Our observation on the Cauchy-Schwarz inequality 

in an inner product space and 2-inner product 

space suggests how the concepts of inner products

and 2-inner products, as well as norms and 2-norms, 

can be generalized to those of $n$-inner products 

and $n$-norms for any $n\in{\bf N}$.

In this paper, we offer a definition of $n$-inner 

products which is simpler than (but equivalent to) 

the one formulated by Misiak \cite{M1}. We also 

reprove the Cauchy-Schwarz inequality and give

a necessary and sufficient condition for the equality.