A general definition is formulated for the Hankel norms as the induced
norms of a strictly proper stable linear time-invariant (LTIV) system
mapping vector-valued L-p-past inputs to vector-valued L-q-future out
puts. Some Hankel norms are derived by the maximum principle and duali
ty, A property termed the system integral invariance is introduced by
the derivation of those Hankel norms. Furthermore, the norm-induced in
itial conditions and the comparison between the Hankel norms and the i
nduced system norms are also presented.