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

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35 views

Proof of Rayleigh trace

I found the following statement without proof: Let us given a self-adjoint Operator $T\colon L^2 \to L^2$ which has n eigenvalues $ \lambda_n \leq \dots \leq\lambda_{n-1} < \lambda_1 =1$ counted ...
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1answer
36 views

If $f$ is identically zero then the coefficients are all zero

I am looking at the space: $$A:=\left\{f(x)=\sum_{k\in\mathbb{Z}}{a_ne^{inx}}:(a_n)_{n\in\mathbb{Z}}\in l^1(\mathbb{Z})\right\}$$ I want to say the following: if $f\equiv0$, then $a_n=0$ for all ...
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1answer
71 views

Show this subpace of a Hilbert space is dense

This is part of an exercise in Rudin's Functional Analysis, in the chapter on Unbounded Operators. Let $H$ be a Hilbert space with orthonormal basis $\{e_n\}$. Let $X$ be the set of all finite sums ...
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30 views

Regularity theory for $H^k$ space

Lat $\Omega$ be a bounded domain in $\mathbf{R}^n$ with smooth boundary. Let $a(u,v)=\int_\Omega \Sigma a_{ij}\partial_iu\partial_jv+cuv$ where $a_{ij}$ and $c$ are smooth functions on $\bar{\Omega}$ ...
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1answer
419 views

If $0\leq A\leq B$ on Hilbert space and $A^{-1}$ exists, show that $A^{-1}\geq B^{-1}$ [duplicate]

Does anyone know how to show this? Let $H$ be a Hilbert space and $A$, $B$ bounded positive operators defined on $H$ such that $A^{-1}: H \rightarrow H$ exists and hence bounded and $A \leq B$. ...
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1answer
203 views

Compact operator in Hilbert spaces $T^2$

I have the following problem: Let H be a Hilbert space a) Prove that if $T: H\to H$ is compact then $T^2$ is compact operator b) Find $S: H\to H$ compact such that $S=T^2$ with T non compact c)If ...
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1answer
40 views

$V=L^2(\Omega,Z)$ is path connected

Let $V=L^2(\Omega,Z)$. Prove that V is path connected by paths of class 1/2 Holder. I would appreciate it if anyone could give me a suggestion. Thank you in advance.
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1answer
44 views

A property of the canonical inclusion $i: L^2(0,1)\to L^1(0,1)$

Prove that the image of the canonical inclusion $i: L^2(0,1)\to L^1(0,1)$ is a countable union of closed sets with empty internal part. Can anyone give me any idea on the solution? Thank you in ...
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196 views

Does a symmetric operator on a Hilbert space have a symmetric adjoint?

Suppose we have a linear operator $T$, densely-defined on some Hilbert space. If $T$ is symmetric (i.e., $T^*$ extends $T$: notationally, $T\subseteq T^*$) does it follow that $T^*$ is also symmetric ...
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84 views

In a separable Hilbert space, can you write an operator from $\mathcal H$ to $\mathcal H$ as a column-finite matrix?

In this question, we are representing an operator $T$ as a matrix with respect to an orthonormal basis $\left\{e_n : n \in \mathbb{N}\right\}$. To do so, we let $t_{ij} = \langle T(e_j),e_i\rangle$. ...
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155 views

Proving that a certain differential operator is self-adjoint

Consider the differential operator $T:u\mapsto -iu'$ for any $u\in D(T):=\{f\in AC[-\pi,\pi]~|~f'\in L^2(-\pi,\pi),f(-\pi)=f(\pi)\}$; we consider $T$ as a densely-defined operator on $L^2(-\pi,\pi)$. ...
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1answer
103 views

what is the advantage of having countable dense subset?

what is the advantage of having countable dense subset (for example of the set $L^2([0,1])$, if i have to prove weak convergence ? edit: to prove is that every sequence $(f_n)_n$ with ...
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1answer
101 views

Properties of orthogonal projections

I have a question about orthogonal projections on a Hilbert space $H$. Can we say that the range of a projection $P\in B(H)$ is closed? Thus we have to ask if the range is a Hilbert space. Moreover: ...
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1answer
84 views

How to understand a Hilbert Space of functions?

Here are some of my understandings of Hilbert Space of functions, I am not sure. L2 space is a Hilbert Space of all square integrable functions. It's easy to understand. And the dimension of this ...
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1answer
40 views

name of matrix of inner products $\langle f_i, f_j\rangle$

Given a Hilbert space $H$ and a number of elements $\phi_i\in H$, does the matrix $M$ with $$ M_{i,j} := \langle\phi_i, \phi_j\rangle $$ have any particular name?
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1answer
136 views

Eigenvalues and adjoint of operator $T(x_k)_{k=1}^{\infty} = (x_{2k})_{k=1}^{\infty}$

Let $T$: $l^2 \rightarrow l^2$ denote the operator \begin{align} T(x_1,x_2,\dots, x_n,\dots) = (x_2,x_4,\dots,x_{2n},\dots). \end{align} There are several questions regarding this operator that I need ...
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1answer
65 views

Please help to understand text about closed operators and extensions

I need help understanding a section of a book I'm reading (Mathematical Foundations Of Quantum Mechanics, by J. von Neumann, Princeton U. Press, 1955, pages 152-153). I have a few questions on two ...
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3k views

Example that in a normed space, weak convergence does not implies strong convergence.

The book "Introductory Functional Analysis with Applications" (Kreyszig) presents the following definitions. A sequence $(x_n)$ in a normed space $X$ is said to be strongly convergent if there is ...
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2answers
452 views

Orthogonal Projections in Hilbert space

I am stuck with the following exercise about projections in Rudin 12.26. Let $H$ be a Hilbert space $P,Q\in B(H)$ self-adjoint projections (A projection has the property that $P^2=P$), then the ...
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1answer
61 views

Closed subspace of Hilbertspace

Let $X$ be a norm closed subspace of a Hilbert space $\mathcal H$. Is it true that if $x_n \in X$ converges weakly to $x \in \mathcal H$, then also $x \in X$ ?
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78 views

Orthogonal projection. [duplicate]

I have found this question in a book, but I don't know how to use that $\left\Vert P\right\Vert =1$. Question: If $P\in\mathcal{L}(H)$ is a projection and $\left\Vert P\right\Vert =1$, show that $P$ ...
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1answer
157 views

Duals of Hilbert Subspace

So I am confused about something very basic. I'm going to outline my confusion, and would love if someone could point out when I'm saying something wrong. Let $H$ be a Hilbert space. It's dual $H^*$ ...
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3answers
491 views

When do inner products of weakly convergent subsequences converge?

If we have 2 weakly convergent subsequences in $L^2(U)$ (for $U$ some bounded open domain with smooth boundary), $u_k\rightharpoonup u$ and $v_k\rightharpoonup v$, under which conditions do we have ...
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2answers
114 views

What is known about this space of parametrised Hilbert spaces?

For each $s \in [0,\infty)$, let $H(s)$ be a Hilbert space. Let us suppose for simplicity that $H(s) = L^2(\Omega_s)$, where $\Omega_s$ is some nice domain that depends on $s$ in a nice way. Define ...
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1answer
49 views

Continous and Discrete basis, Multiplication of Density Matrix and Hamiltonian.

Suppose I have a wave function $\psi(x)$ in position basis. I can make a density function by simply multiplying $\psi(x)$ and its conjugate $\psi^*(x)$. If I operate the density matrix ...
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1answer
107 views

Show that an unbouned normal operator is closed

A linear operator $A$ is called nomal if $\mathcal{D}(A)=\mathcal{D}(A^{*})$ and $\lVert A\phi\rVert =\lVert A^{*}\phi\rVert$ for every $\phi\in \mathcal{D}(A)$. Show that normal operators are closed. ...
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1answer
151 views

Is $L^2(\Omega)$ the only $L^p$ hilbertian space?

I've started today studying Hilbertian spaces, and all of the examples seen in class were about the space $L^2(\Omega)$, where $\Omega$ is a limited domain in $\mathbb{R}^N$ $(N \geq 1)$. Online I ...
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1answer
48 views

subset of hilbert space is weakly bounded iff it is bounded

Let $\mathbb{H}$ be a hilbert space, $E \subset \mathbb{H}$. We say that $E$ is weakly bounded if for every $y \in \mathbb{H}$, there is some $\alpha_{y} \geq 0$ such that $|<x, y>| \leq ...
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56 views

hypothesis on bilinear form

Let $H$ an Hilbert space and $a:H\times H\to \mathbb{R}$ a bilinear form. Let $H_h\subset H$ a finite dimentional subspace and let $\{w_1,\ldots,w_n\}$ a basis of $H_h$. What hypothesis must have on ...
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139 views

Prove that if $\lambda$ is an isolated eigen-value of $T=T^*$, then $\ker(T-\lambda)=E_{\{\lambda\}}H$

Here we have a self-adjoint, densely-defined operator $T$ on a Hilbert space $H$, and $E_M$ is the usual spectral projector for any Borel set $M$, i.e., $E_M=\int_M\text{d}E_t$ (this means, by ...
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1answer
1k views

Dual and adjoint operator

Let $X$ be a Hilbert space with associated canonical isomorphism $I:X\rightarrow X^\ast$ (by the Riesz representation theorem). If $A:X\rightarrow X$ is a linear operator on $X$, then its dual ...
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1answer
59 views

Bounded Closed set in Hilbert space

Let $X$ be a seprable Hilbert space, and $\{e_n\}$ to be an orthonormal basis. Let $$\Omega=\cup_{n=1}^\infty\{(1-t)e_n+te_{n+1};0\leq t\leq 1\},$$$$T:\Omega\to R$$$$T((1-t)e_n+te_{n+1})=n+t,\ ...
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1answer
81 views

Definitions of adjoints (functional analysis vs category thy)

If I have a linear operator $f$ on a Hilbert space, then I define the adjoint of $f$ to be $f^*$ where, $(fx,y)=(x,f^*y)$ for all $x,y$. I am confused because this definitions is very different to ...
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48 views

Completeness of separable solutions to PDEs

Under what conditions will the solutions of a PDE obtained using separation of variables form a complete set for the solution space?
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94 views

Gram Matrix with another inner product

Let $H\subset L^2(\Omega)$ a finite dimentional space with inner product $(\cdot,\cdot)_{0,\Omega}$ and $\{v_i\}_{i=1}^n\subset H$ a basis of $H$. Is the Gram Matrix ...
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24 views

Solve an integral equation in an Hilbert space

Let $V_n\subseteq [H(div;\Omega)]^{2\times 2}$ and $Q_n\subseteq [L^2(\Omega)]^2$ two finte dimentional spaces such that $div(H_n)\subseteq Q_n$. Suppose that $u\in Q_n$ is well known. I must solve ...
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2answers
1k views

Show that a positive operator on a complex Hilbert space is self-adjoint

Let $(\mathcal{H}, (\cdot, \cdot))$ be a complex Hilbert space, and $A : \mathcal{H} \to \mathcal{H}$ a positive, bounded operator ($A$ being positive means $(Ax,x) \ge 0$ for all $x \in ...
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1answer
62 views

compute the norm of a compact operator on $l^2$

Let $a_j\to 0$ and let $T:l^2 \to l^2$ be the operator defined by $ T(s_1,s_2,s_3,...)=(0,a_1s_1,a_2s_2,...)$. Compute the operator norm $||T||$. The hint of the problem is prove that $T$ is a ...
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1answer
117 views

If E is a Hilbert space and $T \in B(E)$ is compact, show $T(E)$ does not contain a closed infinite dimensional subspace

It's the problem from "Essential Results of Functional Analysis," R.J. Zimmer, Chapter 3, problem 3.1. I try to prove this problem and I am confused with the condition "closed infinite dimensional." ...
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65 views

Nilpotents of order 2 are dense in the strong operator topology

I need some help with this homework problem. (The Hilbert space in question is $\ell^2(\mathbb{N})$.) I let $T \in \mathcal{B}(\mathcal{H})$ and looked at a basic nbhd in the strong operator topology ...
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180 views

Spectral theorem and projection

This should be simple, but I'm stuck. Let $A$ be an unbounded self-adjoint operator on a Hilbert space $H$. The spectral theorem says that there is a decomposition of $H$ into a direct sum for which ...
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2answers
213 views

prove that this operator is not compact

Let $g\in C[0,1]$ be a continuous function and $g\ne 0$. Let $G:C[0,1]\to C[0,1]$ the operator defined by: $G(f)(x)=f(x)g(x)$. I proved that the operator is linear and continuous. I want to prove that ...
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1answer
92 views

Stone's theorem for 1-parameter groups of unitary multipliers?

Let $A$ be a nonunital C*-algebra and let $M(A)$ denote its multiplier algebra. Let $(u_t)_{t \in \mathbb{R}}$ be a strictly continuous 1-parameter group of unitary multipliers. That is, $u_t x \to x$ ...
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1answer
45 views

About norm and duality in $\mathcal S(\mathbb R^n)$

Reading the book "Classical and multilinear harmonic analysis, Vol. 1" by Muscalu, Schlag, 2013; I have a problem understanding the first step of the proof of Lemma 11.3. The relevant parts are: ...
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1answer
71 views

$\lambda_k \to 0$ implies $T$ is compact?

I am doing an exercise which asks to show that if $\{\varphi_k\}$ is an orthonormal basis in a Hilbert space with $T$ a bounded operator satisfying $T\varphi_k = \lambda_k \varphi_k$, then $\lambda_k ...
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2answers
164 views

A dense subset of a Hilbert space

I am curious about the following problem: Consider the Hilbert space (a weighted $L^2(\mathbb{R})$ space): $$\mathscr{H}=\bigg\{f: \mathbb{R}\to\mathbb{R}\text{ Lebesgue ...
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1answer
265 views

Is $L^2(\Omega)$ dense in $H^{-1}(\Omega)$?

Is it true that $L^2(\Omega)$, identified with its own dual, is dense in $H^{-1}(\Omega)$? $H^{-1}(\Omega)$ is the dual of $H^1_0(\Omega)$ and $H^1_0(\Omega)$ is the $H^1$-closure of smooth functions ...
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119 views

An orthonormal basis for a Hilbert space

Can anyone give me some hint on the following problem without using any knowledge about complex analysis or Fourier analysis? Thanks a lot! Consider the Hilbert space $$\mathscr{H}:=\bigg\{f\text{ ...
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1answer
231 views

Are two hilbert spaces with the same algebraic dimension (their hamel bases have the same cardinality) isomorphic?

We know that two hilbert spaces tat have orthonormal bases of the same cardinality are isomorphic (as an inner product spaces). my question is what can we say when we know that their hamel bases ...
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1answer
156 views

find a weak solution in an intersection of Sobolev spaces

In using lax-milgram to find a weak solution in an intersection of sobolev spaces the weak solution for $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ was ...