On αŠerstnev probabilistic normed spaces
 Bernardo LafuerzaGuillén^{1} and
 Mahmood Haji Shaabani^{2}Email author
https://doi.org/10.1186/1029242X2011127
© LafuerzaGuillén and Shaabani; licensee Springer. 2011
Received: 3 June 2011
Accepted: 30 November 2011
Published: 30 November 2011
Abstract
In this article, the condition αŠ is defined for α ∈]0, 1[∪]1, +∞[and several classes of αŠerstnev PN spaces, the relationship between αsimple PN spaces and αŠerstnev PN spaces and a study of PN spaces of linear operators which are αŠerstnev PN spaces are given.
2000 Mathematical Subject Classification: 54E70; 46S70.
Keywords
1. Introduction
Šerstnev introduced the first definition of a probabilistic normed (PN) space in a series of articles [1–4]; he was motivated by the problems of best approximation in statistics. His definition runs along the same path followed in order to probabilize the notion of metric space and to introduce Probabilistic Metric spaces (briefly, PM spaces).
For the reader's convenience, now we recall the most recent definition of a Probabilistic Normed space (briefly, a PN space) [5]. It is also the definition adopted in this article and became the standard one, and, to the best of the authors' knowledge, it has been adopted by all the researchers who, after them, have investigated the properties, the uses or the applications of PN spaces. This new definition is suggested by a result ([[5], Theorem 1]) that sheds light on the definition of a "classical" normed space. The notation is essentially fixed in the classical book by Schweizer and Sklar [6].
In the context of the PN spaces redefined in 1993, one introduces in this article a study of the concept of αŠerstnev PN spaces (or generalized Šerstnev PN spaces, see [7]). This study, with α ∈]0, 1[∪]1, +∞[has never been carried out.
Some preliminaries
The set Δ, as well as its subsets, can partially be ordered by the usual pointwise order; in this order, ε_{0} is the maximal element in Δ^{+}. The subset ${\mathcal{D}}^{+}\subset {\mathrm{\Delta}}^{+}$ is the subset of the proper d.f.'s of Δ^{+}.
 (1)
τ(F, ε_{0}) = F,
 (2)
τ(F, G) = τ(G, F),
 (3)
τ(F, G) ≤ τ(H, K) whenever F ≤ H, G ≤ K,
 (4)
τ(τ(F, G), H) = τ(F, τ(G, H)).
for all F, G ∈ Δ^{+} and all x ∈ ℝ [6]. Here, T is a continuous tnorm and T* is the corresponding continuous tconorm, i.e., both are continuous binary operations on [0, 1] that are commutative, associative, and nondecreasing in each place; T has 1 as identity and T* has 0 as identity. If T is a tnorm and T* is defined on [0, 1] × [0, 1] via T*(x, y): = 1  T(1  x, 1  y), then T* is a tconorm, specifically the tconorm of T.
Definition 1.2. A PM space is a triple $\left(S,\mathcal{F},\tau \right)$ where S is a nonempty set (whose elements are the points of the space), $\mathcal{F}$ is a function from S × S into Δ^{+}, τ is a triangle function, and the following conditions are satisfied for all p, q, r in S:
(PM1) $\mathcal{F}\left(p,p\right)={\epsilon}_{0}.$
(PM2) $\mathcal{F}\left(p,q\right)\ne {\epsilon}_{0}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}p\ne q.$
(PM3) $\mathcal{F}\left(p,q\right)=\mathcal{F}\left(q,p\right).$
(PM4) $\mathcal{F}\left(p,\phantom{\rule{2.77695pt}{0ex}}r\right)\ge \tau \left(\mathcal{F}\left(p,\phantom{\rule{2.77695pt}{0ex}}q\right),\phantom{\rule{2.77695pt}{0ex}}\mathcal{F}\left(q,\phantom{\rule{2.77695pt}{0ex}}r\right)\right).$
Definition 1.3. (introduced by Šerstnev [1] about PN spaces: it was the first definition) A PN space is a triple (V, ν, τ), where V is a (real or complex) linear space, ν is a mapping from V into Δ^{+} and τ is a continuous triangle function and the following conditions are satisfied for all p and q in V:
(N1) ν_{ p } = ε_{0} if, and only if, p = θ (θ is the null vector in V);
Notice that condition (Š) implies
(N2) ∀p ∈ V ν_{ p } = ν_{ p }.
Definition 1.4. (PN spaces redefined: [5]) A PN space is a quadruple (V, ν, τ, τ*), where V is a real linear space, τ and τ* are continuous triangle functions such that τ ≤ τ*, and the mapping ν : V → Δ^{+} satisfies, for all p and q in V, the conditions:
(N1) ν_{ p } = ε_{0} if, and only if, p = θ (θ is the null vector in V);
(N2) ∀p ∈ V ν_{ p } = ν_{p};
(N3) ν_{p+q}≥ τ (ν_{p}, ν_{ q });
(N4) ∀ α ∈ [0, 1] ν_{p} ≤ τ* (ν_{α p}, ν_{(1α) p}).
then (V,ν, τ, τ*) is called a Probabilistic PseudoNormed space (briefly, a PPN space). If ν satisfies the conditions (N1) and (N2), then (V,ν, τ, τ*) is said to be a Probabilistic seminormed space (briefly, PSN space). If τ = τ_{ T } and τ* = τ_{ T* } for some continuous tnorm T and its tconorm T*, then (V, ν, τ_{ T }, τ_{ T* }) is denoted by (V, ν, T) and is called a Menger PN space. A PN space is called a Šerstnev space if it satisfies (N1), (N3) and condition (Š).
The system of neighborhoods {N_{ p }(λ): p ∈ V, λ > 0} determines a Hausdorff topology on V, called the strong topology.
A PN space (V,ν, τ, τ*) is said to be strongly complete if every strong Cauchy sequence in V is strongly convergent.
Definition 1.7. [10] A subset A of a PN space (V,ν, τ, τ*) is said to be $\mathcal{D}$compact if every sequence of points of A has a convergent subsequence that converges to a member of A.
where l^{} f(x) denotes the left limit of the function f at the point x and ϕ_{ A }(x): = inf{ν_{p}(x): p ∈ A}.
 (a)
certainly bounded, if R_{ A }(x_{0}) = 1 for some x_{0} ∈]0, +∞ [;
 (b)
perhaps bounded, if one has R_{ A }(x) < 1 for every x ∈]0, ∞ [, and l^{} R_{A}(+∞) = 1.
Moreover, the set A will be said to be $\mathcal{D}$bounded if either (a) or (b) holds, i.e., if ${R}_{A}\in {\mathcal{D}}^{+}$.
Definition 1.9. [12] A subset A of a topological vector space (briefly, TV space) E is topologically bounded, if for every sequence {λ_{ n }}_{ n } of real numbers that converges to 0 as n → ∞ and for every sequence {p_{ n }}_{ n } of elements of A, one has λ_{ n }p_{ n } →θ in the topology of E. Also by Rudin [[13], Theorem 1.30], A is topologically bounded if, and only if, for every neighborhood U of θ, we have A ⊆ tU for all sufficiently large t.
From the point of view of topological vector spaces, the most interesting PN spaces are those that are not Šerstnev (or 1Šerstnev) spaces. In these cases vector addition is still continuous (provided the triangle function is determined by a continuous tnorm), while scalar multiplication, in general, is not continuous with respect to the strong topology [14].
We recall from [15]: for 0 < b ≤ + ∞, let M_{ b } be the set of mtransforms consisting of all continuous and strictly increasing functions from [0, b] onto [0, +∞]. More generally, let $\stackrel{\u0303}{M}$ be the set of nondecreasing leftcontinuous functions ϕ : [0, +∞] [0, +∞], with ϕ (0) = 0, ϕ (+∞) = +∞ and ϕ(x) > 0 for x > 0. Then ${M}_{b}\subseteq \stackrel{\u0303}{M}$ once m is extended to [0, +∞] by m(x) = +∞ for all x ≥ b. Note that a function $\varphi \in \stackrel{\u0303}{M}$is bijective if, and only if, ϕ ∈ M_{+∞}. Sometimes, the probabilistic norms ν and ν' of two given PN spaces satisfy ν' = νϕ for some ϕ ∈ M_{+∞}. not necessarily bijective. Let $\widehat{\varphi}$ be the (unique) quasiinverse of ϕ which is leftcontinuous. Recall from [[6], p. 49] that $\widehat{\varphi}$ is defined by $\widehat{\varphi}\left(0\right)=0,\phantom{\rule{2.77695pt}{0ex}}\widehat{\varphi}\left(+\infty \right)=+\infty $ and $\widehat{\varphi}\left(t\right)=sup\left\{u:\varphi \left(u\right)<t\right\}$ for all 0 < t < +∞. It follows that $\widehat{\varphi}\left(\varphi \left(x\right)\right)\le x$ and $\varphi \left(\widehat{\varphi}\left(y\right)\right)\le y$ for all x and y.
Definition 1.10. A quadruple (V,ν, τ, τ*) is said to satisfy the ϕŠerstnev condition if
$\left(\varphi \mathsf{\text{\u0160}}\right){\nu}_{\lambda p}\left(x\right)={\nu}_{p}\left(\hat{\varphi}\left(\frac{\varphi \left(x\right)}{\mid \lambda \mid}\right)\right)$ for every p ∈ V, for every x > 0 and λ ∈ ℝ\{0}.
A PN space (V,ν, τ, τ*) which satisfies the ϕŠerstnev condition is called a ϕŠerstnev PN space.
Example 1.1. If ϕ(x) = x^{1/α}for a fixed positive real number α, the condition (ϕŠ) takes the form
$\left(\alpha \u0160\right){\nu}_{\lambda p}\left(x\right)={\nu}_{p}\left(\frac{x}{\mid \lambda {\mid}^{\alpha}}\right)$ for every p ∈ V, for every x > 0 and λ ∈ ℝ\{0}.
PN spaces satisfying the condition (αŠ) are called αŠerstnev PN spaces. For α = 1 one has a Šerstnev (or 1Šerstnev) PN space.
where α ≥ 0. Then the pair (V,ν) will be called the αsimple space generated by (V,  · ) and G.
The αsimple space generated by (V,  · ) and G is, as immediately checked, a PSN space; it will be denoted by (V,  · , G; α).
A PSN space (V,ν) is said to be equilateral if there is d.f. F ∈Δ^{+}, different from ε_{0} and from ε_{∞}, such that, for every p ≠ θ, ν_{ p } = F. In Definition 1.11, if α = 0 and α = 1, one obtains the equilateral and simple space, respectively.
It is easily shown that the dominance relation is reflexive and antisymmetric. However, although not, in general, transitive, as examples due to Sherwood [17] and Sarkoci [18] show.
2. Main results (I)αsimple PN space and some classes of αŠerstnev PN spaces
In this section, we give several classes of αŠerstnev PN spaces and characterize them. Also, we investigate the relationship between αsimple PN spaces and αŠerstnev PN spaces.
 (a)
A is $\mathcal{D}$bounded.
 (b)
A is bounded, namely, for every n ∈ N and for every p ∈ A, there is k ∈ N such that ν_{ p/k }(1/n) > 1  1/n.
 (c)
A is topologically bounded.
Theorem 2.2. Let (V,ν, τ, τ*) be an αŠerstnev PN space. Then, for a subset A ⊆ V, the same statements as in Theorem 2.1 are equivalent.
Definition 2.1. The PN space (V,ν, τ, τ*) is called strict whenever $\nu \left(V\right)\subseteq {\mathcal{D}}^{+}$.

If the scalar multiplication η : ℝ × V → V is continuous at the first place with respect to ν, then it is with respect to ν'. If W_{1} is a TV PN space. then it is with W_{2}.
It was proved in [[14], Theorem 4] that, if the triangle function τ* is Archimedean, i.e., if τ* admits no idempotents other than ε_{0} and ε_{∞} [6], and ν_{ p } ≠ ε_{∞} for all p ∈ V, then for every p ∈ V the map from ℝ into V defined by λ α λp is continuous and, as a consequence of [14] the PN space (V,ν, τ, τ*) is a TV space.
Theorem 2.3. [7] Let $\varphi \in \stackrel{\u0303}{M}$ such that ${lim}_{x\to \infty}\phantom{\rule{0.3em}{0ex}}\widehat{\varphi}\left(x\right)=\infty $. A ϕŠerstnev PN space is a TV space if, and only if, it is strict.
Corollary 2.2. An αŠerstnev PN space (V,ν, τ, τ*) is a TV space if, and only if, it is strict.
Corollary 2.3. Let (V,ν, τ, τ*) be an αŠerstnev PN space and τ* be Archimedean and ν_{ p } ≠ ε_{∞} for all p ∈ V. Then the probabilistic norm ν is strict.
 (i)
It is an αŠerstnev PN space.
 (ii)
It is an αsimple PN space.
Theorem 2.5. Every αsimple space satisfies the (αŠ) condition for α ∈]0, 1[∪]1, +∞[.
Proof. Let (V,  · , G; α) be an αsimple PN space with α ∈]0, 1[∪]1, +∞[. From ${\nu}_{p}\left(t\right)=G\left(\frac{t}{\parallel p{\parallel}^{\alpha}}\right)$ for every t ∈ [0, ∞], one has ${\nu}_{\lambda p}\left(t\right)=G\left(\frac{t}{\parallel \lambda p{\parallel}^{\alpha}}\right)=G\left(\frac{t}{\mid \lambda {\mid}^{\alpha}\parallel p{\parallel}^{\alpha}}\right)$ and ${\nu}_{p}\left(\frac{t}{\mid \lambda {\mid}^{\alpha}}\right)=G\left(\frac{\frac{t}{\mid \lambda {\mid}^{\alpha}}}{\parallel p\parallel \alpha}\right)=G\left(\frac{t}{\mid \lambda {\mid}^{\alpha}\parallel p\parallel \alpha}\right)$. Then ${\nu}_{\lambda p}\left(t\right)={\nu}_{p}\left(\frac{t}{\mid \lambda {\mid}^{\alpha}}\right)$ and hence (V,  · , G; α) is an α Šerstnev PN space.
An αsimple space with a ≠ 1 does not satisfy the condition (Š) as seen in the following theorem.
Theorem 2.6. Let (V,  · ) be a normed space, G a d. f. different from ε_{0} and ε_{∞}, and let α be a positive real number different from 1. Then the αsimple space (V,  · , G; α) satisfies the condition (Š) only when G = constant in (0, +∞).
Finally, since the function of β given by β^{ α }+(1 β)^{ α }, being continuous in the compact set [0, 1], takes all values between 1 and 2^{1α}, and $\frac{x}{\parallel p{\parallel}^{\alpha}}$ takes any value in (0, ∞), one concludes that G(x) = G(λx) for every λ ∈ [1, 2^{α1}] (if α > 1) or for every λ ∈ [2^{α1}, 1] (if α < 1). Then G = constant in (0, +∞) and the proof is concluded.
Notice that if G = constant in (0, +∞), then (V,  · , G; α) is a PN space of Šerstnev under any triangle function τ.
Among all αsimple spaces (V,  · , G; α) one has the αsimple PN spaces considered in Theorem 3.2 in [19], i.e., the Menger PN space given by $\left(V,\phantom{\rule{2.77695pt}{0ex}}\nu ,\phantom{\rule{2.77695pt}{0ex}}{\tau}_{{T}_{{G}^{*}}},\phantom{\rule{2.77695pt}{0ex}}{\tau}_{{{T}^{*}}_{{G}^{*}}}\right)$, and in Theorem 3.1 in [19], i.e., the Menger PN space given by $\left(V,\phantom{\rule{2.77695pt}{0ex}}\nu ,\phantom{\rule{2.77695pt}{0ex}}{\tau}_{{T}_{{G}^{*}}},\phantom{\rule{2.77695pt}{0ex}}{\tau}_{{{T}^{*}}_{{G}^{}}}\right)$. From Theorems 3.1 and 3.2 in [19] the following result holds:
Corollary 2.4. Every αsimple PN spaces of the type considered in Theorems 3.1 and 3.2 in [19] are (αŠ) PN spaces of Menger.
Next, we give an example of an αŠerstnev PN space which is also an αsimple PN space.
The preceding example suggests the following theorem.
Theorem 2.7. Let (V,  · ) be a normed space and dim V = 1. Then every αŠerstnev PN space is an αsimple PN space.
and (V,ν, τ, τ*) is an αsimple PN space.
The converse of Theorem 2.5 fails as is shown in the following examples.
Therefore, (ℝ^{2},ν, Π_{ M }, Π_{ M }) is an αŠerstnev PN space.
which is a contradiction.
It is not difficult to show that (ℝ^{2},ν, Π_{Π}, Π_{ M }) is an αŠerstnev PN space, but it is not an αsimple PN space.
Let V be a normed space with dim V > 1 (finite or infinite dimensional) and {e_{ i }}_{i∈I}be a basis for V, where e_{ i } = 1. We can construct some examples on V, similar to Examples 2.3 and 2.4, of αŠerstnev PN spaces which are not αsimple PN spaces.
 (b)Let 0 < α = 1. For p ∈ V, define ν by ν_{ θ } = ε_{0} and${v}_{p}\left(x\right):=\left\{\begin{array}{cc}{e}^{\frac{{\left\lambda \right}^{\alpha}}{x}}\hfill & p=\lambda {e}_{{i}_{0}}\left(\lambda \in \mathbb{R}\backslash \left\{0\right\}\right),\hfill \\ {\epsilon}_{\infty}\left(x\right)\hfill & \mathsf{\text{otherwise}}\hfill \end{array}\right.$
Then (V, ν, Π_{Π}, Π_{ M }) is an αŠerstnev PN space, but it is not an αsimple PN space.
Proposition 2.1. Let (V,ν, τ, τ*) be an αŠerstnev PN space. Then, its completion $\left(\widehat{V},\phantom{\rule{2.77695pt}{0ex}}\nu ,\phantom{\rule{2.77695pt}{0ex}}\tau ,\phantom{\rule{2.77695pt}{0ex}}{\tau}^{*}\right)$ is also an αŠerstnev PN space.
Proof. By [[20], Theorem 3], the completion of a PN space is a PN space.
The following result concerns finite products of PN spaces [21]. In a given PN space (V,ν, τ, τ*) the value of the probabilistic norm of p ∈ V at the point x will be denoted by ν(p)(x) or by ν_{ p }(x).
Then the τ_{ T } product (V_{1} × V_{2}, ν, τ, τ*) is an αŠerstnev PN space under τ and τ*.
for every α ∈]0, 1[∪]1, +∞ [. It is easy to check the axioms (N1) and (N2) hold.
which concludes the proof.
is the Π_{ M } product and it is an αŠerstnev PN space under Π_{Π} and Π_{ M }.
Proof. The conclusion follows from Lemma 2.1 in [22].
3. Main results (II)PN spaces of linear operators which are αŠerstnev PN spaces
Let $\left({V}_{1},\nu ,{\tau}_{1},{\tau}_{1}^{*}\right)$ and $\left({V}_{2},{\nu}^{\prime},{\tau}_{2},{\tau}_{2}^{*}\right)$ be two PN spaces and let L = L(V_{1}, V_{2}) be the vector space of linear operators T : V_{1} → V_{2}.
As was shown in [14], PN spaces are not necessarily topological linear spaces.
We recall that for a given linear map T ∈ L, the map ${\nu}^{A}:L\to {\mathcal{D}}^{+}$ is defined via ${\nu}^{A}\left(T\right):={R}_{TA}^{\prime}$.
where α_{1}, α_{2}, ..., α_{ n } are scalars and u_{1}, u_{2}, ..., u_{ n } belong to H; a subset H of V is a Hamel basis if, and only if, it is a maximal linear independent set [25]. This condition ensures that (L(V_{1}, V_{2}), ν^{ A }, τ, τ*) is a PN space as we can see in [[26], Theorem 3.2].
Corollary 3.1. Let A be an absorbing subset of a PN space $\left({V}_{1},\nu ,{\tau}_{1},{\tau}_{1}^{*}\right)$. If $\left({V}_{2},{\nu}^{\prime},{\tau}_{2},{\tau}_{2}^{*}\right)$ is an αŠerstnev PN space, then $\left(L\left({V}_{1},{V}_{2}\right),{\nu}^{A},{\tau}_{2},{\tau}_{2}^{*}\right)$ is an αŠerstnev PN space; convergence in the probabilistic norm ν^{ A } is equivalent to uniform convergence of operators on A.
Proof. See Theorem 3.1 and [[26], Corollary 3.1].
Corollary 3.2. If V_{2} is α complete αŠerstnev PN space, then $\left(L\left({V}_{1},{V}_{2}\right),{\nu}^{A},{\tau}_{2},{\tau}_{2}^{*}\right)$ is also a complete αŠerstnev PN space.
Proof. See Theorem 3.1 and [[26], Theorem 4.1].
In the remainder of this section, we study some classes of αŠerstnev PN spaces of linear operators. We investigate the relationship between $\left(L\left({V}_{1},{V}_{2}\right),{\nu}^{A},{\tau}_{2},{\tau}_{2}^{*}\right)$, and $\left({V}_{1},\nu ,{\tau}_{1},{\tau}_{1}^{*}\right)$ or $\left({V}_{2},{\nu}^{\prime},{\tau}_{2},{\tau}_{2}^{*}\right)$ and we set some conditions such that $\left(L\left({V}_{1},{V}_{2}\right),{\nu}^{A},{\tau}_{2},{\tau}_{2}^{*}\right)$ becomes a TV space.
Theorem 3.2. Let A be a subset of a PN space $\left({V}_{1},\nu ,{\tau}_{1},{\tau}_{1}^{*}\right)$ that contains a Hamel basis for V_{1} and $\left({V}_{2},{\nu}^{\prime},{\tau}_{2},{\tau}_{2}^{*}\right)$ be an αŠerstnev PN space. If $\left(L\left({V}_{1},{V}_{2}\right),{\nu}^{A},{\tau}_{2},{\tau}_{2}^{*}\right)$ is a TV space, then $\left({V}_{2},{\nu}^{\prime},{\tau}_{2},{\tau}_{2}^{*}\right)$ is a TV space.
Then, ${\nu}^{A}\left(T\right)=\underset{x\to \infty}{lim}inf\left\{{\nu}_{Tp}^{\prime}\left(x\right)\mid p\in A\right\}\le \underset{x\to \infty}{lim}{\nu}_{\lambda q}^{\prime}\left(x\right)<1$. So ${\nu}^{A}\left(T\right)\in {\mathrm{\Delta}}^{+}\backslash {\mathcal{D}}^{+}$ and $\left(L\left({V}_{1},{V}_{2}\right),{\nu}^{A},{\tau}_{2},{\tau}_{2}^{*}\right)$ is not a TV space, which is a contradiction.
The following theorem shows that the converse of the preceding theorem does not hold.
 (i)
If sup{λ : λ ∈ ℝ, λp ∈ A} = ∞ for some p ∈ A and p ≠ θ, then $\left(L\left({V}_{1},{V}_{2}\right),{\nu}^{A},{\tau}_{2},{\tau}_{2}^{*}\right)$ is not a TV space.
 (ii)
If $\left(L\left({V}_{1},{V}_{2}\right),{\nu}^{A},{\tau}_{2},{\tau}_{2}^{*}\right)$ is a TV space, then sup{λ : λ ∈ ℝ, λp ∈ A} < ∞ for every p ∈ A and p ≠ θ.
Then ${\nu}^{A}\left(T\right)\in {\mathrm{\Delta}}^{+}\backslash {\mathcal{D}}^{+}$.
Corollary 3.3. Let $\left({V}_{1},\nu ,{\tau}_{1},{\tau}_{1}^{*}\right)$ be a PN space and $\left({V}_{2},{\nu}^{\prime},{\tau}_{2},{\tau}_{2}^{*}\right)$ be an αŠerstnev PN space. Then $\left(L\left({V}_{1},{V}_{2}\right),{\nu}^{{V}_{1}},{\tau}_{2},{\tau}_{2}^{*}\right)$ is not a TV space.
Example 3.1. Suppose that A is a subset of a PN space $\left({V}_{1},\nu ,{\tau}_{1},{\tau}_{1}^{*}\right)$ that contains a Hamel basis for V_{1}. Let α ∈ ]0, 1] and V_{2} be a normed space. If we define ν : V_{2} → Δ^{+} by ν_{ θ } = ε_{0} and ${\nu}_{p}\left(x\right):=e\frac{\parallel p{\parallel}^{\alpha}}{x}$ for p ≠ θ and x > 0, then (V_{2},ν, Π_{Π}, Π_{ M }) is a TV space. If sup{λ : λ ∈ ℝ, λp ∈ A} = ∞ for some p ∈ A and p ≠ θ, then $\left(L\left({V}_{1},{V}_{2}\right),{\nu}^{A},{\tau}_{2},{\tau}_{2}^{*}\right)$ is not a TV space.
 (a)
If V is a finitedimensional PN space and ${\mathcal{T}}_{1}$, ${\mathcal{T}}_{2}$ are two topologies on V that make it into a TV space, then ${\mathcal{T}}_{1}={\mathcal{T}}_{2}$.
 (b)
If V is a TV PN space and M is a finitedimensional linear manifold in V, then M is closed.
If (X,  · ) is a normed space, we say that A ⊆ X is classically bounded if, and only if, there is an M ∈ ℝ such that for each a ∈ A, a ≤ M. Now, we state the following theorem that we will use it frequently in the rest of this section.
 (a)
V is normable.
 (b)
V is complete.
 (c)
A is $\mathcal{D}$compact if, and only if, it is compact.
 (d)
A is $\mathcal{D}$bounded if, and only if, it is topologically bounded if, and only if, it is classically bounded.
 (e)
A is $\mathcal{D}$compact if, and only if, it is compact if, and only if, it is closed and $\mathcal{D}$bounded.
Proof. (a) Let {e_{1}, e_{2}, ..., e_{ n }} be a Hamel basis for V. Then, for every p in V, there are α_{1}, α_{2}, ..., α_{ n } in ℝ such that p = α_{1}e_{1}+α_{2}e_{2}+···+α_{ n }e_{ n }. If $\parallel p\parallel \phantom{\rule{2.77695pt}{0ex}}:=\sqrt{{\alpha}_{1}^{2}+{\alpha}_{2}^{2}+\cdots +{\alpha}_{n}^{2}}$, then  ·  defines a norm on V. It is easy to check that (V,  · ) is a TV space. By Lemma 3.1, if ${\mathcal{T}}_{1}$ is the strong topology and ${\mathcal{T}}_{2}$ is the norm topology on V which is defined as above, then ${\mathcal{T}}_{1}={\mathcal{T}}_{2}$. So V is normable.
 (i)
A sequence {p_{ n }}_{ n } is a strong Cauchy sequence if, and only if, it is Cauchy sequence in the norm topology.
 (ii)
A sequence {p_{ n }}_{ n } is a strongly convergent to p ∈ V if, and only if, it is convergent to p in the norm topology.
 (b)
Let {p_{ n }}_{ n } be a strong Cauchy sequence. Then {p_{ n }}_{ n } is a Cauchy sequence in the norm topology. Since $\left(V,{\mathcal{T}}_{2}\right)$ is complete, there is p ∈ V such that p_{ n } → p in $\left(V,{\mathcal{T}}_{2}\right)$ as n → ∞. So p_{n →}p in $\left(V,{\mathcal{T}}_{1}\right)$ as n → ∞. Hence, the result follows.
 (c)
Since ${\mathcal{T}}_{1}={\mathcal{T}}_{2}$, the identity map $I:\left(V,{\mathcal{T}}_{1}\right)\to \left(V,{\mathcal{T}}_{2}\right)$ is a homeomorphism. Hence, [[28], Theorem 28.2] and the arguments before part (b) give the desired conclusion.
 (d)
By the fact that ${\mathcal{T}}_{1}={\mathcal{T}}_{2}$ and Theorem 2.2, the results follow.
 (e)
Let (ℝ^{ n }, ·) be Euclidean space and {e_{1}, e_{2}, ..., e_{ n }} be a Hamel basis for V. We define $f:\left(V,{\mathcal{T}}_{2}\right)\to \left({\mathbb{R}}^{n},\parallel \cdot \parallel \right)$ by f(α_{1}e_{1} + a_{2}e_{2} + ··· + α_{ n }e_{ n }) = (a_{1}, a_{2}, ..., a_{ n }). It is clear that f is a homeomorphism. Since a subset in ℝ^{ n } is compact if, and only if, it is closed and bounded, A is compact in the strong topology if, and only if, it is closed and $\mathcal{D}$bounded.
 (a)
$\left(L\left({V}_{1},{V}_{2}\right),{\nu}^{A},{\tau}_{2},{\tau}_{2}^{*}\right)$ is a TV space if, and only if, TA is $\mathcal{D}$bounded for every T∈ L(V_{1}, V_{2}).
 (b)
Let V_{1} = V_{2} = V. If $\left(L\left(V,V\right),{\nu}^{A},{\tau}_{2},{\tau}_{2}^{*}\right)$ is a TV space, then A is $\mathcal{D}$bounded.
 (c)
Let dim V_{1} < ∞. If A is $\mathcal{D}$bounded, then $\left(L\left({V}_{1},{V}_{2}\right),{\nu}^{A},{\tau}_{2},{\tau}_{2}^{*}\right)$ is a TV space.
 (d)
Let dim V_{1} < ∞ and dim V_{1} ≤ dim V_{2}. Then $\left(L\left({V}_{1},{V}_{2}\right),{\nu}^{A},{\tau}_{2},{\tau}_{2}^{*}\right)$ is a TV space if, and only if, A is $\mathcal{D}$bounded.
 (c)
It is enough to show that TA is $\mathcal{D}$bounded for every T ∈ L(V_{1}, V_{2}). Since dim V_{1} < ∞, Theorem 3.4 and [[27], p. 70] imply that T : V_{1} → RangT is continuous for every T ∈ L(V_{1}, V_{2}). Also by [[11], Theorem 2.2], $\u0100$ is $\mathcal{D}$bounded. Hence, Theorem 3.4 concludes that $\u0100$ is compact. Then, $T\u0100$ is compact. Invoking Theorem 3.4, it follows that TA is $\mathcal{D}$bounded.
 (d)
Let $\left(L\left({V}_{1},{V}_{2}\right),{\nu}^{A},{\tau}_{2},{\tau}_{2}^{*}\right)$ be a TV space. Since dim V_{1} < ∞ and dim V_{1} ≤ dim V_{2}, we can define a onetoone linear operator T: V_{1} → V_{2}. Then, by Theorem 3.4 and [[27], p. 70], T : V_{1} → RangT is a homeomorphism. Since TA is $\mathcal{D}$bounded, $\overline{TA}$ is compact. So, ${T}^{1}\left(\overline{TA}\right)$ is compact and therefore A is $\mathcal{D}$bounded.
Conversely, it follows from part (c).
Theorem 3.6. Let $\left({V}_{1},\nu ,{\tau}_{1},{\tau}_{1}^{*}\right)$ and $\left({V}_{2},{\nu}^{\prime},{\tau}_{2},{\tau}_{2}^{*}\right)$ be αŠerstnev PN spaces and A be a subset of V_{1} that contains a Hamel basis for V_{1}. If dim V_{1} < ∞, $\left({V}_{1},\nu ,{\tau}_{1},{\tau}_{1}^{*}\right)$ is a TV space and A is $\mathcal{D}$bounded, then $\left({V}_{2},{\nu}^{\prime},{\tau}_{2},{\tau}_{2}^{*}\right)$ is a TV space if, and only if, $\left(L\left({V}_{1},{V}_{2}\right),{\nu}^{A},{\tau}_{2},{\tau}_{2}^{*}\right)$ is a TV space.
Proof. By Theorems 3.1 and 3.5(c), the proof is obvious.
Example 3.2. Let α ∈]0, 1] and n > m. We define ν : ℝ^{ n } → Δ^{+} by ν_{ θ } = ε_{0} and ${\nu}_{p}\left(x\right):={e}^{\frac{\parallel p{\parallel}^{\alpha}}{x}}$ for p ∈ ℝ^{ n } and x > 0. Also we define ν': ℝ^{ m } → Δ^{+} by ${\nu}_{\theta}^{\prime}={\epsilon}_{0}$ and ${\nu}_{p}\left(x\right):={e}^{\frac{\parallel p{\parallel}^{\alpha}}{x}}$ for p ∈ ℝ^{ m } and x > 0. Hence (ℝ^{ n }, ν, Π_{Π}, Π_{ M }) and (ℝ^{ m }, ν, Π_{Π}, Π_{ M }) are αŠerstnev PN spaces; furthermore, they are TV spaces. Then A is classically bounded in ℝ^{ n } if, and only if, (L(ℝ^{ n }, ℝ^{ m }), ν^{ A }, Π_{Π}, Π_{ M }) is a TV space.
The following example shows that the converse of Theorem 3.3 is not true.
Example 3.3. Let α ∈]0, 1] and n > m. We define (ℝ^{ n },ν , Π_{Π}, Π_{ M }) and (ℝ^{ m }, ν, Π_{Π}, Π_{ M }) in a similar way to the earlier example. If A = {(k, k^{2}, 0, ..., 0): k ∈ N}∪{(1, 0, 0, ..., 0), (0, 1, 0, ..., 0), ..., (0, 0, ..., 0, 1)}, then A is a subset of ℝ^{ n }that contains a Hamel basis for ℝ^{ n }. Although sup{λ : λ ∈ ℝ, λp ∈ A} < ∞ for every p ∈ A and p ≠ θ, (L(ℝ^{ n }, ℝ^{ m }), ν^{ A }, Π_{Π}, Π_{ M }) is not a TV space, because A is not $\mathcal{D}$bounded.
Declarations
Acknowledgements
The authors wish to thank C. Sempi for his helpful suggestions. Bernardo Lafuerza Guillén was supported by grants from Ministerio de Ciencia e Innovación (MTM200908724).
Authors’ Affiliations
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