n [right arrow] [theta] (n [right arrow] [infinity]). For a fixed t > 0, there is an integer N > 0 such that p/n [member of] [N.sub.[theta]](t) when n > N. Putting [delta] = 1/(N + 1) > 0, we obtain [v.sub.[lambda]p] [greater than or equal to] [v.sub.[delta]p] when [absolute value of [lambda]] < [delta], so [lambda]p [member of] [N.sub.[theta]](t). This implies that {p} can be absorbed by [N.sub.[theta]](t) for each t > 0. Therefore [[alpha].sub.p] = 0.

(4) [tau] is continuous, so for each t > 0, there exists an [epsilon] > 0 such that [d.sub.L]([tau](F, G), [[epsilon].sub.0]) < t, i.e. [tau](F, G)(t) > 1 - t, for any F, G [member of] [[DELTA].sup.+] whenever [d.sub.L](F, [[epsilon].sub.0]) < [epsilon] and [d.sub.L](G, [[epsilon].sub.0]) < [epsilon]. For this [epsilon] > 0, because [[alpha].sub.A] = [[alpha].sub.B] = 0, so there is a [delta] > 0 such that [lambda]A [[subset].bar] [N.sub.[theta]]([lambda]) and [lambda]B [[subset].bar] [N.sub.[theta]([lambda]) when [absolute value of [lambda]] < [delta]. So for each p + q [member of] A + B, where p [member of] A and q [member of] B, we get [d.sub.L]([v.sub.[lambda]p], [[epsilon].sub.0]) < [epsilon] and [d.sub.L]([v.sub.[lambda]q], [[epsilon].sub.0]) < [epsilon]. Therefore [v.sub.[lambda](p+q)](t) [greater than or equal to] [tau]([v.sub.[lambda]p], [v.sub.[lambda]q])(t) > 1 - t, i.e. [lambda](p + q) [member of] [N.sub.[theta]](t). Hence [lambda](A + B) [[subset].bar] [N.sub.[theta]](t) when [absolute value of [lambda]] < [delta]. Therefore [[alpha].sub.A+B] = 0.

(5) For each t > [[alpha].sub.B], there is a [delta] > 0 such that [lambda]B [[subset].bar] [N.sub.[theta]](t) when [absolute value of [lambda]] < [delta]. Because A [[subset].bar] B ,so [lambda]A [[subset].bar] [N.sub.[theta]](t), i.e. A can be absorbed by [N.sub.[theta]](t). Following from Definition 3.2, one has [[alpha].sub.A] [less than or equal to] [[alpha].sub.B].

(6) Because A [[subset].bar] A [union] B and B [[subset].bar] A [union] B, so we get [[alpha].sub.A[union]B] [greater than or equal to] [[alpha].sub.A] and [[alpha].sub.A[union]B] [greater than or equal to] [[alpha].sub.B] (5). Therefore [[alpha].sub.A[union]B] [greater than or equal to] max([[alpha].sub.A], [[alpha].sub.B]).

Conversely, for each t > max([[alpha].sub.A], [[alpha].sub.B]), there exist [[delta].sub.A] > 0 and [[delta].sub.B] > 0 such that [[lambda].sub.1]A [[subset].bar] [N.sub.[theta]](t) when [absolute value of [[lambda].sub.1]] < [[delta.sub.A] and [[lambda].sub.2]B [[subset].bar] [N.sub.[theta]](t) when [absolute value of [[lambda].sub.2]] < [[delta].sub.B]. Taking [delta] = min ([[delta].sub.A], [[delta].sub.B]) > 0, then [lambda](A [union] B) [[subset].bar] [N.sub.[theta]](t) when [absolute value of [lambda]] < [delta]. So following from Definition 3.2, one obtains [[alpha].sub.A[union]B] < max ([[alpha].sub.A], [[alpha].sub.B]).

Combining (1)(2)(4) in Theorem 3.3, the following conclusion is obtained:

(a) : Let (V, v, [tau], [[tau].sup.*]) be a PN space and [p.sub.1], [p.sub.2], *** , [p.sub.n] be vectors such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then for any real numbers [k.sub.1], [k.sub.2], *** , [k.sub.n], the absorbed exponent [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The following consequence is rephrased from  (Alsina et al., 1997),

(b) : Let (V, v, [tau], [[tau].sup.*]) be a PN space. Then it is a topological vector space iff the scalar multiplication f (a) = ap is continuous on R for every fixed p [member of] V.

The Euclidean space R and the PN space (V, v, [tau], [[tau].sup.*]) are two topological groups. For a fixed vector p [member of] V, the scalar multiplication f : R [right arrow] (V, v, [tau], [[tau].sup.*]) that is defined by f(a) = ap is continuous iff f is continuous at 0. In other words, f(a) = ap is continuous iff [a.sub.n]p [right arrow] [theta] for arbitrary sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [a.sub.n] [right arrow] 0 (n [right arrow] [infinity]). Combining (a)(b) with (3) in Theorem 3.3, we get the following two theorems.

Theorem 3.4. Let (V, v, [tau], [[tau].sup.*]) be a PN space. Then it is a topological vector space iff [[alpha].sub.p] = 0 for all p [member of] V.

Theorem 3.5. Let (V, v, [tau], [[tau].sup.*]) be a PN space and dim V = n < [infinity]. Then it is a topological vector space iff there exists a base {[p.sub.1], [p.sub.2], ***, [p.sub.n]} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The following theorem gives the expression of the absorbed exponent of a singleton. Theorem 3.6. Let (V, v, [tau], [[tau].sup.*]) be a PN space. Then for each p [member of] V,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Denote [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Because for an arbitrarily fixed t [member of] R, the sequence of number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [0,1] increases with increasing integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] exists.

For each [t.sub.0] > [[beta].sub.p], there is an [epsilon] > 0 such that [t.sub.0] - [epsilon] > [[beta].sub.p] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1 - ([t.sub.0] - [epsilon]). For this [epsilon] > 0, there exists an integer N > 0 such that

[v.sub.p/n]([t.sub.0] - [epsilon]) > 1 - ([t.sub.0] - [epsilon]) - [epsilon]/2 = 1 - [t.sub.0] + [epsilon]/2,

when n > N. So

[v.sub.p/n]([t.sub.0]) [greater than or equal to] [v.sub.p/n]([t.sub.0] - [epsilon]) > 1 - [t.sub.0] + [epsilon]/2 > 1 - [t.sub.0].

Taking [delta] = 1/(N + 1), then [v.sub.[lambda]p]([t.sub.0]) > [v.sub.p]/(N + 1)([t.sub.0]) > 1 - [t.sub.0] when [absolute value of [lambda]] < [delta]. This implies that {p} can be absorbed by [N.sub.[theta]]([t.sub.0]). So one has [[beta].sub.p] [greater than or equal to] [[alpha].sub.p] by definition 3.2.

Conversely, for any t > [[alpha].sub.p], {p} can be absorbed by [N.sub.[theta]](t). So there exists an [delta] > 0 such that [lambda]p [member of] [N.sub.[theta]](t) when [absolute value of [lambda]] < [delta]. Putting N = [[1/[delta]].sub.IP] + 1, where [[1/[delta]].sub.IP] is the integral part of 1/[delta], then 1/n < 1/N < [delta] when n > N. So p/n [member of] [N.sub.[theta]](t), i.e. [v.sub.p/n](t) > 1 - t. Because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] increases with increasing n, so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] . Hence [[beta].sub.p] [less than or equal to] [[alpha].sub.p].

In a PN space (V, v, [tau], [[tau].sup.*]) and for each p [member of] V, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an increasing sequence in metric space ([[DELTA].sup.+], [d.sub.L]). Because ([[DELTA].sup.+], [d.sub.L]) is compact, so there is a distribution function [F.sub.p] [member of] [[DELTA].sup.+] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is convergent to [F.sub.p] in ([[DELTA].sup.+], [d.sub.L]), i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is weakly convergent to [F.sub.p]. The following theorem will show that this convergence is in fact pointwise.

Theorem 3.7. Let (V, v, [tau], [[tau].sup.*]) be a PN space. For each vector p [member of] V, the function [F.sub.p] : [bar.R] [right arrow] [0,1] is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for each t [member of] [bar.R] = R[union]{[+ or - ][infinity]}, then [F.sub.p] [member of] [[DELTA].sup.+].

Proof. For each t [member of] [bar.R], [F.sub.p](t) exists because the sequence of number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [subset].bar] [0,1] is increasing with n. Now we show that [F.sub.p] [member of] [[DELTA].sup.+].

[F.sub.p] is nondecreasing because for each n [member of] [Z.sub.+], [v.sub.p/n](t) is nondecreasing with respect to t [member of] [bar.R].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For arbitrary [epsilon] > 0 and [t.sub.0] [member of] R, there is an integer N > 0 such that [absolute value of [v.sub.p/n]([t.sub.0]) [F.sub.p]([t.sub.0])] < [epsilon]/2 when n > N. [v.sub.p/(N+1)] is left-continuous on R, so there is a [delta] > 0 such that [absolute value of [v.sub.p/(N+1)](t) - [v.sub.p/(N+1)]([t.sub.0])] < [epsilon]/2 when [t.sub.0] - [delta] < t < [t.sub.0]. we get

[absolute value of [v.sub.p/(N+1)](t)-[F.sub.p]([t.sub.0])] [less than or equal to] [absolute value of [v.sub.p/N+1])(t)-[v.sub.p/(N+1)]([t.sub.0])] + [absolute value of [v.sub.p/(N+1)]([t.sub.0])-[F.sub.p]([t.sub.0])] < [epsilon]/2+[epsilon]/2 = [epsilon].

Because 0 [less than or equal to] [v.sub.p/(N+1)](t) [less than or equal to] [F.sub.p](t) [less than or equal to] [F.sub.p]([t.sub.0]) [less than or equal to] 1, therefore

[absolute value of [F.sub.p](t) - [F.sub.p]([t.sub.0])] [less than or equal to] [absolute value of [v.sub.p/N+1)](t) - [F.sub.p]([t.sub.0])] < [epsilon],

whenever [t.sub.0] - [delta] < t < [t.sub.0]. [F.sub.p](t) is left-continuous on R. Hence [F.sub.p] [member of] [[DELTA].sup.+].

Combining Theorem 3. 6 with Theorem 3. 7, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 3.8. Let (V, v, [tau], [[tau].sup.*]) be a PN space. For each vector p [member of] V, the function [F.sub.p] defined in Theorem 3.7 is called the absorbed f unction of p.

In a PN space (V, v, [tau], [[tau].sup.*]), for arbitrary p, q [member of] V and a continuous triangle function [[tau].sub.1], the sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is convergent to [[tau].sub.1] ([F.sub.p], [F.sub.q]) in ([[DELTA].sup.+], [d.sub.L]).

For each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [0,1] is increasing with respect to n because [[tau].sub.1] is nondecreasing. We can prove that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; this only needs changing [v.sub.p/n] to [[tau].sub.1] ([v.sub.p/n], [v.sub.q/n])(t) and [F.sub.p] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in Theorem 3.7. So [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is convergent to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in ([[DELTA].sup.+], [d.sub.L]). Thus, we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 3.9. Let (V, v, [tau], [[tau].sup.*]) be a PN space. Then for a vector p [member of] V, [[alpha].sub.p] = 0 iff [F.sub.p] = [[epsilon].sub.0].

Proof. If [F.sub.p] = [[epsilon].sub.0], then [[alpha].sub.p] = inf {t : [F.sub.p](t) > 1 - t} = inf {t : 1 > 1 - 1} = 0. Conversely, if [[alpha].sub.p] = 0, one has [F.sub.p](t) > 1 - t for each t> 0. If [F.sub.p] [not equal to] [[epsilon].sub.0], then [F.sub.p] < [[epsilon].sub.0]. There is a [t.sub.0] > 0 such that [F.sub.p]([t.sub.0]) < 1. For any t > 0 such that t < min{[t.sub.0],1 - [F.sub.p]([t.sub.0])}, we get [F.sub.p](t) > 1 - t > 1 - (1 - [F.sub.p]([t.sub.0])) = [F.sub.p]([t.sub.0]); this is a contradiction because [F.sub.p] is increasing.

Theorem 3.10. Let (V, v, [tau], [[tau].sup.*]) be a PN space and p, q [member of] V. Then

(1) [F.sub.kp] = [F.sub.p] for each k [not equal to] 0;

(2) [F.sub.p+q] [greater than or equal to] [tau]([F.sub.p], [F.sub.q]);

(3) [F.sub.p] is an idempotent of [[tau].sup.*]. This implies that the number of the absorbed exponents of vectors in V is no greater than the number of idempotents of [[tau].sup.*].

Proof.

(1) Following from Theorem 2.3, for each real number a such that [absolute value of a] [member of] (0,1] and n [member of] [Z.sub.+], one obtains [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let n [right arrow] [infinity], then [F.sub.p] = [F.sub.ap]. So [F.sub.kp] = [F.sub.p] for any k [not equal to] 0.

(2) For each positive integer n, we get [v.sub.(p+q)/n] [greater than or equal to] [tau]([v.sub.p/n], [v.sub.q/n]). So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(3) For arbitrary a [member of] (0, 1), one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So [[tau].sup.*]([[epsilon].sub.0], [F.sub.p]) = [F.sub.p] [less than or equal to] [[tau].sup.*]([F.sub.p], [F.sub.p]) [less than or equal to] [[tau].sup.*]([[epsilon].sub.0], [F.sub.p]). Hence [F.sub.p] = [[tau].sup.*]([F.sub.p], [F.sub.p]).

Remark 3.11. When [[tau].sup.*] is Archimedean, i.e. [[tau].sup.*] only has [[epsilon].sub.0] and [epsilon.sub.[infinity] as it's idempotents in [[DELTA].sup.+], if [v.sub.p] [not equal to] [epsilon.sub.[infinity] for every p [member of] V, we obtain [F.sub.p] = [[epsilon].sub.0] for arbitrary p [member of] V because [F.sub.p] [greater than or equal to] [v.sub.p]. This implies that [[alpha].sub.p] = 0 for each vector P in V. Therefore this space is a topological vector space. This conclusion is consistent with that given in  and proved by a different method.

Theorem 3.12. Let (V, v, [tau], [[tau].sup.*]) be a PN space. For arbitrary p, q [member of] V, if [[alpha].sub.p] = 0, then [[alpha].sub.p+q] - [[alpha].sub.q].

Proof. Following from Theorem 3.9 and Theorem 3.10,

[F.sub.p+q] [greater than or equal to] [tau]([F.sub.p], [F.sub.q]) = [tau]([[epsilon].sub.0], [F.sub.q]) = [F.sub.q].

Conversely, because q = (p + q) + ( - p) and [[alpha].sub.-p] = [[alpha].sub.p] = 0, so we get [F.sub.q] [greater than or equal to] [F.sub.p+q].

Therefore [F.sub.p+q] = [F.sub.q].

Theorem 3.13. Let (V, v, [tau], [[tau].sup.*]) be a PN space. Then for each p [member of] V, [[alpha].sub.p] [not member of] [T.sub.p] = {t : [F.sub.p](t) > 1 - t}.

Proof. If [[alpha].sub.p] = 0, then [[alpha].sub.p] [not member of] [T.sub.p] is natural. When [[alpha].sub.p] > 0, if [[alpha].sub.p] [member of] [T.sub.p] ,then [F.sub.p]([[alpha].sub.p])+[[alpha].sub.p] [greater than or equal to] 1. Because [F.sub.p](t) + 1 is left-continuous on R, so there is a [delta] > 0 such that [F.sub.p]([alpha]) > 1 - [alpha] when [[alpha].sub.p] - [delta] < [alpha] < [[alpha].sub.p], which implies that [[alpha].sub.p] = inf {t : [F.sub.p](t) > 1 - t} < [[alpha].sub.p], this is a contradiction.

4. Strong Topology and Strong Completeness

Each PN space has a linear structure and a strong topological structure. But this space may not be a topological vector space. If a linearly isomorphic mapping from one PN space ([V.sub.1], [v.sub.1], [[tau].sub.1], [[tau].sup.*.sub.1]) onto another PN space ([V.sub.2], [v.sub.2], [[tau].sub.2], [[tau].sup.*.sub.2]) is also a homeomorphic mapping, then both the linear structure and the strong topological structure are the same on these two spaces. In this case, we will call this mapping an LS mapping, means the linearly homeomorphic with respect to the strong topology, in order to distinguish the notion of the linearly homeomorphic in topological vector space in which the linear structure and the topological structure are compatible with each other. If there is an LS mapping from ([V.sub.1], [v.sub.1], [[tau].sub.1], [[tau].sup.*.sub.1]) to ([V.sub.2], [v.sub.2], [[tau].sub.2], [[tau].sup.*.sub.2]), we will say these two space are linearly and strong topologically equivalent (briefly, LS equivalent). Moreover, the linear homeomorphism between two topological vector PN spaces ([V.sub.1], [v.sub.1], [[tau].sub.1], [[tau].sup.*.sub.1]) and ([V.sub.2], [v.sub.2], [[tau].sub.2], [[tau].sup.*.sub.2]) will be considered as a special LS equivalence in this paper.

Theorem 4.1. Let (V, v, [tau], [[tau].sup.*]) be a PN space. A relation in V is defined by:

x ~ y iff [[alpha].sub.x-y] = 0 for vectors x, y [member of] V,

then this relation is an equivalence relation. [x] is used to denote equivalence class of x, a binary operation [direct sum] is defined in the set G(V) = V/~ = {[x]: x [member of] V} by [x] [direct sum] [y] = [x + y], then (G(V), [direct sum]) is an Abelian quotient group of V.

Proof. x ~ x because [[alpha].sub.x-x] = 0. If x ~ y, then y ~ x because [[alpha].sub.y-x] = [[alpha].sub.x-y] = 0. If x ~ y and y ~ z, then [[alpha].sub.x-z] = [[alpha].sub.(x-y)+(y-z)] = 0, so x ~ z. Therefore the relation ~ is an equivalence relation.

Suppose [x.sub.1], [x.sub.2] and [y.sub.1], [y.sub.2] are arbitrary representative elements of equivalence classes [x] and [y] respectively, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This implies that the operation [direct sum] can not be influenced by selecting different representative elements of an equivalence class.

For arbitrary x, y, z [member of] V, [[theta]] [direct sum] [x] = [x] = [x] [direct sum] [[theta]], so [[theta]] is an identity element in G(V). If [y] [direct sum] [x] = [y + x] = [x], then [[alpha].sub.y] = [[alpha].sub.(y+x)-x] = 0, so [y] = [[theta]], this implies that the identity element is unique. [x] [direct sum] [ - x] = [[theta]] = [ - x] [direct sum] [x], so [ - x] is an inverse element of [x]. If [x] [direct sum] [y] = [x + y] = [[theta]], then ay-(-x) = ax+y = 0, so [y] = [ - x], this implies that the inverse element is unique. ([x] [direct sum] [y]) [direct sum] [z] = [x + y] [direct sum] [z] = [x + y + z] = [x] [direct sum] [y + z] = [x] [direct sum] ([y] [direct sum] [z]), so [direct sum] is associative. [x] [direct sum] [y] = [x + y] = [y] [direct sum] [x], so [direct sum] is commutative. Therefore G(V) is an Abelian group.

Define a mapping [phi] : V [right arrow] G(V) by [phi](x) = [x] for each x [member of] V. For arbitrary y, z [member of] V, we have [phi] (y + z) = [y + z] = [y] [direct sum] [z] = [phi] (y) [direct sum] [phi](z). So [phi] is a surjective homomorphism. Hence G(V) [congruent to] V/[[phi].sup.-1]([[theta]])]. Therefore G(V) is a quotient group of V.

From the proof of Theorem 4.1, we see that the set {p [member of] V : [[alpha].sub.p] = 0} = [[phi].sup.-1]([[theta]])is a normal subgroup of V. In fact, it is a vector space, which follows from (a) below Theorem 3.3. The group G(V) will be called the absorbed exponent group of (V, v, [tau], [[tau].sup.*]).

Definition 4.2.  A category C consists of the following data:

(1) A set Ob(C), whose elements are called the objects of C;

(2) For any A, B [member of] Ob(C), a set H(A, B) whose elements are the morphism in C from A to B;

(3) For any A, B, C [member of] Ob(C), and any f [member of] H(A, B) and g [member of] H(B, C), a morphism h [member of] H(A, C) called the composite of f and g, and written gf;

(4) For any A [member of] Ob(C) a morphism [1.sub.A] [member of] H(A, A), called the identity morphism.

The above data must satisfy the conditions:

h(gf) = (hg)f for f [member of] H(A, B), g [member of] H(B, C) and h [member of] H(C, D);

and

f[1.sub.A] = [1.sub.B]f = f for f [member of] H(A, B).

Two objects A and B are isomorphic if there exist morphisms f [member of] H(A, B) and g [member of] H(B, A) such that gf = [1.sub.A] and fg = [1.sub.B].

Definition 4.3.  A covariant functor from a category C to a category D consists of two maps (denoted by the same letter), a map F : Ob(C) [member of] Ob(D), and for any A, B [member of] Ob(C) a map F : H(A, B) [member of] H(F(A), F(B)), satisfying the conditions:

F([1.sub.A]) = [1.sub.F(A)] for A [member of] Ob(C)

and F(fg) = F(f )F(g) whenever fg is defined in C.

Theorem 4.4.  A functor maps isomorphic objects to isomorphic objects.

Now we construct a category. Each PN space is Considered as an object of C. For arbitrary two PN spaces ([V.sub.1], [v.sub.1], [[tau].sub.1], [[tau].sub.1.sup.*]) and ([V.sub.2], [v.sub.2], [[tau].sub.2], [[tau].sub.2.sup.*]), H(([V.sub.1], [v.sub.1], [[tau].sub.1], [T.sub.1.sup.*]), ([V.sub.2], [v.sub.2], [[tau].sub.2], [[tau].sub.2.sup.*])) is the set of all linearly continuous operators from ([V.sub.1], [v.sub.1], [[tau].sub.1], [[tau].sub.1.sup.*]) to ([V.sub.2], [v.sub.2], [[tau].sub.2], [[tau].sub.2.sup.*]). From topological group theories, it is the set of all continuous homomorphisms from topological group ([V.sub.1], [v.sub.1], [[tau].sub.1], [[tau].sub.1.sup.*]) to ([V.sub.2], [v.sub.2], [[tau].sub.2], [[tau].sub.2.sup.*]). For any f [member of] H([V.sub.1], [v.sub.1], [[tau].sub.1], [[tau].sub.1.sup.*]), ([V.sub.2], [v.sub.2], [[tau].sub.2], [[tau].sub.2.sup.*]) and g [member of] H(([V.sub.2], [v.sub.2], [[tau].sub.2], [[tau].sub.2.sup.*]), ([V.sub.3], [v.sub.3], [[tau].sub.3], [[tau].sub.3.sup.*])), the composite of morphisms f and g is defined as the composite of two linear maps between two linear spaces. This composite is also continuous. The identity morphism 1V of (V, v, [tau], [[tau].sup.*]) is the identity map from (V, v, [tau], [[tau].sup.*]) to itself. Then C is a category; it is called the category of probabilistic normed spaces and denoted by Pn. This category is a subcategory of the category of topological groups [T.sub.op][G.sub.p]. Considering another category, the category of Abelian groups [A.sub.b][G.sub.p], in which the objects are all Abelian groups and H(A, B) is the set of all group homomorphisms. We construct a covariant functor F from [P.sub.n] to [A.sub.b][G.sub.p]. For each object (V, v, [tau], [[tau].sup.*]), let F((V, v, [tau], [[tau].sup.*])) = G(V). For any f [member of] H(([V.sub.1], [v.sub.1], [[tau].sub.1], [[tau].sub.1.sup.*]), ([V.sub.2], [v.sub.2], [[tau].sub.2], [[tau].sub.2.sup.*])), F(f): G([V.sub.1]) [right arrow] G([V.sub.2]) is defined by F(f)[[x].sub.1] = [[f(x)].sub.2] for each x [member of] [V.sub.1], where [[x].sub.1] and [[f(x)].sub.2] are equivalence classes of x and f (x) with respect to the relation ~ which is defined in Theorem 4.1 in [V.sub.1] and [V.sub.2] respectively. If p [member of] [[[theta]].sub.1], the absorbed function Ff(p) = [[epsilon].sub.0] because f is continuous. This implies that f (p) [member of] [[[theta]].sub.2]. So the definition of F(f) can't be influenced by selecting different represent elements in an equivalence class. For any x, y [member of] [V.sub.1], F(f)([[x].sub.1] [direct sum] [[y].sub.1]) = F(f)[[x + y].sub.1] = [[f(x + y)].sub.2] = [[f(x) + f(y)].sub.2] = [[f(x)].sub.2] [direct sum] [[f(y)].sub.2] = F(f)[[x].sub.1] [direct sum] F(f)[[y].sub.1], so F(f) is a homomorphism from G([V.sub.1]) to G([V.sub.2]). For any object (V, v, [tau], [[tau].sup.*]), F(1V)[x] = [1Vx] = [x] for each x [member of] V, so F([1.sub.V]) = [1.sub.G(V)] = [1.sub.F(V)]. For arbitrary f [member of] H(([V.sub.1], [v.sub.1], [[tau].sub.1], [[tau].sup.*]), ([V.sub.2], [v.sub.2], [[tau].sub.2], [[tau].sup.2.sup.*]) and g [member of] H(([V.sub.2], [v.sub.2], [[tau].sub.2], [[tau].sub.2.sup.*]), ([V.sub.3], [v.sub.3], [[tau].sub.3], [[tau].sub.3.sup.*])), one has F(gf )[[x].sub.1] = [[gf(x)].sub.3] = F(g)[[f(x)].sub.2] = F(g)F(f)[[x].sub.1] for each x [member of] [V.sub.1], so F(gf) = F(g)F(f). F is a covariant functor from [P.sub.n] to [A.sub.b][G.sub.p], we will call F the absorbed exponent functor.

Theorem 4.5. The absorbed exponent group G (V ) is an LS equivalence invariant for PN spaces.

Proof. In category [P.sub.n], two objects ([V.sub.1], [v.sub.1], [[tau].sub.1] , [[tau].sub.1.sup.*]) and ([V.sub.2], [v.sub.2], [[tau].sub.2], [[tau].sub.2.sup.*]) are isomorphic iff they are LS equivalent. Considering the absorbed exponent functor F, the conclusion is directly derived from Theorem 4.4.

To reveal strong topological structures on n-dimensional PN spaces (V, v, [tau], [[tau].sup.*]), we need only considering PN spaces ([R.sup.n], v, [tau], [[tau].sup.*]).

In fact, let (V, v, [tau], [[tau].sup.*]) be an arbitrary n-dimensional PN space, where V [not equal to] [R.sup.n]. Taking any arbitrary bases [p.sub.1], [p.sub.2], *** , [p.sub.n] of V and [e.sub.1], [e.sub.2], *** , [e.sub.n] of [R.sup.n]. Define a map [mu] : [R.sup.n] [right arrow] [[DELTA].sup.+] such that [mu] [summation over (1 [less than or equal to] i [less than or equal to] n)] [k.sub.i][e.sub.i] = v [summation over (1 [less than or equal to] i [less than or equal to] n)] [k.sub.i][p.sub.i] for any [k.sub.i] [member of] R. Then ([R.sup.n], [mu], [tau], [[tau].sup.*]) is a PN space. Considering the linear map L : V [right arrow] [R.sup.n], where L([summation over (1 [less than or equal to] i [less than or equal to] n)] [k.sub.i][p.sub.i]) = [summation over (1 [less than or equal to] i [less than or equal to] n)] [k.sub.i][p.sub.i] for any [k.sub.i] [member of] R. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for each t [member of] (0,1), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are neighborhoods of origins [[theta].sub.V] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in V and [R.sup.n] respectively. So L is an LS mapping from (V, v, [tau], [[tau].sup.*]) to ([R.sup.n], [mu], [tau], [[tau].sup.*]). Hence these two spaces are LS equivalent. Therefore they have the same strong topological structures.

First, we study the strong topological structures and strong completeness of 1dimensional PN spaces.

Theorem 4.6. Let (R, v, [tau], [[tau].sup.*]) be a PN space. If there is a p [member of] R such that [[alpha].sub.p] [not equal to] 0, then [N.sub.q](t) = {q} for any t [member of] (0, [[alpha].sub.p]] and q [member of] R, i.e. the strong topology on this PN space is the discrete topology on the set R.

Proof. For each 0 [not equal to] r [member of] R, we get [[alpha].sub.r] = [[alpha].sub.p] [not equal to] 0 following from (2) in Theorem 3.3. The neighborhood [N.sub.0]([[alpha].sub.p]) = {q : [v.sub.q]([[alpha].sub.p]) > 1 - [[alpha].sub.p]}. For each 0 [not equal to] q [member of] R, one obtains q [not member of] [N.sub.0]([[alpha].sub.p]); otherwise, [[alpha].sub.q] = [[alpha].sub.p] [member of] [T.sub.q] = {t : [F.sub.q](t) > 1 - t}, it's a contradiction with Theorem 3.13. It's obvious that 0 [member of] [N.sub.0]([[alpha].sub.p]). So [N.sub.0]([[alpha].sub.p]) = {0}. For each t [member of] (0, [[alpha].sub.p]], {0}[[subset].bar] [N.sub.0](t) [[subset].bar] [N.sub.0]([[alpha].sub.p]) = {0}. Therefore [N.sub.0](t) = {0} and [N.sub.q](t) = q + [N.sub.0](t) = {q} for each q [member of] R.

If a PN space (R, v, [tau], [[tau].sup.*]) is also a topological vector space, then it is strongly complete  (Saadati, Amini, 2005). The following Lemma will show that each nontopological vector PN space (R, v, [tau], [[tau].sup.*]) is also strongly complete.

Lemma 4.7. Let (R, v, [tau], [[tau].sup.*]) be a PN space. If there is a p [member of] R such that [[alpha].sub.p] not equal to] 0, then this space is strongly complete.

Proof. Because [[alpha].sub.p] [not equal to] 0, so [N.sub.0](t) = {0} for any t [member of] (0, [[alpha].sub.p]]. Suppose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a strong Cauchy sequence, then there exists an integer N > 0 such that [p.sub.n] - [p.sub.m] [member of] [N.sub.0]([[alpha].sub.p]) = {0} when n, m > N, i.e. [p.sub.n] = [p.sub.N+1] when n > N. So this sequence is strongly convergent to [p.sub.N+1].

Theorem 4.8. Let (R, v, [tau], [[tau].sup.*])bea PN space. Then it is strongly complete.

Proof. The conclusion is directly derived from Lemma 4.7 and the statement above this lemma.

Lemma 4.9. Let (R, [v.sub.1], [[tau].sub.1], [[tau].sub.1.sup.*]) and (R, [v.sub.2], [[tau].sub.2], [[tau].sub.2.sup.*]) be two PN spaces. If there are two vectors p, q [member of] R such that [[alpha].sup.1.sub.p] = 0 and [[alpha].sup.2.sub.q] = 0, where [[alpha].sup.i] denotes the absorbed exponent in i-th PN space, i = 1,2, then these two spaces are LS equivalent.

Proof. The strong topologies on these two PN spaces are all discrete topologies on R, which follows from Theorem 4.6. The identity map I : R [right arrow] R is an LS mapping.

Lemma 4.10. Let (R, [v.sub.1], [[tau].sub.1], [[tau].sub.1.sup.*]) and (R, [v.sub.2], [[tau].sub.2], [[tau].sub.2.sup.*]) be two PN spaces. If there is a p [member of] R such that [[alpha].sub.p] [not equal to] 0 in the second space and the first space is a topological vector space, then these two space are not LS equivalent.

Proof. G((R, [v.sub.1], [[tau].sub.1], [[tau].sub.1.sup.*])) [congruent to] 0 and G((R, [v.sub.2], [[tau].sub.2], [[tau].sub.2.sup.*])) [congruent to] R, so these two spaces are not LS equivalent, which follows from Theorem 4.5.

We give the following classification for strong topological structures on 1- dimensional PN spaces. Here we need a conclusion from  (Kothe, 1969), (c): A n- dimensional [T.sub.2] topological vector space is linearly homeomorphic to n-Euclidean [R.sup.n].

Theorem 4.11. There are two equivalence classes of PN spaces (R, v, t, [[tau].sup.*]), denoted by [C.sup.0.sub.1] and [C.sup.1.sub.1], in the sense of LS equivalence, where [C.sup.1.sub.1] is the class of PN spaces which are homeomorphic to 1-Euclidean space, and [C.sup.0.sub.1] is the class of PN spaces which are homeomorphic to the discrete topology on the set R.

Proof. Because each PN space is a [T.sub.2] space, so (R, [v.sub.1], [[tau].sub.1], [[tau].sub.1.sup.*]) is LS equivalent to (R, [v.sub.2], [[tau].sub.2], [[tau].sub.2.sup.*]) when they are all topological vector spaces. Combining with Lemma 4.9 and Lemma 4. 0, the conclusion is derived.

Next, we study the strong topological structures on general n-dimensional PN spaces by using results obtained for 1-dimensional PN spaces.

Let ([R.sup.n], v, [tau], [[tau].sup.*])bea PN space. For any p [member of] [R.sup.n] and p [not equal to] [theta], we consider the relative topology on the linear subspace [R.sub.p] = {kp : k [memer of] R} of [R.sup.n]. This topology is just the strong topology on ([R.sub.p], v, [tau], [[tau].sup.*]). We will use [equivalent to] to denote LS equivalence.

Considering the n-Euclidean [R.sup.n] as a special Menger PN space ([R.sup.n], v, M), where v is defined by [v.sub.p] = [epsilon][parallel]p[parallel] for each p [member of] [R.sup.n]. Then the strong topology on ([R.sup.n], v, M) is the same with the Euclidean topology on the set [R.sup.n]. The relation ([R.sup.k], v, [tau], [[tau].sup.*]) [equivalent to] [R.sup.k] appears in the following part means that the PN space ([R.sup.k], v, [tau], [[tau].sup.*])is LS equivalent to ([R.sup.k], v, M).

Theorem 4.12. Let([R.sup.n], v, [tau] , [[tau].sup.*]) be a topological vector PN space. Then for an arbitrary base {[e.sub.1], *** , [e.sub.n]}, the strong topology on ([R.sup.n], v, [tau], [[tau].sup.*]) is the topological product of the strong topologies on ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], v, [tau], [[tau].sup.*]), 1 [less than or equal to] i [less than or equal to] n.

Proof. Because ([R.sup.n], v, [tau], [[tau].sup.*]) is a topological vector space, so for an arbitrary base {[e.sub.1], ***, [e.sub.n]}, we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Following from the conclusion (c), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here, 'x' represents the topological product. ?

Lemma 4.13. Let ([R.sup.n], v, [tau], [[tau].sup.*]) be a PN space. If [[alpha].sub.p] [not equal to] 0 for every [theta] [not equal to] p [member of] [R.sup.n], then [beta] = inf {[[alpha].sub.p] : p [member of] [R.sup.n], p [not equal to] [theta]} > 0.

Proof. By induction, the case n = 1 holds because [beta] = [[alpha].sub.p] > 0 for each p [not equal to] 0. Assume the conclusion holds for any cases n [less than or equal to] k. When n = k + 1, because [[alpha].sub.p] > 0 for each p [not equal to] [theta] ,so [beta] [greater than or equal to] 0. Assume [beta] = 0, then for any base {[e.sub.1], [e.sub.2], *** , [e.sub.k+1]}, there is a sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We can let [a.sup.m.sub.1][e.sub.1] + [a.sup.m.sub.2][e.sub.2] = 0 for each m [member of] [Z.sub.+], because [[beta].sub.1] = inf {[[alpha].sub.p] : p [member of] Span([e.sub.2], *** , [e.sub.k+1]), p [not equal to] [theta]} > 0. Let [p.sub.m] = [[bar.p].sub.m]/[a.sup.m.sub.1] = [e.sub.1] + ([a.sup.m.sub.2][e.sub.2]/[a.sup.m.sub.1] + ... + ([a.sup.m.sub.k+1][e.sub.k+1])/[a.sup.m.sub.1]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Putting [[tau].sub.1] = 1/2, there is a positive integer [N.sub.1] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so there is a [[lambda].sub.1] > 0 such that [[lambda].sub.1]p[N.sub.1] [member of] [N.sub.[theta]](1/2). Putting [t.sub.2] = 1/[2.sup.2], there is an integer [N.sub.2] > [N.sub.1] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so there is a [[lambda].sub.2] > 0 such that [[lambda].sub.2]p[N.sub.2] [member of] [N.sub.[theta]](1/[2.sup.2]). Following from Theorem 2.3, we can let 0 < [[lambda].sub.2] < [[lambda].sub.1] . Go on this process, we can get two ordered sequences

0 < [N.sub.1] < [N.sub.2] < [N.sub.3] < *** < [N.sub.m] < ...

and

[[lambda].sub.1] > [[lambda].sub.2] > [[lambda].sub.3] > ... > [[lambda].sub.m] > ... > 0,

m [member of] [Z.sub.+], which satisfy the following condition:

[[lambda].sub.m]P[N.sub.m] [member of] [N.sub.[theta]](1/[2.sup.m]), i.e. [[lambda].sub.m]P[N.sub.m] [right arrow] [theta](m [right arrow] [infinity]).

Because [[lambda].sub.m+1] < [[lambda].sub.m], so [[lambda].sub.m+1]p[N.sub.m] [member of] [N.sub.[theta]](1/[2.sup.m]), we also have [[lambda].sub.m+1]p[N.sub.m] [right arrow] [theta] (m [right arrow] [infinity]). Because ([R.sup.k+1], v, [tau], [[tau].sup.*]) is a topological group, so [[lambda].sub.m+1]p[N.sub.m+1] - [[lambda].sub.m+1]p[N.sub.m] [right arrow] [theta] (m [right arrow] [infinity]) also holds. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

But [[beta].sub.1] = inf {[[alpha].sub.p] : p [member of] Span([e.sub.2], ... , [e.sub.k+1]), p [not equal to] [theta]} > 0 and [[lambda].sub.m] > 0 for each m [member of] [Z.sub.+]. one has [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for each 2 [less than or equal to] i [less than or equal to] k + 1 and m [member of] [Z.sub.+], i.e. p[N.sub.m] = p[N.sub.1] for any m [member of] [Z.sub.+], which implies that [[lambda].sub.m]p[N.sub.m] = [[lambda].sub.m]p[N.sub.1] [right arrow] [theta]. It's a contradiction because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So the assumption does not hold, we have [beta] > 0.

Theorem 4.14. Let ([R.sup.n], v, [tau], [[tau].sup.*]) be a PN space. If [[alpha].sub.p] [not equal to] 0 for every [theta] [not equal to] p [member of] [R.sup.n], then [N.sub.q](t) = {q} for arbitrary 0 < t < [beta] = inf{[[alpha].sub.p] : p [member of] [R.sup.n], p [not equal to] [theta]} and q [member of] [R.sup.n], i.e. the strong topology on this PN space is the discrete topology on the set [R.sup.n].

Proof. For each 0 < t < [beta] = inf{[[alpha].sub.p] : p [member of] [R.sup.n], p [not equal to] [theta]}, we get q [member of] [N.sub.[theta]](t) for any q [not equal to] [theta]; otherwise, [v.sub.q](t) > 1 - t, so [v.sub.q]([[alpha].sub.q]) [greater than or equal to] [v.sub.q]([beta]) [greater than or equal to] [v.sub.q](t) > 1 - t > 1 - [[alpha].sub.q], then [[alpha].sub.q] [member of] [T.sub.q] = {t : [F.sub.q](t) > 1 - t}, this is a contradiction with Theorem 3.13. Hence [N.sub.[theta]](t) = {[theta]} and [N.sub.q](t) = q + [N.sub.[theta]](t) = {q}.

Theorem 4.15. Let ([R.sup.n], v, [tau], [[tau].sup.*]) be a PN space. If [[alpha].sub.p] = 0 for every [theta] = p [member of] [R.sup.n], then the strong topology on ([R.sup.n], v, [tau], [[tau].sup.*]) is the topological product of strong topologies on all subspaces ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], v, [tau], [[tau].sup.*])(1 [less than or equal to] i [less than or equal to] n), where {[e.sub.1], [e.sub.2], ..., [e.sub.n]} is an arbitrary base of the vector space [R.sup.n].

Proof. A discrete topology on [R.sup.n] is naturally the product topology of n discrete topologies on R, proof is complete following from Theorem 4.6 and Theorem 4.14.

We give the following classification for the strong topological structures on n- dimensional PN spaces.

Theorem 4.16. All PN spaces ([R.sup.n], v, [tau], [[tau].sup.*]) are included in n + 1 equivalence classes, denoted by [C.sup.0.sub.n], [C.sup.1.sub.n], ... , [C.sup.n.sub.n], where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

0 [less than or equal to] i = dim([[phi].sup.-1][[theta]]) [less than or equal to] n for each ([R.sup.n], v, [tau], [[tau].sup.*]) [member of] [C.sup.i.sub.n], R represents 1-Euclidean space and [2.sup.R] represents the discrete topology on the set R.

Proof. [[phi].sup.-1][[theta]] is a subspace of ([R.sup.n], v, [tau],[[tau].sup.*]). Take a base of [[phi].sup.-1][[theta]] and denoted {[e.sub.1], ... , [e.sub.m]}. It can be extended to a base of [R.sup.n], denoted {[e.sub.1], ..., [e.sub.m], [e.sub.m+1], ..., [e.sub.n]}.

If 0 < m < n, the relative topology on [R.sup.c.sub.n-m] = Span([e.sub.m+1], ..., [e.sub.n]) is the discrete topology and [beta] = inf{[[alpha].sub.p] : p [member of] [R.sup.c.sub.n-m], p [not equal to] [[theta].sup.c.sub.n-m]} > 0, where [[theta].sup.c.sub.n-m] represents the null vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in [R.sup.c.sub.n-m]. For each t < [beta], [N.sub.[theta]](t) in ([R.sup.n], v, [tau], [[tau].sup.*]) is the same with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in ([R.sup.s.sub.m], v, [tau], [[tau].sup.*]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in ([R.sup.c.sub.n-m], v, [tau], [[tau].sup.*]), where [R.sup.s.sub.m] = Span([e.sub.1], ..., [e.sub.m]) = [[phi].sup.-1][[theta]] and [[theta].sup.s.sub.m] represents the null vector [theta] in [R.sup.s.sub.m]. Considering the linear bijection H : [R.sup.n] [right arrow] [R.sup.s.sub.m] x [R.sup.c.sub.n-m],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ,

where [R.sup.s.sub.m] x [R.sup.c.sub.n-m] is the product of two topological group [R.sup.s.sub.m] and [R.sup.c.sub.n-m], H is a group homomorphism. Each PN space is an [A.sub.1] space, {[N.sub.[theta]](l/l): l [member of] [Z.sub.+] is a neighborhood base of [theta]. When 1/l < [beta], one obtains [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore {[N.sub.[theta]](l/l): l [member of] [Z.sub.+], 1/l < [beta]} and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are neighborhood bases of [theta] and [[theta]sup.s.sub.m] x [[theta]sup.c.sub.n-m] respectively, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This implies that H and [H.sup.-1] are continuous at [theta] and [[theta].sup.s.sub.m] x [[theta].sup.c.sub.n-m] respectively, so H is a topological group homeomorphism. Therefore H is an LS mapping and

([R.sup.n], v, [tau], [[tau].sup.*]) [equivalent to] ([R.sup.s.sub.n-m], v, [tau], [[tau].sup.*]).

By Theorem 4.12, one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By Theorem 4.15 and 4.6 , we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If m = n, then ([R.sup.n], v, [tau], [[tau].sup.*]) is a topological vector space. By Theorem 4.12, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If m = 0, then ([R.sup.n], v, [tau], [[tau].sup.*]) is a discrete space. By Theorem 4.15 and 4.6, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because finite topological products are associative and commutative in the sense of homeomorphism, so each [C.sup.i.sub.n] for 0 [less than or equal to] i [less than or equal to] n is a equivalence class of PN spaces. For any 0 [less than or equal to] i [not equal to] j [less than or equal to] n,([R.sup.n], [v.sub.i], [[tau].sub.i], [[tau].sub.i.sup.*]) [member of] [C.sup.i.sub.n] and ([R.sup.n], [v.sub.j], [[tau].sub.j], [[tau].sup.*.sub.j]]) [member of] [C.sup.j.sub.n], we get G(([R.sup.n], [v.sub.i], [[tau].sub.i], [[tau].sup.*.sub.i]])) = [R.sup.n]/[[phi].sup.-1][[theta]] = [R.sup.n]/[R.sup.s.sub.i] = [R.sup.c.sub.n- i] [congruent to] [R.sup.n-i] and G(([R.sup.n], [v.sub.j], [[tau].sub.j], [[tau].sup.*.sub.j])) = [R.sup.n]/[[phi].sup.-1][[theta]] = [R.sup.n]/[R.sup.s.sub.j] = [R.sup.c.sub.n- j] [congruent] [R.sup.n-j]. Following from Theorem 4.5, [C.sup.i.sub.n] and [C.sup.j.sub.n] are different equivalence classes.

Finally, we show that each finite dimensional PN space is strongly complete.

Theorem 4.17. Let ([R.sup.n], v, [tau], [[tau].sup.*]) be a PN space. Then it is strongly complete.

Proof. We continue to use the notations and terminologies in Theorem 4.16.

If m = dimp[[phi].sup.-l][[theta]] = n, then ([R.sup.n], v, [tau], [[tau].sup.*]) is a topological vector space. Because each PN space is a [T.sub.2] space, so ([R.sup.n], v, [tau], [[tau].sup.*]) is linearly homeomorphic to n-Euclidean ([R.sup.n], [parallel].[parallel]), which follows from the conclusion (c). This implies that the countable neighborhood bases {[N.sub.[theta]](1/n): n [member of] [Z.sub.+]} and {[B.sub.[theta]](1/n): n [member of] [Z.sub.+]} of [theta] are contained with each other, where [B.sub.[theta]](1/n) = {p : [parallel]p[parallel] < 1/n} for each n [member of] [Z.sub.+]. Taking [[lambda].sub.0] = 1/2, there is a positive integer [K.sub.1] [greater than or equal to] 2 such that

[B.sub.[theta]](1/[K.sub.1]) = {p : [parallel]p[parallel] < 1/[K.sub.1]} [[subset].bar] [N.sub.[theta]]([[lambda].sub.0]) = [N.sub.[theta]](1/2).

For this [B.sub.[theta]](1/[K.sub.1]), there exists a positive integer [K.sub.2] [greater than or equal to] max{[2.sup.2], [K.sub.1]} such that

[N.sub.[theta]](1/[K.sub.2]) [[subset].bar] [B.sub.[theta]](1/[K.sub.1]) [[subset].bar] [N.sub.[theta]](1/2).

For this [N.sub.[theta]](1/[K.sub.2]), there exists a positive integer [K.sub.3] [greater than or equal to] max{[2.sup.3], [K.sub.2]} such that

[B.sub.[theta]](1/[K.sub.3]) [[subset].bar] [N.sub.[theta]](1/[K.sub.2]) [[subset].bar] [B.sub.[theta]](1/[K.sub.1]) [[subset].bar] [N.sub.[theta]](1/2).

Go on this process, we can get an ordered sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[[subset].bar] *** [[subset].bar] [N.sub.[theta]](1/[K.sub.2]) [[subset].bar] [B.sub.[theta]](1/[K.sub.1]) [[subset].bar] [N.sub.[theta]](1/2).

Where [K.sub.s] [greater than or equal to] max{[2.sup.s], [K.sub.s-1} for any positive integer s. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a neighborhood base of [theta] under Euclidean topology [T.sub.E] , and {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a neighborhood base of [theta] under strong topology T. Taking any strong Cauchy sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in ([R.sup.n], v, [tau], [[tau].sup.*]), for any positive integer l, there is a positive integer N such that

[p.sub.s] - [p.sub.t] [member of] [N.sub.[theta]] (1/[K.sub.2l]) [[subset].bar] [B.sub.[theta]](1/[K.sub.2l-1]),

when s, t> N .So {[p.sub.s]}s[member of]N is also a Cauchy sequence in ([R.sup.n], [parallel].[parallel]). Because ([R.sup.n], [parallel]. [parallel])is complete, there is a p [member of] [R.sup.n] such that [p.sub.s] [right arrow] p (s [right arrow] [infnity]) with respect to [T.sub.E]. So for any positive integer l, there is a positive integer N such that

[p.sub.s] - p [member of] [B.sub.[theta]](1/[K.sub.2l+1]) [[subset].bar] [N.sub.[theta]] (1/[K.sub.2l]),

when s > N. Hence [p.sub.s] [member of] [N.sub.p](1/[K.sub.2l]) when s > N. So {[p.sub.s]}s[member of]N is strongly convergent to p. Therefore ([R.sup.n], v, [tau], [[tau].sup.*]) is strongly complete.

If m = dimp[[phi].sup.-l][[theta]] = 0, then ([R.sup.n], v, [tau], [[tau].sup.*]) is a discrete space and [N.sub.[theta]](t) = {[theta]} for arbitrary t < [beta] = inf{[[alpha].sub.p] : p [member of] [R.sup.n], p [not equal to] {[theta]}. For any strong Cauchy sequence {[p.sub.s]}s[member of][Z.sub.+], there is a N > 0 such that [p.sub.s] - [p.sub.l] [member of] [N.sub.[theta]](t) = {[theta]} when s, l > N. This implies that [p.sub.s] [right arrow] [p.sub.N+1] (s [right arrow] [infinity]). ([R.sup.n], v, [tau], [[tau].sup.*]) is strongly complete.

If 0 < m = dim[[phi].sup.-l][[theta]] < n, for each strong Cauchy sequence {[p.sub.s] = [k.sup.s.sub.1][e.sub.1] + ... + [k.sup.s.sub.m][e.sub.m] + [k.sup.s.sub.m+1][e.sub.m+1] + *** + [k.sup.s.sub.n][e.sub.n]}s[member of][Z.sub.+], because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for each t < [[beta].sup.1] = inf{[[a.sub.p] : p [member of] [R.sup.c.sub.n-m], p [not equal to] [[theta].sup.c.sub.n-m]} and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , so there is a N > 0 such that [k.sup.l.sub.i] = [k.sup.N.sub.i] for any l > N and m + 1 [less than or equal to] i [less than or equal to] n. {[p.sup.m.sub.s] = [k.sup.s.sub.1][e.sub.1] + ... + [k.sup.s.sub.m][e.sub.m]} is a strong Cauchy sequence in topological vector space ([R.sup.s.sub.m], v, [tau], [[tau].sup.*]). So there is a vector [p.sup.m] [member of] [R.sup.s.sub.m] such that [p.sup.m.sub.s] is strongly convergent to [p.sup.m] in ([R.sup.s.sub.m], v, [tau], [[tau].sup.*]). Let p = [p.sup.m] + [k.sup.N.sub.m+1][e.sub.m+1] + ... + [k.sup.N.sub.n][e.sub.n]. Then [p.sub.s] [right arrow] p (n [right arrow] [infinity]). So ([R.sup.n], v, [tau], [[tau].sup.*]) is strongly complete.

Acknowledgment

It is a pleasure to acknowledge the supports from National Natural Science Foundation of China (no: 11072204), the Fundamental Research Funds for the Central Universities (no: SWJTU11ZT15), Science and Technology Department of Sichuan Province (no: 2010JY0079), and Research Project Funds of Kunming University for introducing the talent (no: YJL12005).

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L. Li

School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China E-mail: LDouble@yahoo.cn

X.H. Xu

Civil Department, Jinjiang College, Sichuan University, Chengdu 620860, China E-mail: xuxuhua888@163.com

M.X. Zhang

College ofMathematics, Chengdu University ofInformation Technology, Chengdu 610225, China E-mail: mxzhang01@sina.com

Y.H. Li

School ofMechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China E-mail: yinghui.li@home.swjtu.edu.cn

Q.K. Liu

Department ofMathematics, Kunming University, Kunming 650214, China E-mail: qkliu888@yahoo.com.cn