Understanding Hilbert Space in Quantum Mechanics: A Beginner's Guide

[dextercioby]

It cannot be normalized,simply because it's not a state vector.It's not an eigenvector of momentum operator,because the latter does not have eigenvectors,as it doesn't have discrete spectrum...

Pay attention with such details.Mathematical details mean very much in QM.

Daniel.

 

[C*-algebras]

I'm sorry,but your "proof" is useless.In the previous post you stated that for a topological metric space everywhere continuity is completely equivalent to boundness.
The Theorem of Riesz applies ONLY TO LINEAR FUNCTIONAL EVERYWHERE CONTINUOS IN H,thind which,by virtues of topology,means that the functionals need to be bounded on H.So assuming them unbounded and applying the Riesz Representation Theorem makes no sense,okay...??

So yeah,i was wrong,$$\tilde{H}$$ comprises only bounded (hence everywhere continuous) linear functionals on H.

Daniel.

Riesz thm indeed makes great sense. But I guess a more insightful version would be using the GNS construction and Gelfand-Naimark theorem where Hilbert space is constructed through non-commutative structure rather than "postulated" as in Dirac-Neumann axiomatic system.
In this sense, I prefer QM be essentially an ALGEBRAIC theory... prob. somehow becomes a byproduct.