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

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Weak Solutions to PDES

I am working through some practice problems for my PDE class and I came across the following: Let $U\in \mathbb{R}^n$ be a smooth, bounded, connected open set. Let $\Gamma_1$, $\Gamma_2$, be two ...
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47 views

Two operators $X$ and $Z$ in an infinite dimensional Hilbert space satisfying $X^2=Z^2=I$ and $\{X,Z\}= 0$

I am seeking to extend the following theorem to the case of infinite dimensional Hilbert space: Suppose we have two Hermitian operators $X$ and $Z$ in a finite dimensional Hilbert space $\mathcal H$. ...
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1answer
43 views

Projection Theorem

I've been trying to apply the projection theorem to the following problem with no success. I've spent a few hours on this today, any help would be appreciated. Let H be a finite dimensional Hibert ...
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3answers
103 views

Natural ways in which the *complex* valued L-integral and *complex* Hilbert spaces come up

I have two questions regarding how two concepts that involve complex valued functions may come up in a natural way. (Non-natural ways are: These concept come up in order to present a unified theory, ...
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45 views

How can we use theory from $L^2(\mathbb{R})$ on a sequence of numbers (discrete signal)

In have problems understanding connection between theory that is done in $L^2(\mathbb{R})$ and its application on discrete signal. look at this paper http://home.ustc.edu.cn/~zhanghan/cs/...
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57 views

Show that $\langle y_i, y_j\rangle = 0 \forall i \neq j.$

Let $y_1, y_2, . . .$ be a sequence in a Hilbert space. Let $V_n$ be the linear span of $\{y_1, y_2, . . . , y_n\}$. Assume that for $n \ge 1, ∥y_{n+1}∥ \le ∥y − y_{n+1}∥$ for each $y \in V_n$. ...
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36 views

Flat Extrinsic Vs. Intrinsic Distance

Context: Let $\Psi: \mathbb{R}^d \rightarrow \mathscr{H}$ be a $C^k$-embedding of $\mathbb{R}^d$ into a Hilbert space $\mathscr{H}$. We may view $\mathscr{M}:=Im(\Psi)$ as a submanifold of $\...
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Direct sum decomposition of $L^2(\mathbb{R})$ using Fourier Transform

Let $L_+^2(\mathbb{R})=\{f\in L^2(\mathbb{R}):supp \hat{f}\subset\mathbb{R^+}\}$ and $L_-^2(\mathbb{R})=\{f\in L^2(\mathbb{R}):supp \hat{f}\subset\mathbb{R^-}\}$, where $\hat{f}$ denotes the Fourier ...
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25 views

If $A$ is the Laplacian on $H^2(0,1)∩H_0^1(D)$, then the fractional power space $\mathfrak D(A^{r/2})=H_0^r(D)$ for all $r\in\mathbb R$

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for }...
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If $G$ is the Green's function of the Laplacian $A$ and $L$ is the integral operator with kernel $G$, then $L$ is the inverse of $A$

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for }...
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25 views

How do we compute the Green's function of the Laplacian?

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for }...
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29 views

Is a bounded, linear, nonnegative and symmetric operator with finite trace on a Hilbert space Hilbert-Schmidt?

Let $U=(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space and $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\rangle$ $Q$ be a bounded, linear, ...
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10 views

Property related to weak convergence

Let $H$ be a real Hilbert space and $F:H\rightarrow H$ satisfying $$ \lim\langle F(u_k),v_k-u_k\rangle=\lim\langle F(u),v-u\rangle, $$ for all sequences $\{u_k\},\{v_k\}$ such that $u_k$ converges ...
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34 views

norm of orthogonal projection of some vector in Hilbert space

Let $H$ be Hilbert space and $u_1,u_2,...u_n \in H$ (vectors dont have to be orthogonal) $V=span\{u_1,u_2,...u_n\}\subset H$ and $S$ is unit sphere in $V$. $P_V$ is orthogonal projection on V. Now ...
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22 views

Orthonormal Basis of Hermitian matrices for a Hilbert Space of Operators

Consider a set of operators $O$ on a Hilbert space $V$ of dimension $d$. I could prove that $O$ is also a Hilbert space with dimension $d^2$ (inner product being $(A,B) = tr(A^\dagger B))$. Now I am ...
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1answer
44 views

How does this argument show continuity?

I want to show that the unitary group $U(\mathcal H)$ of a Hilbertspace $\mathcal H$ is a topological group wrt the strong operator topology. For the standard proof it is most convenient to use that ...
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67 views

The strong topology on $U(\mathcal H)$ is metrisable

The strong operator topology on a Banach space $X$ is usually defined via semi-norms: For any $x \in X$, $|\cdot|_x: B(X) \to \mathbb R, A \mapsto \|A(x)\|$ is a semi-norm, the strong topology is the ...
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32 views

Relation between complete spaces and usage of calculus

I have been studying about metric spaces and completion of metric spaces. While reading into Hilbert spaces, I discovered this phrase on their Wikipedia webpage:" Hilbert spaces are complete: there ...
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56 views

Using direct sums, construct an inseparable Hilbert space with an uncountable orthonormal basis

Using direct sums, construct an inseparable Hilbert space with an uncountable orthonormal basis. This is Problem 13 in Chapter II in Reed & Simon, and I'm really stuck on this one. Would ...
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24 views

Meaning of completeness for a sequence

I am studying some course notes and came across the following proposition: "An orthonormal sequence in a Hilbert space H is complete iff the only vector in H which is orthogonal to each of the ...
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An application of the Closed Graph Theorem [duplicate]

Let $T:L^2([0,1]) \to L^2([0,1])$ be a bounded linear map of Hilbert spaces such that if $f\in L^2([0,1])$ is continuous then so is $Tf$. Show that there is a positive constant C such that $$sup_{x\...
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21 views

Extension of Lipschitz Continuous Operators on arbitrary sets in Hilbert Spaces

Let $T: D(T) \subset X \to Y $ be a Lipschitz continuous operator on an arbitrary set $D(T)$ in the Hilbert Space $X$. Show that $T$ can be extended to an operator $\tilde{T}:X \to Y$ which is ...
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30 views

Is the range of a self-adjoint operator stable by its exponential?

Let $H$ be an Hilbert space, and $A \in L(H)$ be a bounded linear self-adjoint operator on $A$. We assume that $R(A)$, the range of $A$, is not closed. Is it true or not that $R(A)$ is stable by $e^{-...
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57 views

If $Q$ is an operator on a Hilbert space with $Qe_n=λ_ne_n$ for all $n$, then $Q^{-\frac 12}e_n=\frac 1{\sqrt{λ_n}}e_n$ for all $n$ with $λ_n>0$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $\mathfrak L(U)$ be the set of bounded and linear operators on $U$ $Q\in\mathfrak L(U)$ be nonnegative and symmetric $(e_n)_{...
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18 views

Are spherical harmonics a basis for $H^1$?

We know that spherical harmonics are a complete orthonormal system for $L^2(\mathbb{S}^2)$. Is it true that they are also a complete orthonormal system for $H^1(\mathbb{S}^2)$? Furthermore, is it ...
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51 views

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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37 views

What are eigenvalues/eigenfunctions of a “pointwise product” operator

Let us consider the Hilbert space $l^2([0,1])$ with inner product $<u,v>=\int_0^1 u(x)v(x)\mathrm dx$. We define a pointwise product operator $A$ as $(A\circ u)(x)=a(x)\cdot u(x)$, where "$\cdot$...
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Proof of equivalence of $\lambda$ norms in Sobolev space $H_0^1(\Omega)$

Consider the following metrics in $H_0^1(\Omega)$ with $\Omega$ a bounded domain: $$\| u\|_\lambda=\left( \int_\Omega |\nabla u|^2+\lambda\int_\Omega u^2\right)^{\frac{1}{2}}$$ and $$\| u\|_0=\left( \...
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Explanation of the proof of Theorem 2.13 in Young, “An introduction to Hilbert Space”

Let $\lVert\cdot\lVert$ be any norm on the vector space $E$ and let $\rho\left(\sum^n_{j=1}\lambda_je_j\right)=\left(\sum^n_{j=1}|\lambda_j|^2\right)^{1/2}$ where $(e_j)$ is a basis for $E$. Now ...
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28 views

Is there a sample-path continuous stochastic process whose sample paths do not almost surely lie in an RKHS?

Let $f$ be a mean zero second-order stochastic process with continuous covariance function $k$, that is indexed on a separable metric space $\mathcal{X}$ and that is sample-path continuous. Can we ...
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34 views

Characterizing Bounded Symmetric Bilinear Functions on Hilbert Spaces

Context: I am reading about Sobolev spaces and the Poisson equation from Eberhard Zeidler's Applied Functional Analysis book/article, and a key tool seems to be what Zeidler calls the "Main theorem ...
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39 views

Given a linear Hilbert-Schmidt embedding $ι$ between Hilbert spaces, prove that $ιι^*$ is a bounded, linear operator with finite trace

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $U_0:=Q^{\frac 12}(U)$, $$\langle u,v\rangle_0:=\langle Q^...
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Convergence in Hilbert spaces from $L^2$ to pointwise convergence

Let $H:=L^2(]0,1[)$ and consider $(u_k)$ a Hilbert basis of $H$, and $\alpha _k \in \mathbb{R}$ satisfying $\displaystyle \sum_{k=1}^{\infty} |\alpha _k |^2 < + \infty$. I know that $\sum_{k \...
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36 views

If $Q$ is an operator on a Hilbert space $U$, $U_0:=Q^{1/2}(U)$ and $(e_n)_n$ is a basis of $U_0$, then $u↦\sum_na_n(u,e_n)_0e_n$ is an embedding

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space and $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\rangle$ $Q$ be a bounded, linear, ...
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physical or geometrical meaning of orthogonality of function

basically I'm not a mathematician. I have three doubts in relating discrete space and continuous space. We know that if two vectors are orthogonal then we can say, their dot product is zero $$\vec{a}...
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50 views

Nonlinear contraction on Hilbert space

Let $C\subset H$ be a nonempty closed convex subset of a Hilbert space $H$ and let $T:C\rightarrow C$ be a nonlinear contraction; i.e. $$|Tu-Tv|\leq|u-v|\quad\forall u,v\in C.$$ Let $(u_n)$ be a ...
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15 views

Existence of nuclear dominating positive definite kernel

Let $\mathcal{X}$ be a metric space and $k: \mathcal{X} \times \mathcal{X} \to \mathbb{R}$ a continuous positive definite kernel. Can we always find a positive definite kernel $r$ such that $r \gg k$ (...
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20 views

How can the inverse of an operator between Hilbert spaces H,K be defined on the dual of H?

I need some help to understand the following statement. Let $A$ be an operator defined as follows: $Av = -\Delta v - \nabla \text{div} u$ It is known that the operator $A$ is positive self-adjoint ...
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1answer
27 views

Dissipativity for Hilbert spaces

I want to prove that an operator $A:D(A)\to X$ is dissipative $\iff$ $\text{Re}\langle Ax,x\rangle\le 0$ $\forall x\in D(A)$. The proof for this is actually sketched on the Wikipedia page for ...
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22 views

What do Fourier Series for Other Symmetric Operators Look Like?

I understand that Fourier analysis works (up to constant multiples) by considering the inner-product space $E$ of smooth functions $[-\pi,\pi] \to \mathbb C$ with inner product. . . $\displaystyle (...
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83 views

Every Hilbert space is isometrically isomorphic with $\ell^2$

Let $H$ be a hilbert space and let $\{u_\alpha\}_{\alpha \in A}$ be a orthornormal basis ($A$ is not supposed to be countable a priori). Then there is an isometric isomorphism between $H$ and $\...
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43 views

How to define properly the radial and angular dependence of a function?

Recently I've came across the following situation when studying Quantum Mechanics: suppose we have two operators $A,B$ on the space of functions $L^2(\mathbb{R}^3)$ and suppose they have the following ...
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1answer
44 views

How to write this down in a rigorous manner?

I'm studying Quantum Mechanics and there's something I'm in doubt in how to write it down in a rigorous way. The idea is: consider a Hilbert space $\mathcal{H}$ and one hermitian operator $A\in \...
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12 views

Unbounded self-adjoint operator on pre-Hilbert spaces

If $H$ is a Hilbert space and $A:H\to H$ is a self-adjoint operator is simple to prove that $A$ is bounded. But if $H$ is pre-Hilbert, is there a unbounded self-adjoint operator $A:H\to H$? Everyone ...
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1answer
48 views

$\sigma$-algebra generated by weak topology in Hilbert Space

In general, if we have $H$ Hilbert space, and equipped with the weak topology, say $\tau^\ast$, is $\sigma(\tau^*)=\mathcal{B}$?, where $\mathcal{B}$ is the usual Borel $\sigma$-algebra I suspect it ...
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1answer
30 views

Orthogonal complement of a subspace in $l^2$

Consider $l^2$ as a Hilbert space with the usual inner product. It's quite easy to see that the subspace $X$ consisting of the sequences $(x_n)_{n\ge1}$ such that $x_{2n} = nx_{2n-1}$ for all $n\ge1$ ...
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48 views

Definition of essential spectrum?

Suppose we have a Hilbert space $\mathscr{H}$ and a bounded linear map $T\in\mathscr{B(H)}$ NOT necessarily self-adjoint. There seems to be loads of definitions of the essential spectrum of $T$. My ...
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67 views

Does $AB=(AB)^{\ast}$ and $A=A^{\ast}$ implies $B=B^{\ast}$?

Suppose that we have $AB=(AB)^{\ast}$ and $A=A^{\ast}$, does this implies that $B=B^{\ast}$? ($A^{\ast}$ is the Hermitian adjoint of $A$.) I have a feeling that they might not be equal in general. ...
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2answers
34 views

Finding distance of $h(t)=t$ from a closed subspace $Y$ of $\pi$-periodic functions in $L^2(-\pi,\pi)$

Let $Y=\{f\in L^2(-\pi,\pi):f(t-\pi)=f(t) \text{for almost all $t\in(0,\pi)$} \}$ Show that there exists $g\in Y$ such that $$\|h-g\|_2=\inf \{\|h-f\|_2:f\in Y\}$$ where $h(t)=t$. Compute $\|h-...
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1answer
64 views

Parturbations of orthonormal bases [closed]

Suppose that $(e_n)_{n=1}^{\infty}$ is an orthonormal basis in a Hilbert space $H$, and let $(f_n)$ be an orthonormal sequence in $H$ such that $$\sum_{n=1}^\infty \|e_n-f_n\|<\infty.$$ How can ...