On bicomplex 𝔹ℂ-modules <i>l<sub>p</sub> </i> <sup>𝕜</sup>(𝔹ℂ) and some of their geometric properties
Ni̇lay Deği̇rmen, Birsen Sağır
Abstract
Abstract In this paper, we examine the validity of bicomplex versions of some crucial inequalities with respect to the hyperbolic-valued norm <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mo>⋅</m:mo> <m:msub> <m:mo stretchy="false">|</m:mo> <m:mi mathvariant="normal">𝕜</m:mi> </m:msub> </m:mrow> </m:math> {|\cdot|_{\Bbbk}} 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∥</m:mo> <m:mo>⋅</m:mo> <m:msub> <m:mo>∥</m:mo> <m:mrow> <m:mi>𝔻</m:mi> <m:mo>,</m:mo> <m:mo>⋅</m:mo> </m:mrow> </m:msub> </m:mrow> </m:math> {\|\cdot\|_{\mathbb{D},\cdot}} by defining the concept of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔻</m:mi> </m:math> {\mathbb{D}} -normed Banach bicomplex A -module and constructing <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔻</m:mi> </m:math> {\mathbb{D}} -normed Banach bicomplex <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝔹</m:mi> <m:mo></m:mo> <m:mi>ℂ</m:mi> </m:mrow> </m:math> {\mathbb{BC}} -modules <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>l</m:mi> <m:mi>p</m:mi> <m:mi mathvariant="normal">𝕜</m:mi> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>𝔹</m:mi> <m:mo></m:mo> <m:mi>ℂ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {l_{p}^{\Bbbk}(\mathbb{BC})} .