# The Cauchy-Schwarz Inequality in Complex Normed Spaces

## TL;DR

This paper introduces a generalized product in complex normed spaces, proves the Cauchy-Schwarz inequality within this context, and offers a new proof that relies solely on the norm, extending understanding beyond inner product spaces.

## Contribution

It defines a new product in complex normed spaces and proves the Cauchy-Schwarz inequality without using linearity of the inner product.

## Key findings

- The generalized product satisfies the Cauchy-Schwarz inequality.
- A new proof of the inequality that depends only on the norm.
- Properties of the generalized product are explored.

## 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.