# Tagged Questions

For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

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### How to prove, that the ordering on positive bounded operators agrees with ordering of their ranges?

Hypothesis: Assume, that $A$ and $B$ are positive bounded operators (on some Hilbert space $H$) and $A\geq B \geq 0$. Then ${\rm range}(A) \supset {\rm range}(B)$. The textbook "$C^*$-algebras by ...
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### Representation of a vector

$(l^2,\|\cdot\|_2)$ is a Hilbert space with scalar product $\langle x,y\rangle=\sum^{\infty}_{k=1}x_ky_k$. How can I show that every vector $x\in l^2$ can be written in a form ...
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### Prove or disprove existence of a sequence converging weakly to $0$ in an infinite dim Hilbert space

This is a problem on an old analysis qual, the prompt is: "Prove or give a counter example: if $H$ is an infinite dimensional Hilbert space and $0$ is the zero vector in $H$, then there exists a ...
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### What is $H^1([0,1]) \otimes H^1([0,1])$?

Let $H^1([0,1])$ denote the Sobolev space $H^1$ on the interval $[0,1]$. What is $H^1([0,1]) \otimes H^1([0,1])$? Here, $\otimes$ the tensor product of Hilbert spaces. In particular, how is that ...
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### The unit ball in a Hilbert space

I have a request for any ideas to prove: If $H$ is a Hilbert space, then any unit vector is an extreme point of the unit ball of $H$. Every isometry is an extreme point of the unit ball of the ...
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### Hermitian matrices and great circles

I am considering parameterised curves in an $n$-dimensional complex vector space, given by the solution to the discrete Schrödinger equation: $$|\psi\rangle(t) = e^{-iHt}|\psi_0\rangle,$$ Where $H$ ...
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### A theorem about operators in Hilbert sapce

For ench $n\geq1$, $B(\mathcal{H})$ is $\ast$-isomorphic to $\mathbb{M}_n(B(\mathcal{H}))$. Thanks to the one who tell me the proof or tell me where I can find the proof.
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### Showing $\mathcal{H}$ is a hilbert space.

So this is an early exercise in Conway's A Course In Functional Analysis. I'm trying to get to grips with this upto open mapping and closed graph to see if I want to do any more functional analysis. ...
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Is there some sort of natural Hilbert space structure on $\mathbb{C}[z]$ so that $\{\frac{z^k}{\sqrt{k!}}\}$ are orthonormal? Can this structure be extended to $\mathbb{C}[z_1]\otimes ... 2answers 132 views ### The eigenvalues of a certain integral operator are square-summable Let$(X,\Omega,\mu)$be a measure space, and$k\in L^2(X\times X, \Omega\times \Omega,\mu\times\mu)$. Then it is well-known that $$(Kf)(x)=\int k(x,y)f(y)\ d\mu(y)$$ is a compact operator with norm ... 2answers 153 views ### Norm of a$2\times 2$matrix as a Hilbert space operator Work in the Hilbert space$\mathbb C^2$. Let $$A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$$ be a matrix with entries in$\mathbb C$, and let$A$... 1answer 131 views ### An isometry of Hilbert spaces using the Radon-Nikodym derivative Let$(X,\Omega)$be a measurable space and let$\mu, \nu$be two$\sigma$-finite measures on$(X,\Omega)$. Suppose$\nu \ll \mu$and let$\phi$be the Radon-Nikodym derivative of$\nu$with respect to ... 2answers 340 views ### Net convergence and norm-convergence in Hilbert spaces Let$\mathcal H$be a Hilbert space which is not necessarily separable. Given a sequence of element$h_n$indexed by$\mathbb N$, we say their sum converges in norm if the sequence$\{\sum_{n=1}^k h_n ...
Conway, in A Course in Functional Analysis, leaves the following corollary (2.11) to the reader. If $\mathcal S$ is a linear manifold in $\mathcal H$, then $\mathcal S$ is dense in $\mathcal H$ ...
The following is problem 1.10 in chapter 1 of Conway's A Course in Functional Analysis. Let $G$ be an open subset of $\mathbb C$ and show that if $a\in G$, then $\{f\in L^2_a(G): f(a)=0\}$ is ...