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

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1answer
69 views

Reproducing Kernel and the continuous of the evaluation functional $e_{t}$

I'm working on Reproducing Kernel Hilbert Spaces and I had a problem proving the the continuity of the evaluation functional $e_{t}$ ($e_{t}(\phi) = \phi(t)$). Theorem A Hilbert Space of complex ...
2
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2answers
447 views

Is fractional Sobolev space $H^s$ Hilbert?

For $s \in (0,\infty)$ a fractional number, define $H^s(\Omega) = W^{s,2}(\Omega)$ on good domain $\Omega$. Every textbook doesn't say that $H^s$ is Hilbert. Is it? I have only seen this fact when ...
2
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1answer
59 views

Riesz Representation Theorem: isomorph

Riesz' Representation Theorem states that every linear functional can be represented by a vector. This shows that the Dual can be ANTILINEARLY and norm preserving identified with the Hilbert Space ...
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0answers
64 views

How to find a hilbert basis of a given subspace considering a given inner product

Let $X$ be the space of continuous functions on $[-1;1]$ to $\mathbb{R}$ with the inner product: $$\langle f,\ g\rangle = \int_{-1}^{1} \! f(x)g(x) \, dx$$ and let $U$ be a subspace of $X$ with $U := ...
0
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1answer
47 views

showing that a sequence is converging.

suppose $\left \{ T_{k} \right \}$ is a collection of bounded operators on Hilbert space $H$ ,with $\left \| T_{k} \right \|\leq 1$ for all $k$ .suppose also that $$T_{k}T_{j}^{*}=T_{k}^{*}T_{j}=0 ...
7
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1answer
194 views

Dense subspace of $L^{2}[0,1]$

I know that $C[0,1]$ is dense in $L^{2}[0,1]$ but is $\{f\in C^{2}[0,1]:f(0)=f(1)=0\}$ dense in $L^{2}[0,1]$?
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1answer
41 views

Are eigenspaces in a Hilbert space rays?

It may sound as a dumb question but I just want to be sure that I understand all the terminology: The eigenspaces corresponding to a (non-degenerate) eigenvalue of a operator on a Hilbert space are ...
4
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2answers
94 views

Upper bound for norm of Hilbert space operator

It is a standard result that for a bounded self-adjoint operator $T$ on a complex Hilbert space $H$, we have $||T||=\sup_{||x||=1}|\langle Tx,x\rangle|:=M$. It seems that for any bounded operator on ...
2
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0answers
58 views

$\gamma-$radonifying operators.

I am reading about $\gamma$-radonifying operators, and came across 2 similar definitions. And I hoped to understand why they are equivalent. Let $H$ be a seperable real Hilbertspace, $E$ banach ...
2
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1answer
104 views

Closed range in Hilbert Space

If $H$ is a Hilbert Space. Let $A: H \rightarrow H$ be a one-to-one bounded operator with the additional property that $\beta||u|| \leq ||Au||$. How would you show that $R(A)$ (the range of A) is ...
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1answer
90 views

About eigenspaces

In the context of a Hilbert space $H$, when an operator $A$ is diagonalizable we usually decompose the Hilbert space into direct sum of eigenspaces $$H=\bigoplus\limits_{n=1}^\infty E_n$$ where $E_n$ ...
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37 views

Help on a Hilbert Space theory utilization.

I need some help here concerning the Hilbert Spaces theory. Below, you can see a part of Olivier Chapelle's paper: "Training a Support Vector Machine in the Primal". As you can see below, in Eq.(8) ...
2
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1answer
77 views

Norm of the dual of the Tensor product of Hilbert spaces

Let $V$ and $W$ be Hilbert spaces, we can define inner product and induced norm on Tensor product of these spaces as: Let $v_1,v_2 \in V$,and $w_1,w_2 \in W$. then $(v_1 \otimes w_1, v_2 \otimes ...
6
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1answer
294 views

Approximation of bounded and continuous mappings

Does anyone know if we can approximate a bounded (i.e. bounded sets in V are mapped to bounded sets in V': for every bounded $U\subseteq V$ and $x\in U$, there exists $K_U>0$ such that ...
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0answers
33 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
33 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
68 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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0answers
29 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 $\bar{\Omega}$ ...
2
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1answer
350 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
157 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
38 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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2answers
163 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 ...
4
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1answer
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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0answers
125 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 $L^2(-\pi,\pi)$. ...
0
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1answer
93 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 ...
3
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1answer
94 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
80 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?
2
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1answer
127 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
62 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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2answers
2k 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 ...
2
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2answers
387 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
52 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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68 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
128 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 $H^*$ ...
4
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3answers
380 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 ...
2
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1answer
39 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 ...
6
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1answer
95 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
124 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
43 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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1answer
54 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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2answers
123 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 ...
2
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1answer
676 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
48 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,\ ...
4
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1answer
79 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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0answers
40 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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87 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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0answers
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 ...