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 outputs. Some Hankel norms are derived by the maximum principle and duality, A property termed the system integral invariance is introduced by the derivation of those Hankel norms. Furthermore, the norm-induced initial conditions and the comparison between the Hankel norms and the induced system norms are also presented.