The Cauchy-Schwarz Inequality in Complex Normed Spaces

2017/01/21 by Volker Thürey, Thürey, Volker Wilhelm
Mathematics · #46B99 #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Functional Equations Stability Results #Mathematical Inequalities and Applications

paper · pdf · doi:10.48550/arxiv.1701.06031

openalex publication_date 2017/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a product in all complex normed vector spaces, which generalizes the inner product of complex inner product spaces. Naturally the question occurs whether the Cauchy-Schwarz inequality is fulfilled. We provide a positive answer. This also yields a new proof of the Cauchy-Schwarz inequality in complex inner product spaces, which does not rely on the linearity of the inner product. The proof depends only on the norm in the vector space. Further, we present some properties of the generalized product.

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