# Hilbert space for Density Operators (instead of Banach spaces)

Is it possible to construct a well defined inner-product (and therefore orthonormality) within the set of self-adjoint trace-class linear operators? In the affirmative case, dynamics could be analyzed in Hilbert space, which seem way more simple that Banach spaces.

Which is the fundamental reason why this is not possible?

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$$\langle A|B\rangle := tr(A^\dagger B)\:.$$
The space of Hilbert Schmidt operators is made of all bounded operators $A$ in the considered Hilbert space, such that $A^\dagger A$ is trace class. It is in fact possible to reformulate all QM using that notion.
The Hibert-Schmidt ideal has Hilbert space structure, while the others ideals (like the trace class one) have not. They are all subsets of the compact operators, and satisfy inequalities similar to the $L^p$ functions (e.g. Hölder). A very complete reference is the book of Barry Simon Trace ideals and their applications. –  yuggib Aug 15 '14 at 10:12