  In this article, we continue to explore Probability Bracket Notation (PBN),
proposed in our previous article. Using both Dirac vector bracket notation
(VBN) and PBN, we define induced Hilbert space and induced sample space, and
propose that there exists an equivalence relation between a Hilbert space and a
sample space constructed from the same base observable(s). Then we investigate
Markov transition matrices and their eigenvectors to make diffusion maps with
two examples: a simple graph theory example, to serve as a prototype of
bidirectional transition operator; a famous text document example in IR
literature, to serve as a tutorial of diffusion map in text document space. We
show that the sample space of the Markov chain and the Hilbert space spanned by
the eigenvectors of the transition matrix are not equivalent. At the end, we
apply our PBN and equivalence proposal to Thermophysics by associating sample
(phase) space with the Hilbert space of a single particle and the Fock space of
many-particle systems.
