Traditional \(L_p\) spaces are fundamental in functional analysis, demarcated by the relationship \(1/p + 1/q = 1\) . This research pioneers the concept of \(\theta\) -Lebesgue space, stemming from a simultaneous weakening of both the classical \(L_p\) relation and its \(\theta\) -variant, \(1/(\theta (p)) + 1/(\theta (q)) = 1\) . This conceptual shift addresses a gap in existing mathematical frameworks, aiming to create a space that may encompass a broader range of mathematical purposes. The primary objective is to rigorously designate the \(\theta\) -Lebesgue space within this new context, explore its foundational properties, and articulate its theoretical significance in the realm of functional analysis. The focus is on establishing the theoretical underpinnings and potential implications of this generalized space. Adopting a detailed analytical methodology, this paper demarcates the \(\theta\) -Lebesgue space under the relaxed conditions. It examines its normative and topological properties and how these properties differentiate from and extend beyond traditional \(L_p\) spaces. The study involves a deep dive into the space’s inherent theory and its implications for functional analysis. The paper reveals that the dual relaxation of \(L_p\) space relations leads to a unique set of properties within the \(\theta\) -Lebesgue space. Notably, it presents a generalized form of Hölder’s inequality and explores the nuanced aspects of duality and reflexivity in this new context.
