How does one define "predual" and the surrounding notions? More specifically:
Why must there be only one predual of $X$ when $X$ is a Banach space? What is the correct notion of similarity here that gives this uniqueness? Is it isomorphic homeomorphism of Banach spaces? I'm also interested in the corresponding algebraic statement. Is it true that if $V$ is a vector space, then it has at most one predual?
I have noticed from looking online that the predual of $B(H)$ is the trace class operators, and the predual of that is the compact operators, which strangely enough means that taking preduals doesn't always reduce the "size" of the space (I'm not able to be precise since I don't know the true meaning of the uniqueness of predual), even though in the algebraic setting, one always has the usual injection of a vector space into its dual. I suppose that this discrepancy is because in the analytic definition of dual, we require continuity, so that the dual vector of a vector $x$ in $X$ when $X$ is a Banach space need not actually be in the continuous dual $X^*$ of $X$?