  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.
