In this paper, we introduce the different ways of proving the triangle inequality ku − vk ≤ kuk + kvk, in the Hilbert space. Thus, we prove this triangle inequality through the binomial inequality and also, prove it through the Euclidean norm. The first generalized procedure for proving the triangle inequality is feasible for any even positive integer n. The second alternative proof of the triangle inequality establishes the Euclidean norm of any two vectors in the Hilbert space.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.