Inner Product Approach to Generalize the Notion of Pythagoras Theorem for Normed Spaces

Abstract

The Pythagorean Theorem, a fundamental result in Euclidean geometry, traditionally relates the lengths of the sides of a right-angled triangle. In this paper, we extend the classical Pythagorean Theorem into the context of normed vector spaces, using the concept of inner products. We explore how the theorem manifests in higher-dimensional spaces and provide a generalized version applicable to normed spaces beyond two dimensions. This generalization not only reinforces the geometric interpretation of the theorem but also connects it to broader mathematical frameworks such as vector spaces, norms, and inner products. The results presented here demonstrate the versatility of the Pythagorean Theorem and its relevance across various fields of mathematics, highlighting its applications in both theoretical and applied contexts.