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In this work, it is assumed that the norm over bicomplex numbers is the hyperbolic (픻 -valued) norm. In this paper, we provide an overview of bicomplex Lebesgue spaces and investigate some of their geometric properties, including 픹 ⁢ ℂ -convexity, 픹 ⁢ ℂ -strict convexity, and 픹 ⁢ ℂ -uniform convexity. Moreover, the basic inequalities such as 픻 -Hölder's inequality and 픻 -Minkowski inequality for bicomplex Lebesgue spaces are presented, used to show geometric properties. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the notions of the Köthe–Toeplitz duals of sequence spaces by introducing the concepts of bicomplex α-dual, bicomplex β-dual and bicomplex γ-dual, and also we compute them for some bicomplex sequence spaces l p ⁢ ( 픹 ⁢ ℂ ) {l_{p}(\mathbb{BC})} for 1 ≤ p ≤ ∞ {1\leq p\leq\infty} , c 0 ⁢ ( 픹 ⁢ ℂ ) {c_{0}(\mathbb{BC})} and c ⁢ ( 픹 ⁢ ℂ ) {c(\mathbb{BC})} . Furthermore, we define a concept of bicomplex multiplier space as the bicomplex version of multiplier space of two sequence spaces and support this definition with examples. [ABSTRACT FROM AUTHOR]

In this paper, we examine the validity of bicomplex versions of some crucial inequalities with respect to the hyperbolic-valued norm | ⋅ | 한 and we discuss some topological and geometric concepts such as completeness, convexity, strict convexity and uniform convexity in the bicomplex setting with respect to the hyperbolic-valued norm ∥ ⋅ ∥ 픻 , ⋅ by defining the concept of 픻 -normed Banach bicomplex A-module and constructing 픻 -normed Banach bicomplex 픹 ⁢ ℂ -modules l p 한 ⁢ (픹 ⁢ ℂ) . [ABSTRACT FROM AUTHOR]