In this paper, we have defined rough statistical convergence in intuitionistic fuzzy normed spaces which is an useful characterization in the field of statistical convergence. We have proved some properties related to rough convergence which provides some new functional tools in the situation of uncertainty like intuitionistic fuzzy normed spaces. Further, we have established the relationship between the set of statistical limit points and set of cluster points of rough statistically convergent sequences in these spaces.

In the present paper, we use the difference operator ∆m (Iλ )-summability to define some new summability concepts on neutrosophic normed spaces. We also introduce concepts of generalized limit point, and cluster point and obtain some relationships among these notions. Finally, we define generalized Cauchy sequences on these spaces and present a characterization of a new summability method that preserves linear operators on neutrosophic normed spaces.

For an admissible ideal I ⊆ P(N) and a lacunary sequence θ = (ks), the aim of the present paper is to introduce the concept of I(S)-summability and I(Sθ)-summability w.r.t neutrosophic norm (G,B,Y). We also investigate some relations among these notions and prove that these are equivalent if and only if 1 < lim infs qs ≤ lim sup s qs < ∞.

In this article, we aim to define ℐ 2 ⁢ ( 𝒮 ) {\mathcal{I}_{2}(\mathcal{S})} -summability and ℐ 2 ⁢ ( 𝒮 θ p , r ) {\mathcal{I}_{2}(\mathcal{S}_{{\theta}_{p,r}})} -summability, and obtain interesting relationships among these by imposing certain conditions on p and r. Finally, we show that the space ℐ 2 ⁢ ( 𝒮 θ p , r ⁢ ( G , B , Y ) ) ∩ l ∞ 2 ⁢ ( V ) \mathcal{I}_{2}(\mathcal{S}_{{\theta}_{p,r}}(G,B,Y))\cap l^{2}_{\infty}(V) is a closed subspace of l ∞ 2 ⁢ ( V ) {l^{2}_{\infty}(V)} .

In this paper, we define and study three novel summability concepts – strong deferred 𝜎-summability, deferred 𝜎-statistical summability, and 𝜎-statistical summability in intuitionistic fuzzy 𝑟-normed linear spaces (briefly called IF-𝑟-NLS) by using 𝜎-mean. We also provide an example in support of the new notions and uncover some interesting relationships. Additionally, we study deferred 𝜎-statistical summability in the context of two pairs of sequences of positive integers, namely, α n , γ n , and u n , v n satisfying α n ≤ u n < v n ≤ γ n .

In present paper, we introduce the concepts of neutrosophic boundedness, neutrosophic compactness in neutrosophic 2-normed spaces and analyze some of their topological properties. We show that if the space is finite dimensional then any two neutrosophic 2-norms are equivalent. Finally, we defined neutrosophic boundedness and neutrosophic continuity of linear operators and study some of their properties.