# Tagged Questions

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

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### Compactness in Hilbert spaces

Let $H$ be a Hilbert space with orthonormal basis $\{h_n:n\in \Bbb N\}$. Let $P_n$ be the orthogonal projection to $\operatorname{span}\{h_1,\cdots, h_n\}$. Claim: A bounded subset $U\subset H$ is ...
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### Exponent of an Exponential Operator

There is a problem in my textbook that asks me to prove the following: For a bounded operator $A$ on a Hilbert space, prove that: $$(e^A)^n = e^{An}$$ for any natural number, $n$. However upon ...
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### Eigenvectors Operators and Unilateral Shifts

We showed that a (non-zero) compact self-adjoint operator on a Hilbert space always has an eigenvector. Let $V:l^2(\mathbb{N})\to l^2(\mathbb{N})$ be the unilateral shift, the unique bounded operator ...
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### Vectors in a Hilbert space are countably supported with respect to any orthonormal basis

Let $\{e_i\}_{i\in I} \subset \mathcal{H}$ be an orthonormal set in the Hilbert space $\mathcal{H}$. For any vector $x\in \mathcal{H},$ let $$I_x=\{i\in I|\,\langle x,e_i\rangle \neq 0\}.$$ How can we ...
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### Bounded Operators: Topological Dual

Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider the bounded operators: $$\mathcal{B}(\mathcal{H},\mathcal{K}):=\{T:\mathcal{H}\to\mathcal{K}:\|T\|<\infty\}$$ Regard the linear ...
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### Approximating step functions by Haar wavelets

Let $\psi = \chi_{[0,1/2)} - \chi_{[1/2,1)}$, then $\psi_{n,k}(t) = 2^{n/2}\psi(2^nt-k)$ with $n \in \mathbb{N}$ and $k \in \{0,1,\dots,2^n-1\}$ defines the Haar-Wavelets on $L^2(0,1)$. Let $S$ be the ...
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### Exponential of a self-adjoint operator

Let $\mathcal{H}$ be an Hilbert space. Firstly, I shall define some notions as their definitions may vary: A spectral resolution is a function $E:\mathbb{R}\to\mathcal{L}(\mathcal{H})$ (the space ...
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### orthonormal basis in $L^2$ space

Let $\{\phi_i (x)\}_{i=1}^\infty$ be an orthonormal basis for $L^2 (S)$. Prove that $\{\psi_{ij} (x,y) = \phi_i (x) \phi_j (y)\}_{i,j=1}^\infty$ is an orthonormal basis for $L^2 (S \times S)$. Thanks ...
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### Are quantum operators associative?

Let H be the Hamiltonian representing the total energy of the potential and kinetic component. But because all classical dynamical variables can be written as a function of position, x, and momentum, ...
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### Every partially defined isometry can be extended to a isometry

I know that the following theorem holds true: Let $S$ be a subset of $\mathbb R^n$, and let $f:S\to \mathbb R^n$ a map such that $d(p,q)=d(f(p),f(q))$ for every $p,q \in S$ (here $d$ is the usual ...
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### What really is ''orthogonality''?

I know that we can define two vectors to be orthogonal only if they are elements of a vector space with an inner product. So, if $\vec x$ and $\vec y$ are elements of $\mathbb{R}^n$ (as a real ...
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Let $\Omega$ be the open unit ball in $\mathbb{R}^n$, and $\Gamma := \Omega \cap \{x_n=0\}$. Let $\Omega_1 = \{ x \in \Omega: x_n > 0 \}$ and $\Omega_2 = \{ x \in \Omega: x_n < 0 \}$. Define $... 1answer 42 views ### Homogeneous and Inhomogeneous Function Spaces I would like a general explanation on the difference between homogenous and inhomogeneous function spaces, there doesn't seem to be a very good explanation online. I know that for Sobolev spaces for ... 3answers 46 views ### Is faithful positive sesqulinear form an inner product? As in the title: does a positive sesqulinear form need to be conjugate-symmetric? Background: The question comes from an attempt to understand the proof of the Stinespring representation theorem. It ... 1answer 32 views ### Norm of a self adjoint operator Let$T$be a (bounded) self-adjoint operator on a Hilbert space. Is it true that$||T^k|| = ||T||^k$for all positive integers$k$? It's true for$k=1,2$, and I'm wondering if this could be ... 2answers 38 views ### Why can't a Hilbert curve be used to put the real numbers into a listable format? There's a very good chance this question will make absolutely no sense, as my understanding of Hilbert curves is very superficial. But let me explain where my question is coming from. From my ... 1answer 186 views ### When is a function of the largest eigenvalue continuous and/or differentiable? I want to understand why the following function, the largest eigenvalue of a symmetric linear operator, is continuous and Gâteaux differentiable. \begin{equation*} \lambda(V)=\sup_{f \in \ell^2(I):\ \... 1answer 30 views ### Let$A$be a non-separable$C^*$-algebra. Is it possible that there is a faithful representation$\pi:A\to L(H)$on a separable hilbert space$H$? Let$A$be a non-separable$C^*$-algebra. Is it possible that there is a faithful representation$\pi:A\to L(H)$on a separable hilbert space$H$? I know that if$A$is separable, one can choose$H$... 1answer 37 views ### Does an essentially self-adjoint operator have the same kernel as its closure? Let$H$be a Hilbert space and let$A : D(A) \subset H \to H$be an essentially self-adjoint operator. Let$\overline A$be the unique self-adjoint extension of$A$. Question: Is it true that$\...
I 'm stuck with the definition of block diagonal operators on hilbert spaces. Def.: A bounded linear operator $T$ on a hilbert space $H$ is called block diagonal if there exists an increasing ...
Let $H$ be a Hilbert Space (over $\mathbb{R}$ or $\mathbb{C}$ but maybe is valid for any field) and $E$ a continuous operator. Suppose $E$ is idempotent, i.e.,$E^2=E$ and positive definite, i.e. \$(Ev,...