# Questions tagged [hilbert-spaces]

A Hilbert space $H$ is a real or complex vector space endowed with an inner product such that $H$ is a complete metric space when endowed with the norm induced by this inner product.

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### Trace inequality normal derivative

For $v(\Omega) \in W^1_2$ and $\Omega \in C^1$ we have a trace inequality: $$\Vert v \Vert _{L_2(\partial \Omega)} \leq C_\Omega \Vert v \Vert _{W_2^1},$$ which can be found in many places in the ...
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### Is $F: B(H)\to B(H\otimes K)$, $X\mapsto X\otimes \mathbb{1}_K$ completely contractive?

Take Hilbert spaces $H$ and $K$. Consider a linear map $F: B(H)\to B(H\otimes K)$, $X\mapsto X\otimes \mathbb{1}_K$. Is it true that $F$ is completely contractive? If it is, I would be very grateful ...
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### Is this subspace of $B(\mathcal{H})$ known?

Let $\mathcal{H}$ be a Hilbert space. Suppose that I take a fixed ONB of $\mathcal{H}$ let us call it $\{ e_i \}_{i\in \mathbb{N}}$ and then I define \begin{align*} \|T \|_{D} = \sup_{l_i, m_i} \sum_{...
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### Expressing the sum of two squared inner products more compactly: is it possible to lift the dimension? [closed]

Let $v_1,v_2\in\mathbb{R}^d$ be two fixed vectors, and $\langle \cdot,\cdot\rangle_{\mathbb{R}^d}$ be the usual Euclidean inner product in $\mathbb{R}^d$. My question is as follows. Is there an (...
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### Mean squared absolute value of inner product of unit vectors

Given a nonempty finite subset $S$ of the unit sphere of $d$-dimensional complex Hilbert space, let $\lambda(S) = \frac{1}{\lvert S \rvert^2} \sum_{x,y \in S} \lvert \langle x,y \rangle \rvert^2$ be ...
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### Embedding random variables in infinite-dimensional spaces

Let $H$ be a reproducing kernel Hilbert space of functions $f:E\to F$ with kernel $k$. A point in $E$ may be embedded into $H$ via the canonical embedding $x\mapsto k(x,\cdot)$. Similarly, a random ...
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### What is a $C^\infty$ diffeomorphism from $\ell_2\setminus\{0\}$ to $\ell_2$ which is the identity outside a ball?

Let $\ell_2:=\{x=(x_n)_{n\in\mathbb N}:\ \|x\|^2:=\sum_n|x_n|^2<\infty\}$ with its natural norm. According to Wikipedia https://en.wikipedia.org/wiki/Kuiper%27s_theorem and to other sources, it is ...
Consider the Banach algebra $B_2(H)$ of Hilbert Schmidt operators on a Hilbert space $H$ with Hilbert Schmidt norm. We know that $B_2(H)$ is a Hilbert space as well with $\left<A,B\right>=tr(B^*... 0answers 74 views ### Is it possible that the dimension of the intersection of a nested sequence of Hilbert space is 1? Let$H$be an infinite dimensional separable Hilbert space over$\mathbb{C}$Let$\{h_n\}_{n \in \mathbb{N}} \in H$be a sequence of linearly independent vectors in$H$Let$$V= \bigcap_{n=1}^\... 1answer 89 views ### Concentration of measure on finite powers of$S^\infty$I am wondering about a natural generalization of theorem 1.4 in the article Dvoretzky's theorem — Thirty years later by Milman. My first thought was to look at Milman's paper that he cites for the ... 2answers 201 views ### Spectral theory in non-separable Hilbert Spaces I am wondering about what can be said about the spectral theorem for unbounded, self-adjoint operators in a non-separable Hilbert space. There is a comment in this sense to the question "Does spectral ... 1answer 211 views ### Weak convergence of Hilbert Schmidt operators So I am stuck at this situation. Let$\{A_n\}$be a weakly convergent sequence in$B_2(H)$converging to$0$in the weak topology on$B_2(H)$. Which means that$\left<A_n,D\right>=\operatorname{...
Given a set of $N$ operators $\mathcal{O}$ with a known set of Lie Algebra group multiplication rules $\mathcal{G}$ that can be programmed into a classical computer, is there a classical poly($N$) ...