A linear model of a technical or physical system is often described by a so called transfer matrix, which is considered as
an element of an appropriate Hardy space. It is also often assumed that the input and output of the modelled system
are elements of another Hardy space. In this work a detailed mathematical introduction to the theory of Hardy spaces
is given. Properties of Hardy spaces which are important for the description of technical or physical systems are
highlighted. An important question is how robust a system is under input disturbances. This question can be answered
if the gain of the system is known. The gain of a system is the maximum ratio between output and input and is
therefore not easy to determine (since every possible input must be considered). The main result of this work is, that,
under certain assumption on the model of the system, one can calculate the gain in a much simpler way. This result is
well known in the literature, but to the author's best knowledge a convincing proof is not available. The gaps in existing
proofs are discussed and a complete proof is given.