# Tagged Questions

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

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### Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space

Let $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be open $H:=L^2(\Omega,\mathbb R^d)$ $U$ be a separable $\mathbb R$-Hilbert space $Q:U\to H$ ...
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### $ι:U→V$ is an embedding, $Q:=ιι^*$, $L∈𝓛(ℝ^d)$, $Φ∈\text{HS}(U,ℝ^d)$ $⇒$ $\text{tr}LΦ\sqrt Q(Φ\sqrt Q)^*$ doesn't depend on $ι$

Let$^1$ $U$ and $V$ be separable $\mathbb R$-Hilbert spaces $\iota\in\operatorname{HS}(U,V)$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ $u:\mathbb R^d\to\mathbb R$ be twice Fréchet ...
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### Hilbert space is orthornormality needed for representation?

In a Hilbert space $H$ with countable basis, if I know there is a countable basis $\{h_n\}$ of $H$ then can I express every element $h\in H$ therein as: h = \sum_n \langle h,h_n\...
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### Denseness of polynomial in reproducing kernel Hilbert space

Let $\mathcal H_K$ be a reproducing kernel Hilbert space and $K$ be the associated reproducing kernel on $\mathbb D \times \mathbb D.$ Further assume that $K(z, 0)= 1$ for all $z\in \mathbb D$ and ...
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### Basis of convex and concave functions

Let $g(t)$ be a positive function. I'm trying to show that the set $\{ |t|^ne^{\frac{g(t)}{t^n}}\}_{n\in \mathbb{N}}$ is linearly independent in the space of measurable functions on $\mathbb{R}$ (ie: ...
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### positive operators

Consider two Hilbert spaces $H$ and $K$ and a bounded operator $T$ on $H\oplus K$. I know that there is a theorem which says the following: $T\ge 0$ (in the operator sense) if and only if $T$ can be ...
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### Proving Sobolev space on [0,1] is RKHS

My aim is to prove that the space: $\mathcal{H}$ = {$f:[0,1] \to \mathbb{R}: f\;is\;absolutely\;continuous,\;f(0)=f(1)=0,\;f'\in L^2[0,1]$} is a reproducing kernel hilbert space. Now assuming an ...
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### Confused about Domain of Unbounded Operators (Hilbert Spaces)

I understand that a bounded (linear) operator $A$ on a Hilbert space $H$ satisfies a condition $||Av|| \le c ||v||$ for some fixed real number $c$ and for all $v \in H$. So, the domain of a bounded ...
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### Smooth Approximations in $L^2((0,1))$

Let $L^2((0,1))$ be as usual the Lebesgue space of measurable complex-valued functions $f:(0,1) \rightarrow \mathbb{C}$ such that $\int |f(x)|^2 dx < \infty$. It is a well known fact (see e.g Lieb ...
### How is this function a member of $L^{1}(0, \frac{1}{b})$?
The function in question is: $$F_{n}(x) = \sum_{k \in \mathbb{Z}} f\left(x-\frac{k}{b}\right) g^{\ast}\left(x - na - \frac{k}{b}\right)$$ Where $\ast$ denotes complex conjugation, $f, g \in L^{2}$,...