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In this present work, the topological structure of 2-normed spaces is studied from the viewpoint of topological vector spaces. A separating family of seminorms is induced from a given 2-norm, and the criteria for metrizability and normability of 2-normed spaces are investigated using this family. A sufficient condition for metrizability and a necessary and sufficient condition for normability of 2-normed spaces are deduced during this investigation.

The history of functional analysis is not very old. The idea grew up in early twentieth century. Researchers felt the necessity of this subject during the studies of integration theory and integral equations. However, Banach was the pioneer of formal functional analysis. In 1922, he defined normed linear spaces as a set of axioms. Since then, mathematicians keep on trying to find a proper generalization of this concept. The first notable attempt was by Vulich [10]. He introduced K-normed space in 1937. In another process of generalization, Gähler [1] introduced 2-metric in 1963. As a continuation of his research, Gähler [2] proposed a mathematical structure, called 2-normed space, as a generalization of normed linear spaces. As a further extension, he introduced n-metric and n-norm in his subsequent works [3,4,5] and regarded normed linear spaces as 1-normed spaces. However, many researchers disagree to consider 2-norm and n-norm as generalization of norm. In spite of this disagreement, several researchers have worked on this topic for decades. They have found out many interesting properties of this space and lots of fixed point theorems are established.

In this article, we have studied 2-normed spaces from the viewpoint of topological vector spaces. A separating family of seminorms is induced from the 2-norm and used it as a one-way bridge between 2-normed spaces and topological vector spaces. Our method gives an alternative proof of the statement that every 2-normed space is a locally convex topological vector space. The line of proof gives more insight of the space and helps to find a sufficient condition for metrizability. Additionally, we have been able to find out a necessary and sufficient condition for normability of a 2-normed space. As a complementary result, a norm is derived for such normable 2-normed space. We also have shown that one can derive a norm from a given 2-norm in such a way that convergence of sequences for both 2-norm and the derived norm are equivalent if and only if the 2-norm is normable. This answers an open problem given by Gunawan assnd Mashadi [7].

is called the natural topology or the topology generated by 2-metric in X. The set of all possible finite intersections of such neighbourhoods forms a base for this topology. Gähler considered this as the topology of 2-metric space.

So a 2-normed space can be treated as a 2-metric space with the induced 2-metric \(\sigma\) defined in Theorem 1. The topology of a 2-normed space is defined as the topology of the 2-metric space induced by that 2-norm. Moreover, if \(\sigma\) satisfies property (K), then it is said that the 2-norm satisfies property (K).

[2] A 2-normed space \((X,\Vert \cdot ,\cdot \Vert )\) has property (K) if and only if for all \(a^*,b^*\in X\) with \(\Vert {a^*},{b^*}\Vert \ne 0\) the norm on X defined by

A sequence \(\lbrace x_n\rbrace\) in a 2-normed space \((X,\Vert \cdot ,\cdot \Vert )\) is called a Cauchy sequence if for every \(y\in X\) the following holds: \(\lim \limits _{m,n\rightarrow \infty }\Vert {x_n-x_m},{y}\Vert =0\).

[6] A sequence \(\lbrace x_n\rbrace\) in a 2-normed space \((X,\Vert \cdot ,\cdot \Vert )\) is said to converge to \(x\in X\) if \(\lim \limits _{n\rightarrow \infty }\Vert {x_n-x},{y}\Vert =0\) for all \(y\in X\).

[11] A topology \(\uptau\) on a vector space X over the field \(\mathbb {K}\) (where \(\mathbb {K}=\mathbb {R}\) or \(\mathbb {C}\)) is called a vector topology if the following two conditions hold:

Our next theorem leads to the conclusion that a 2-normed space is a topological vector space. This was already done by Gähler in [2]. But, here we prove it in a different way. Our method is more constructive in nature. It helps us to find out a necessary and sufficient condition for normability which is discussed in Sect. 4 of this article.

Let \((X,\Vert \cdot ,\cdot \Vert )\) be a 2-normed space. Then, the topology of \((X,\Vert \cdot ,\cdot \Vert )\) coincides with the topological vector space induced by the separating family of seminorms \({\mathscr {P}}\).

Therefore, \({\mathscr {U}}\subseteq \uptau _{\mathscr {P}}\). Thus, the smallest topology containing the elements of \({\mathscr {U}}\), which is the natural topology of the 2-normed space, is a subset of \(\uptau _{\mathscr {P}}\).

Let \((X,\Vert \cdot ,\cdot \Vert )\) be a 2-normed space with a countable Hamel basis \({\mathscr {B}}\). Then, the set \({\mathcal {B}}_0\) defined in Proposition 5 is countable. Hence, the topology of \((X,\Vert \cdot ,\cdot \Vert )\) has a countable local base, and so, by Theorem 3, \((X,\Vert \cdot ,\cdot \Vert )\) is metrizable. \(\square\)

A 2-normed space \((X,\Vert \cdot ,\cdot \Vert )\) is said to satisfy property \((K_m)\) for some \(m\in \mathbb {N}\) if there exist linearly independent elements \(a_1,a_2,\ldots ,a_m\) in X such that for every sequence of points \(x,x_1,x_2,x_3,\ldots\) of X,

There exists a Hamel basis \({\mathscr {B}}\) of X such that \(\lbrace a_1,a_2,\ldots ,a_m\rbrace\) is a subset of \({\mathscr {B}}\). From Proposition 4 and Theorem 5, we can conclude that the topology of \((X,\Vert \cdot ,\cdot \Vert )\) coincides with the locally convex topological vector space generated by the separating family of seminorms

Let X be a 2-normed space, with Hamel basis \({\mathscr {B}}\), which is normable. Then, there exists a bounded open set G containing 0. For some \(a_i\in {\mathscr {B}}\), \(n_i,m\in \mathbb {N}\),

So far, a necessary and sufficient condition for normability of 2-normed spaces is discussed. In this section, we derive a norm from a normable 2-norm such that both have the same topology. We start with the following proposition.

Let \((X,\Vert \cdot ,\cdot \Vert )\) be a 2-normed space and consider a linearly independent subset \(\lbrace a_1,a_2,\ldots ,a_k\rbrace\) of X. Define a family of functions \(\Vert \cdot \Vert _p\) from X to \(\mathbb {R},\) where \(1\le p\le \infty\) by

Theorem 9 is useful to check the normability of a 2-normed space, and Theorem 10 is useful to construct a norm for a normable 2-normed spaces. So both those Theorems may be treated as complement of each other.

Let \((X,\Vert \cdot ,\cdot \Vert )\) be a finite-dimensional 2-normed space and \(a_1\) and \(a_2\) be two linearly independent vectors from X. Then, the topology of the norm \(\Vert \cdot \Vert _\infty\) on X defined by

X is finite-dimensional. Therefore, Corollary 1 concludes that \((X,\Vert \cdot ,\cdot \Vert )\) is normable. Let \(\Vert \cdot \Vert\) be such a norm for \((X,\Vert \cdot ,\cdot \Vert )\). Now, \(\Vert \cdot \Vert _\infty\) is also a norm on X. But all norms are equivalent over a finite-dimensional vector space. So \(\Vert \cdot \Vert\) and \(\Vert \cdot \Vert _\infty\) are equivalent. Hence, the topologies of \(\Vert \cdot \Vert _\infty\) and \((X,\Vert \cdot ,\cdot \Vert )\) are same. \(\square\)

Let \((X,\Vert \cdot ,\cdot \Vert )\) be a 2-normed space and \(\Vert \cdot \Vert _\infty\) be the derived norm defined in Proposition 6. Then, convergence of sequences in \((X,\Vert \cdot ,\cdot \Vert )\) and \((X,\Vert \cdot \Vert _\infty )\) is equivalent if and only if both the spaces have same topology.

Let \((X,\Vert \cdot ,\cdot \Vert )\) be a 2-normed space and \(\Vert \cdot \Vert _\infty\) be the derived norm defined in Proposition 6. Then, the following are equivalent,

This clearly indicates that convergence of sequences with respect to a 2-norm and its derived norm are not equivalent if the 2-normed space is not normable. However, Example 1 assures the existence of 2-normed spaces which are not normable. Hence, we get a negative answer to Question 1.

In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.

This norm-induced topology also makes ( X , τ d ) {\displaystyle \left(X,\tau _{d}\right)} into what is known as a topological vector space (TVS), which by definition is a vector space endowed with a topology making the operations of addition and scalar multiplication continuous. It is emphasized that the TVS ( X , τ d ) {\displaystyle \left(X,\tau _{d}\right)} is only a vector space together with a certain type of topology; that is to say, when considered as a TVS, it is not associated with any particular norm or metric (both of which are "forgotten"). This Hausdorff TVS ( X , τ d ) {\displaystyle \left(X,\tau _{d}\right)} is even locally convex because the set of all open balls centered at the origin forms a neighbourhood basis at the origin consisting of convex balanced open sets. This TVS is also normable, which by definition refers to any TVS whose its topology is induced by some (possibly unknown) norm.

There is another notion of completeness besides metric completeness and that is the notion of a complete topological vector space (TVS) or TVS-completeness, which uses the theory of uniform spaces. Specifically, the notion of TVS-completeness uses a unique translation-invariant uniformity, called the canonical uniformity, that depends only on vector subtraction and the topology τ {\displaystyle \tau } that the vector space is endowed with, and so in particular, this notion of TVS completeness is independent of whatever norm induced the topology τ {\displaystyle \tau } (and even applies to TVSs that are not even metrizable). Every Banach space is a complete TVS. Moreover, a normed space is a Banach space (that is, its norm-induced metric is complete) if and only if it is complete as a topological vector space. If ( X , τ ) {\displaystyle (X,\tau )} is a metrizable topological vector space (such as any norm induced topology, for example), then ( X , τ ) {\displaystyle (X,\tau )} is a complete TVS if and only if it is a sequentially complete TVS, meaning that it is enough to check that every Cauchy sequence in ( X , τ ) {\displaystyle (X,\tau )} converges in ( X , τ ) {\displaystyle (X,\tau )} to some point of X {\displaystyle X} (that is, there is no need to consider the more general notion of arbitrary Cauchy nets). 2b1af7f3a8

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