2.1. INTRODUCTION 11
signals. Strictly speaking, signals in H2or H⊥
2are not defined on the ω axis. However
we usually considerthem to be by taking a limitas we approach the axis.
A slightly more specialized set is RL2, the set of real rational functions in L2. These are
strictly proper functions with no poles on the imaginaryaxis. Similarly we can consider
RH2as strictly proper stable functions and RH⊥
2as strictly proper functions with no
polesin Re(s)<0. The distinction between RL2and L2is of little consequence for the
sorts of analysis we will do here.
The concept of a unit ball will also come up in the following sections. This is simply the
set of all signals (or vectors,matrices or systems) with norm less than or equal to one.
The unit ball of L2, denoted by BL2, is therefore defined as,
BL2=x(t)kx(t)k2<1.
Now let’s moveonto norms of matrices and systems. As expected the norm of a matrix
gives a measure of its size. We will again emphasize only the norms which we will
consider in the following sections. Consider defining a norm in terms of the maximum
gain of a matrix or system. This is what is known as an induced norm. Consider a
matrix, M, and vectors, uand y,where
y=Mu.
Define, kMk,by
kMk=max
u,kuk<∞
kyk
kuk.
Because Mis obviously linear this is equivalentto,
kMk=max
u,kuk=1kyk.
The properties of kMkwill depend on how we define the norms for the vectors uand y.
If we choose our usual default of the Euclidean norm then kMkis given by,
kMk=σmax(M),