Some generalized sequence space defined by Musielak–Orlicz function

In this paper, we introduce and investigate new classes of sequence spaces defined by a Musielak–Orlicz function Π = (πk) over a linear space (V, q), where q is a seminorm. By combining the structural flexibility of Musielak–Orlicz functions with generalized averaging techniques inspired by the classical M(ϕ)-space, we define the sequence spaces ℓ1[Π, q], ℓ∞[Π, q], and M[Π, ϕ, q]. These spaces extend several well-known Orlicz-type and modulus-based sequence spaces and provide a unified framework for studying coordinate-wise variable modular growth conditions. The space M[Π, ϕ, q] is constructed as a natural generalization of the classical M(ϕ) sequence space by incorporating a Musielak–Orlicz modular structure. We investigate the fundamental algebraic and topological properties of these spaces, including linearity,solidity, symmetricity, monotonicity, and completeness. Suitable Luxemburg-type seminorms are introduced,and it is shown that these spaces become complete seminormed spaces whenever the underlying seminormed space (V, q) is complete. Furthermore, we establish several inclusion relations between the newly defined spaces under appropriate growth conditions on the sequence ϕ = (ϕs) and on Musielak–Orlicz functions satisfying the Δ2-condition. In particular, we prove that ℓ1[Π, q] ⊆ M[Π, ϕ, q] ⊆ ℓ∞[Π, q],and obtain additional inclusion results involving compositions and sums of Musielak–Orlicz functions. These results generalize earlier works on Orlicz sequence spaces and contribute to the further development of modular and generalized sequence space theory, opening new directions for the study of dual spaces, matrix transformations, and operator-theoretic properties in Musielak–Orlicz-type settings.