L¥ and H¥ Spaces

§*L**¥** ***(***j***R****) Space:** *L**¥** ***(***j***R****) **or simply *L**¥* is a Banach space of matrix-valued (or scalar-valued) functions
that are (essentially) bounded on *j***R**, with norm

§*H**¥*** Space: ***H**¥* is a (closed) subspace of *L**¥* with functions that are analytic and bounded in the open
right-half plane. The *H**¥* norm is defined as

The
second equality can be regarded as a generalization of the maximum modulus theorem for matrix
functions. See Boyd and Desoer [1985] for a proof.