# Tagged Questions

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

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I'm stuck on an exercise about the spectrum of a selfadjoint operator on a Hilbert space. The problem is the following: Let $(X,\langle \cdot, \cdot\rangle)$ a Hilbert space and let $A \in B(H)$ a ...
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### Trace class norm and rank inequality

I am quite new to operators in Hilbert spaces and I have been trying to show that for any linear and bounded operator $T : \mathcal{H} \rightarrow \mathcal{H}$ \vert \vert T \vert \...
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### $W_0^{1,\:p}(\Lambda)$ is dense in $L^2(\Lambda)$

Let $d\in\mathbb N$ $\lambda$ be the Lebesgue measure on $\mathbb R^d$ $\Lambda\subseteq\mathbb R^d$ be open with $\lambda(\Lambda)<\infty$ $p\ge 2$ $W^1(\Lambda)$ denote the set of weakly ...
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### Possible master thesis [on hold]

I am searching a topic for my master thesis. My interests are especially probability theory (something with brownian motion would be nice) and fourier analysis (also in an abstract Hilbert space ...
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### Physical (Quantum Mechanical) Significance of completeness of Hilbert Spaces.

I'm not sure if the question is very 'mathematical',but I'm asking any way. I have a basic knowledge of quantum mechanics and I'm studying Hilbert spaces. I was wondering what is the physical ...
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### Proving that a space is complete

There is something that bugs me about the proof I've been shown that $C(\Omega)$ (the space of continuos function on $\Omega$, a compact subset of $\mathbb R^n$) with the $\sup$ norm is complete. ...
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### Orthogonal of an Hilbert subspace and density

If $V$ is a subspace of an Hilbert space $H$, I know that the orthogonal of $V$, $V$$^o, is (Vclosed)^o, even if V is not closed. Does this mean that V is always dense in V$$^o$? Thanks!...
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### If $ι:U→V$ is a Hilbert-Schmidt embedding and $(v_n)_{n∈ℕ}$ is an orthonormal basis of $V$, then $(ιι^*v_n)_{n∈ℕ}$ is an orthonormal basis of $ιU$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle_U)$ and $(V,\langle\;\cdot\;,\;\cdot\;\rangle_V)$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $T:=\iota\iota^\ast$ ...
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### If $Q$ is an operator on a Hilbert space $U$, $(e_n)$ is an ONB of $U$ consisting of eigenvectors of $Q$, then $(Q^{1/2}e_n)$ is an ONB of $Q^{1/2}U$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ ...
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### weighted shift operator for complex Hilbert space

I am trying to solve that if H is a complex Hilbert space with orthonormal basis $\{e_n\}_{n=1}^{\infty}$ and let $\{a_n\}_{n=1}^{\infty}$ be a sequence with $\lim_{n\rightarrow}a_n = 0$. Define the ...
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### the spectral radius of normal operator

Let $H$ be a Hilbert space and $T$ be linear bounded operator in $H$. Prove that if $T$ is normal then the spectral radius of $T$, $$r(T)=\|T\|.$$ Is this TRUE?
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### Prove that $\|T\|=\sup_{\|x\|=1}|\langle x,T(x)\rangle|$. [closed]

Let $T$ be a self adjoint bounded linear operator in a Hilbert space $H$. Prove that $$\|T\|=\sup_{\|x\|=1}|\langle x,T(x)\rangle|$$
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### Show that usual $L^2$ norm is equivalent to arbitrary norm $||\cdot||$ that satisfies 'convergence condition'

Given $L^2(\mathbb{R})$ consider a norm $||\cdot||$ on $L^2$ such that $(L^2,||\cdot||)$ is a Banach space and every $||\cdot||$-convergent sequence has a subsequence that converges almost everywhere. ...
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### Prove or disprove: $\lVert T\rVert=\sup_{\lVert x\rVert=1}|\langle Tx,x\rangle|$, where $H$ is a Hilbert space and $T$ is bdd linear operator.

Edit: To clarify, note that $T:H\to H$. This is a problem on an old preliminary exam in Analysis that I'm working through to prep for my own prelim. My initial thought was to disprove it, but I can'...
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### Generalized Absolute Value II

Let $x$ be an operator in $B(H)$. We say a pair $(c,y)$ forms a polar decomposition for $x$ if $y$ is a positive operator, $c$ in $B(H)$ with $x=cy$ such that the restriction of $c$ on $\overline{yH}$...
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### Example of Hilbert space non isomorphic to $L2$

I'm looking for an example of a Hilbert space that can't be seen as the countable direct sum of $L^{2}(X,\mu)$ spaces nor subespaces of $L^{2}(X,\mu)$. Some idea to start? Thanks everyone.
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### The difference between an isometric operator and a unitary operator on a Hilbert space?

It seems that both isometric and unitary operators on a Hilbert space have the following property: $U^*U = I$ ($U$ is an operator and $I$ is the identity operator) What is the difference between ...