We discuss some aspects of M\"obius gyrovector spaces from viewpoints of basic theory of functional analysis. Any open ball centered at the origin of a complex inner product space endowed with a slightly modified M\"obius addition is a gyrocommutative gyrogroup. A M\"obius scalar multiplication by complex numbers can be naturally introduced so that some of the axioms of real inner product gyrovector spaces are satisfied. For every linear operator between inner product spaces whose operator norm is less than or equal to one, the restriction to the M\"obius gyrovector space is Lipschitz continuous with respect to the Poincar\'e metric. Moreover, the Lipschitz constant is precisely the operator norm.