Banach Space representations of physical systems I think most physicists mostly model physical systems as some kind of Hilbert space. 
Hilbert spaces are a strict subset of Banach spaces.
Questions:


*

*Can physical systems really have non-compact topologies, as a Banach
space has?

*Does anyone have an example of physics which requires a physical space
which is Banach and not Hilbert?
 A: Sobelev spaces (in which the norm of a function is a weighted sum of $L^p$ norms of the function and its derivatives) are Banach spaces, and they do come up in the analysis of PDE.  Some physicists might regard the use of such spaces as intrinsically mathematics rather than physics.
A: (Classical) field theory lives naturally in more general spaces than Hilbert (and even more general than Banach). The space of smooth sections of a fiber bundle is a Fréchet manifold (if the basespace is compact).
If this fiber bundle thing is new to you, you can consider your fields just as smooth functions $\psi: \text{spacetime} \rightarrow R$. In field theory you want to say something about the behavior of any field, so you are more or less forced to look at the space $C^\infty(M)$ (which is neither a Hilbert- nor a Banachspace)
A: Hilbert spaces are occur everywhere where the Lagrangian\Hamiltonian is quadratic in derivatives. If the Lagrangian is non-quadratic then the Hilbert spaces are no longer so convenient. In particular in analysis of Navier–Stokes equations the Banach spaces(not Hilbert spaces) are active used.
