Hilbert spaces Definition and 57 Threads

Under what circumstances is a (linear) operator \mathcal{H} \to \mathcal{H} between a Hilbert space and itself diagonalizable? Under what circumstances does (number of distinct eigenvalues = dimension of H), i.e., there exists a basis of eigenvectors with distinct eigenvalues? Although I am...

Let \mathcal{H} be a Hilbert space over \mathbb{C} and let T \in \mathcal{B(H)}. I want to prove that \|Tx\| = \|x\| \, \Leftrightarrow \, T^{\ast}T = I for all x \in \mathbb{H} and where I is the identity operator in the Hilbert space. Since this is an if and only if statement I began...

A Hilbert Space is a complete inner product space. My first question: From the definition above, is it safe to say that every sequence in a Hilbert Space converges? And so can we say that Hilbert Spaces only contain Cauchy sequences? Second question: These 'sequences' that we talk about...

Let U, V, W be inner product spaces. Suppose that T:U\rightarrow V and S:V\rightarrow W are bounded linear operators. Prove that the composition S \circ T:U\rightarrow W is bounded with \|S\circ T\| \leq \|S\|\|T\|

I browsed a book by Byron & Fuller "Math. Physics" and read the following: Algebra, Geometry & Analysis are joined when functions are treated as vectors in a vector space. This makes Hilbert spaces extremely useful in QM.(paraphrased but that's the jist of it) Comments on this? If it's...

...For abut two weeks I've searching the internet for a good online (i.e.free for download)course on the basis of quantum mechanics,that means the mathematical background of this theory.I found a few,especially from American colleges,but all of them seemed to have serious problems regarding...

Well, you all know that LQG has different kinds of Hilbert spaces (4 in the sake of truth). You start with the kinematical Hilbert space that is the vector space of all possible quantum states of spacetime. However, all these spacetimes are not physically real, not all of them make sense. Then...