# Tagged Questions

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

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### Lower bound on gap between consecutive eigenvalues on $L_2(\mathbb{R}^3)$

A similar version of this question was originally posted by me in the physics community, but it was suggested that I ask the mathematicians instead. So I have tried to strip off most of the physics ...
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### Duality Pairing and self-adjoint operators notation

Let $X$ be a Hilbert space and let $X'$ be the dual space of $X$ with respect to the duality pairing $\langle\cdot,\cdot\rangle$. Let $A: X \mapsto X'$ be a bounded linear operator. We assume that $A$...
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### Eigenvalue of every eigenvector is an eigenvalue of element of o.n eigenvector basis

I need to prove (or disprove) that given a bounded operator $A$ on a Hilbert space with orthonormal basis of eigenvectors $\left\{ e_i \right\}$, if $Av=\lambda v$ for $v\neq 0$ then $\lambda$ is an ...
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### Prove multiplication by sequence is a compact operator

Let $c_0(\mathbb N)$ be the space of sequence in $\mathbb C$ whose limit is zero, equipped with the $\ell^\infty$ norm. Let $u_n$ be a sequence in $\mathbb C$ and define the operator $A$ taking a ...
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### $(k\otimes h^\ast)^\ast=h\otimes k^\ast$?

Let $H,K$ be Hilbert spaces with $h\in H,k\in K$. Let $k\otimes h^\ast(g)= \left\langle g,h \right\rangle k$. I'm supposed to prove $(k\otimes h^\ast)^\ast=h\otimes k^\ast$, but I don't see how this ...
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### Integral operator convergence study in $L^2(\mathbb{R})$

Exercise I want to study in $\mathcal{H}=L^2(\mathbb{R})$ the convergence of $$A_nf(x)=\log n \int_\mathbb{R} \frac{1}{1+n(x-y)^2}f(y)dy.$$ Solving a) Pointwise convergence : For $n \to \infty$ ...
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### Dimension and separability of $\ell^2(I)$?

Let $\ell^2(I)= \left\{ x:I\rightarrow \mathbb C\mid \sum _{i\in I} |x_i|^2<\infty \right\}$ with inner product $\sum_{i\in I}x_i \bar y_i$. I am supposed to find the dimension of this space and ...
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### Does the Parseval identity imply the completeness of an orthonormal system?

Let $V$ be an inner product space which is not complete. Can there by an orthonormal system satisfying the Parseval identity for each vector but which is not complete i.e the only vector orthogonal to ...
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### Prove $\exists x$ such that $\|Ax-b\|$ is minimal and this is unique if $A$ is invertible

Suppose $A\in \mathbb R^{m\times n}$. I'm supposed to prove $\exists x$ such that $\|Ax-b\|$ in minimal and this $x$ is unique if $A$ is invertible, in which case I also need to exhibit a formula. I ...
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### Metric projection onto nonnegative sequences

I need to find the metric projection onto non-negative sequences in $\ell^2$. Intuitively I'm thinking each element in the sequence should be sent to the maximum between it and zero, since that should ...
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