1 Introduction
Locally convex probabilistic normed spaces are an interesting topic. In fact, some papers [
1‐
3] discussed the subject, and we enjoy the topic too. On the basis of these papers, we try to search more concepts and properties about locally convex probabilistic normed spaces. In this article, we show our results.
Probabilistic normed spaces (briefly, PN spaces) were introduced by Šerstnev [
4] by means of a definition that was closely modeled in the theory of normed spaces. Here we consistently adopt the new and, in our opinion, convincing definition of a PN space given Alsina, Schweizer, and Sklar [
5], from which we further use the notation and concepts. On the basis of this classical work, continuity properties, linear operators, and nonlinear operators on PN spaces are studied in detail [
6‐
8], and contraction maps, boundedness property, finite and countable infinite products, and probabilistic quasi-normed spaces are deeply discussed [
9‐
13]. In order to understand the new advances on PN spaces, we refer to [
14].
We recall the definition, properties, and examples of probabilistic normed spaces. Let Δ be the space of distribution functions, and
\(\Delta^{+}:=\{F\in \Delta:F(0)=0\}\) be the subset of distance distribution functions [
15]. The space Δ can be metrized in several equivalent ways so that the metric topology coincides with the topology of weak convergence for distribution functions. Here, we assume that Δ is metrized by the Sibley metric
\(d_{S}\), which is the same metric denoted by
\(d_{L}\) in [
15]. We also consider the subset
\(D^{+}\subset \Delta^{+}\) of the proper distance distribution functions, that is, those
\(F\in \Delta^{+}\) for which
\(\lim_{x\longrightarrow +\infty} F(x)=1\).
A triangle function is a mapping
\(\tau: \Delta^{+}\times \Delta^{+}\longrightarrow \Delta^{+}\) that is commutative, associative, nondecreasing in each variable and has
\(\varepsilon_{0}\) as the identity, where
\(\varepsilon_{a}\) \((a\leq +\infty)\) is the distribution function defined by
$$\varepsilon_{a}(t):= \textstyle\begin{cases} 0, & t\leq a,\\ 1, & t>a. \end{cases} $$
Given a nonempty set S, a mapping
\(\mathcal {F}\) from
\(S\times S\) into
\(\Delta^{+}\) and a triangle function
τ, a probabilistic metric space (briefly a PM space) is the triple
\((S, \mathcal {F}, \tau)\) with the following properties, where we set
\(F_{p, q}:=\mathcal {F}_{p, q}\):
(PM1)
\(F_{p, q}=\varepsilon_{0}\) if and only if \(p=q\);
(PM2)
\(F_{p, q}=F_{q, p} \) for all p and \(q \in S\);
(PM3)
\(F_{p, r}\geq \tau(F_{p, q}, F_{q, r}) \) for all p, q, \(r \in S\).
A probabilistic normed space (briefly, a PN space) is a quadruple
\((\mathcal{V}, \upsilon, \tau, \tau^{*})\), where
\(\mathcal{V}\) is a vector space,
τ and
\(\tau^{*}\) are continuous triangle functions such that
\(\tau\leq \tau^{*}\), and
υ is a mapping from
\(\mathcal{V}\) into
\(\Delta^{+}\), called the probabilistic norm, such that for every choice of
p and
q in
\(\mathcal{V}\), the following conditions hold:
(PN1)
\(\upsilon_{p}=\varepsilon_{0}\) if and only if \(p=\theta\) (θ is the null vector in \(\mathcal{V}\));
(PN2)
\(\upsilon_{-p}=\upsilon_{p}\);
(PN3)
\(\upsilon_{p+q}\geq\tau(\upsilon_{p}, \upsilon_{q})\);
(PN4)
\(\upsilon_{p}\leq \tau^{*}(\upsilon_{\lambda p}, \upsilon_{(1-\lambda)p})\) for every \(\lambda \in [0, 1]\).
When there is a continuous
t-norm
T (see [
7,
15]) such that
\(\tau=\tau_{T}\) and
\(\tau^{*}=\tau_{T^{*}}\), where
$$\begin{aligned} &{T^{*}(x, y):=1-T(1-x, 1-y),} \\ &{\tau_{T}(F, G) (x):=\sup_{s+t=x} T \bigl(F(s), G(t) \bigr),\quad\mbox{and}\quad \tau_{T^{*}}(F, G) (x):=\inf _{s+t=x} T^{*} \bigl(F(s), G(t) \bigr),} \end{aligned}$$
the PN space
\((\mathcal{V}, \upsilon, \tau_{T}, \tau_{T^{*}})\) is called a Menger PN space and is denoted by
\((\mathcal{V}, \upsilon, T)\).
A PN space is called a Šerstnev space if it satisfies (PN1), (PN3), and the following condition, which implies both (PN2) and (PN4):
For any \(p\in\mathcal{V}\), \(\alpha \in \mathbb{R}\backslash\{0\}\), and \(x>0\), \(\upsilon_{\alpha p}(x)=\upsilon_{p}(\frac{x}{|\alpha|})\).
If
\((\mathcal{V}, \upsilon, \tau, \tau^{*})\) is a PN space and a mapping
\(\mathcal {F}: \mathcal{V}\times \mathcal{V}\longrightarrow \Delta^{+}\) is defined as
$$ \mathcal {F}(p, q):=\upsilon_{p-q} $$
(1)
then
\((\mathcal{V}, \mathcal {F}, \tau)\) is a probabilistic metric space. Every PM space can be endowed with strong topology; this topology is generated by the
strong neighborhoods defined as follows: for every
\(t>0\), the neighborhood
\(N_{p}(t)\) at a point
p of
\(\mathcal{V}\) is defined by
$$N_{p}(t):= \bigl\{ q\in \mathcal{V}:d_{S}( \upsilon_{p-q}, \varepsilon_{0})< t \bigr\} = \bigl\{ q\in \mathcal{V}:\upsilon_{p-q}(t)>1-t \bigr\} . $$
2 Main Results
(I) Š-probabilistic seminorm and locally convex Š-probabilistic semi-normed spaces
Proof
(1) For any
\(\alpha \in\mathbb{R}\),
\(\alpha\neq 0\), we have
$$\begin{aligned} \mathscr{P}_{\lambda}(\alpha p) =&\inf \bigl\{ t\geq 0;\upsilon_{\alpha p}(t)>1- \lambda \bigr\} \\ =&\inf \biggl\{ t\geq 0;\upsilon_{p} \biggl(\frac{t}{|\alpha|} \biggr)>1- \lambda \biggr\} \\ =& |\alpha|\inf \bigl\{ t\geq 0;\upsilon_{p}(t)>1-\lambda \bigr\} \\ =& |\alpha|\mathscr{P}_{\lambda}(p). \end{aligned}$$
It is obvious that from
\(\alpha=0 \) we get
\(\mathscr{P}_{\lambda}(0\cdot p)=0 \cdot \mathscr{P}_{\lambda}(p)\) and
\(\mathscr{P}_{\lambda}(p)\geq 0\). According to the definition of
\(\mathscr{P}_{\lambda}\), for any
\(\varepsilon>0\), we have
\(\upsilon_{p}(\mathscr{P}_{\lambda}(p)+\frac{\varepsilon}{2})>1-\lambda\) and
\(\upsilon_{q}(\mathscr{P}_{\lambda}(q)+\frac{\varepsilon}{2})>1-\lambda\). By condition (ŠPSN3),
\(\upsilon_{p+q}(\mathscr{P}_{\lambda}(p)+\mathscr{P}_{\lambda}(q)+\varepsilon)>1-\lambda\). Therefore,
$$\begin{aligned} \mathscr{P}_{\lambda}(p+q) =&\inf \bigl\{ t\geq 0;\upsilon_{p+q}(t)>1- \lambda \bigr\} \\ \leq& \mathscr{P}_{\lambda}(p)+\mathscr{P}_{\lambda}(q)+\varepsilon. \end{aligned}$$
Letting
\(\varepsilon\to0\), we have
\(\mathscr{P}_{\lambda}(p+q)\leq\mathscr{P}_{\lambda}(p)+\mathscr{P}_{\lambda}(q)\),
\(\lambda\in(0, 1]\).
Conclusion (1) is proved.
(2) Firstly, it is easy to show that
\(W(p, \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}, \lambda)\) is convex. In fact, for any
\(p, q\in W(p, \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}, \lambda)\),
$$\mathscr{P}_{\lambda_{i}}(p)< \lambda\quad\mbox{and}\quad \mathscr{P}_{\lambda_{i}}(q)< \lambda. $$
Then, for every
\(t \in [0, 1]\),
$$\begin{aligned} \mathscr{P}_{\lambda_{i}} \bigl(tp+(1-t)q \bigr) \leq& \mathscr{P}_{\lambda_{i}}(tp)+ \mathscr{P}_{\lambda_{i}} \bigl((1-t)q \bigr)= t\mathscr{P}_{\lambda_{i}}(p)+(1-t)\mathscr{P}_{\lambda_{i}}(q) < t\lambda+(1-t)\lambda = \lambda. \end{aligned}$$
Thus,
$$tp+(1-t)q\in W(p, \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}, \lambda). $$
Secondly, we consider the system
$$\mathcal{W}(p)= \bigl\{ W(p, \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}, \lambda):\lambda>0, \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\in(0, 1] \bigr\} , $$
in which
\(W(p, \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}, \lambda)=\{p\in\mathcal{V}:\mathscr{P}_{\lambda_{i}}(p)<\lambda, \lambda_{i}\in(0, 1], i=1, 2, \ldots, n\}\). By Lemma
2.1 we know that if
\(W_{1}=W(p, \lambda_{1}^{\prime}, \lambda_{2}^{\prime}, \ldots, \lambda_{n}^{\prime}, \lambda^{\prime})\),
\(W_{2}=W(p, \lambda_{1}^{\prime\prime}, \lambda_{2}^{\prime\prime}, \ldots, \lambda_{m}^{\prime\prime}, \lambda^{\prime\prime})\),
\(\lambda=\min (\lambda^{\prime}, \lambda^{\prime\prime})\), and
\(W_{3}=W(p, \lambda_{1}^{\prime}, \lambda_{2}^{\prime}, \ldots, \lambda_{n}^{\prime}, \lambda_{1}^{\prime\prime}, \lambda_{2}^{\prime\prime}, \ldots, \lambda_{m}^{\prime\prime}, \lambda)\), then
\(W_{3}\subset W_{1}\cap W_{2}\), so that property (a) is satisfied.
If \(\alpha\in \mathbb{R}\) and \(|\alpha|\leq 1\), then from \(\mathscr{P}_{\lambda_{i}}(p)<\lambda\), we get \(\mathscr{P}_{\lambda_{i}}(\alpha p)<\lambda\), that is, the set \(W(p, \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}, \lambda)\) is balanceable, so that property (b) is satisfied.
Let \(q\in \mathcal{V}\) and denote \(W(p, \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}, \lambda)\) by \(W_{0}\). Let \(\mu\in \mathbb{R}\) be such that \(0<|\mu|<\lambda\), and let \(\sigma=\max_{1\leq i\leq n} \mathscr{P}_{\lambda_{i}}(q)\). If \(q \notin W_{0}\) and \(\alpha=\mu\sigma^{-1}\), then \(\mathscr{P}_{\lambda_{i}}(\alpha q)=|\mu|\sigma^{-1}\mathscr{P}_{\lambda_{i}}(q)\leq|\mu|<\lambda\), that is, \(\alpha q\in W_{0}\), so that property (c) is satisfied.
Let \(W_{1}=W(p, \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}, 2^{-1}\lambda)\). Then \(W_{1}+W_{1}=\frac{1}{2} W_{0}+\frac{1}{2} W_{0}=W_{0}\), and we see that \(\mathcal{W}(p)\) satisfies property (d).
Since
\(W_{0}=W(p, \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}, \lambda)\) and
\(\alpha \in \mathbb{R}\),
\(\alpha\neq 0\), we have
$$\begin{aligned} \alpha W_{0} =& \bigl\{ \alpha p|\mathscr{P}_{\lambda_{i}}(p)< \lambda, i=1, 2, \ldots, n \bigr\} \\ =& \bigl\{ p|\mathscr{P}_{\lambda_{i}}(p)< |\alpha|\lambda, i=1, 2, \ldots, n \bigr\} \\ =& W(p, \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}, |\alpha|\lambda), \end{aligned}$$
that is,
\(\alpha W_{0}\in \mathcal{W}(p)\), so
\(\mathcal{W}(p)\) satisfies property (e).
Conclusion (2) is proved.
(3) Next, we prove that
\(U(\lambda, \lambda_{i})=W(p, \lambda_{i}, \lambda)\) (
\(i=1, 2, \ldots, n\)). Let
\(p\in U(\lambda, \lambda_{i})\). Then
\(\upsilon_{p}(\lambda)>1-\lambda_{i}\). Since the distribution function
\(\upsilon_{p}\) is left continuous, there exists
\(\lambda^{\prime}\in (0, \lambda)\) such that, for each
\(i=1, 2, \ldots, n\),
$$\upsilon_{p}(\lambda)\geq \upsilon_{p} \bigl( \lambda^{\prime} \bigr)>1-\lambda_{i}. $$
Hence,
$$\inf \bigl\{ t\geq 0;\upsilon_{p}(t)>1-\lambda_{i} \bigr\} \leq \lambda^{\prime}< \lambda, $$
which implies that
\(p\in W(p, \lambda_{i}, \lambda)\). Conversely, let
\(p\in W(p, \lambda_{i}, \lambda)=\{p\in\mathcal{V}:\mathscr{P}_{\lambda_{i}}(p)<\lambda\}\). Then
\(\mathscr{P}_{\lambda_{i}}(p)=\)
\(\inf\{t\geq 0;\upsilon_{p}(t)>1-\lambda_{i}\}<\lambda\),
\(\upsilon_{p}(\lambda)>1-\lambda_{i}\), that is,
\(p\in U(\lambda, \lambda_{i})\). Thus, we get the conclusion
\(U(\lambda, \lambda_{i})=W(p, \lambda_{i}, \lambda)\) (
\(i=1, 2, \ldots, n\)).
On the other hand, for each
\(W_{0}=W(p, \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}, \lambda) \in\mathcal{W}(p)\), we have
$$\begin{aligned} W(p, \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}, \lambda) =& \bigl\{ p\in\mathcal{V}:\mathscr{P}_{\lambda_{i}}(p)< \lambda, \lambda_{i}\in(0, 1], i=1, 2, \ldots, n \bigr\} \\ =&\bigcap_{i=1}^{n} W(p, \lambda_{i}, \lambda) = \bigcap_{i=1}^{n} U(\lambda, \lambda_{i}), \end{aligned}$$
which implies that
\(\mathcal{W}(p)\) coincides with
\(\mathcal{N}_{0}\). Therefore, the topologies induced by them are equivalent. This completes the proof. □
(II) Probabilistic seminorm and locally convex probabilistic seminormed spaces
It is easy to prove that following lemma.
(III) The cases of simple spaces and
α-simple spaces
In view of Theorem
2.5, we easily get the corollary.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.