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

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11 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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1answer
19 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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15 views

Prove that the standard orthonormal sequence $(e_n)^\infty_{1}$ is complete in $l^2$. [on hold]

Prove that the standard orthogonal sequence $(e_n)^{\infty}_{1}$ is complete in $l^{2}$. Where $(e_{n})$ is the sequence with nth component equal to 1 and all others zero.
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1answer
17 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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0answers
18 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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1answer
65 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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1answer
37 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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17 views

No canonical evaluation map $H^*\hat{\otimes}H \to \mathbb{K}$ in $\mathsf{Hilb}$?

Is it true that there is no canonical continuous evaluation $H^* \hat{\otimes}H \to \mathbb{K}$ on the Hilbert space tensor product coming from the natural pairing $H^* \times H \to \mathbb{K}$? At ...
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1answer
38 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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7 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
33 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
14 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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0answers
74 views

Continuous inclusions in Hilbert-Sobolev space $H^s(\mathbb{R}^n) \hookrightarrow \mathcal{E}^k(\mathbb{R}^n)$

I have to prove that $H^s(\mathbb{R}^n) \hookrightarrow \mathcal{E}^k(\mathbb{R}^n)$ with $s \in \mathbb{R}$, $k \in \mathbb{N}$ and $s-k > n/2$, where $\mathcal{E}^k(\mathbb{R}^n):=\lbrace u: ...
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0answers
19 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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1answer
65 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
31 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 ...
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1answer
58 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 ...
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0answers
21 views

Function with both easy to find Fourier and Hermitian coefficient

I'm writing some notes on Spectral theory and I would like to make a simple example finding the generalized fourier coefficient of a function in respect of two different bases. I was thinking about ...
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11 views

Norm and Inner Product Inequality in Hilbert spaces

Let $H$ be a Hilbert space, and suppose that $C \subset H$ is closed, convex and nonempty. Then, for $y_{j}=P_{C}(x_{j})$, $j=1,2$ where $P_{C}$ is the metric projection onto $C$ and $x_{1},x_{2} \in ...
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0answers
24 views

Distance from image of bounded operator

Let $A:H\rightarrow H$ be a bounded linear operator on Hilbert space $H$. Suppose we have $x\in H$ and $r>\mathrm{dist}(x,A(H))=\inf\{\|x-Ah\|,\ h\in H\}$. How to prove that then there exist ...
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0answers
51 views

Properties of weakly convergent series in Hilbert space

Let $H$ be a Hilbert space and $\{x_n\}_{n=1}^{\infty}$ given sequence of vectors from $H$. Suppose that for every $\{\alpha\}_{n=1}^{\infty}\in \ell^2$ series $\sum_{n=0}^{\infty}\alpha_nx_n$ is ...
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1answer
23 views

What does it mean for an inner product to be conjugate linear in the second entry?

Let $G$ be a group and $L^2(G) = \{f: G \rightarrow \mathbb{C} \}$. Now define an inner product on $L^2(G)$ by $$\langle f, g \rangle = \sum_{x \in G}f(x)\overline{g(x)}$$ Where $\overline{g(x)}$ is ...
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1answer
47 views

Explanation of “weight function” of inner product in Hilbert space

I am a physicist so I am sorry if the following is not written in a rigorous(or even completely right) way. As Quantum Mechanics is formed in Hilbert space, I would like to know what the weight ...
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4answers
53 views

Compact operator in Hilbert spaces reach the maximum in the sphere.

I found the following question in my textbook: (QUESTION) Let $\mathcal{H}$ a Hilbert space and $T: \mathcal{H} \rightarrow \mathcal{H}$ a compact operator. Show that exists $x \neq 0$ in ...
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1answer
43 views

Is there a nice way to express $\psi_1$ using this orthonormal sequence?

Suppose that $H$ is a separable Hilbert space and $(\psi_n)_{n=1}^{\infty}$ is a complete orthonormal sequence in $H$. We define a sequence $(\phi_n)_{n=1}^{\infty}$ by $$ \phi_n=\psi_1+\psi_{n+1}\ ;\ ...
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0answers
15 views

Prove that a function is continuous (square integrability)

I need help for the following proof of continuity: Let $E=L_2([t_0,t_1],\mathbb R)$ be a Hilbert space of square-integrable real-valued functions on $[t_0,t_1]$. Let ...
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0answers
30 views

orthonormalization for a hilbert space

sincerely, I'm stuck. Then, I have two questions: if we take $V=\{v\in H^1(0,1) ; v(0)=0\}$ and $Q=\{ w_1,w_2\}$ is a lineary independent set where $w_1 = \frac{*}{\Vert *\Vert_{V\cap H^2(0,1)}}$ and ...
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0answers
23 views

Uncountable basis in Hilbert space

For a (uncountable dimension) Hilbert space $\mathcal{H}$, suppose we have uncountably many vectors $K_x$, only $0$ is orthogonal to all of $K_x$. (Specifically, a reproducing kernel Hilbert space ...
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1answer
38 views

Tensor Product on Hilbert Spaces (well definedness)

Let $H_1$, $H_2$,...,$H_n$ be $n$ Hilbert Spaces. For each $\phi_i \in H_i$, Let $$ \phi_1 \otimes \phi_2 \otimes... \otimes \phi_n:= \text{Conjugate multilinear form which acts on $H_1 \times H_2.. ...
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1answer
20 views

Form of the Polar decomposition for $M_{\varphi}$

Polar Decomposition:Let ‎$‎‎v$ ‎be a‎ ‎continuous ‎linear ‎operator ‎on a‎ ‎Hilbert ‎space ‎‎$‎‎H$.then ‎there ‎is a‎ ‎uniqe ‎partial ‎isometry ‎‎$‎‎u\in B(H)$ ‎such ‎‎$‎‎v=u‎‎\mid ‎v‎\mid‎‎‎$ ‎and ...
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1answer
22 views

Compact operator with no non-zero eigenvalues is zero?

Suppose we have a Hilbert space $H$ and a compact operator $T$ acting on $H$. If $T$ has no non-zero-eigenvalues, is it necessarily the zero operator? Secondly, if I decompose $H$ into eigenspaces of ...
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0answers
21 views

How should $A^α$ be defined for real $α ∈ [0,∞)$ and $A\in M_n(\mathbb C)$?

Let $A\in M_n(\mathbb C)$ be arbitrary. I'm interested to know How should $A^{\alpha}$ be defined for real $\alpha\in [0,\infty)$? When $A$ is nonsingular, we can define $A^{\alpha}=\exp(\alpha ...
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1answer
44 views

Example of a self-adjoint bounded operator on a Hilbert space with empty point spectrum

Good day, I wanted to find a self-adjoint bounded operator on a Hilbert space with empty point spectrum i.e. $$ T = T^* ~\text{but}~ \sigma_p(T)= \emptyset $$ Some definitions and results of the ...
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0answers
27 views

Mercer's expansion on Sinc function

I hope to know about the Mercer's expansion on $K(x,y) = \frac{\sin(x-y)}{\pi(x-y)}$, which is the reproducing kernel for a Hilbert space of band-limited functions. By Mercer's theorem, it can be ...
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1answer
22 views

Image of a dense set through unbounded operator

Let $T$ be a densely defined, closed operator on a Hilbert space $H$ such that $T^*T$ remains densely defined. Obviously, $\sigma(I+T^*T)\subset [1,\infty)$, which in particular implies this operator ...
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0answers
26 views

On the sequence of orthonormal basic

I have a question : Let $0 \leq a \leq b \leq +\infty $, supposing that ${\phi_n(x,t)}_{n \geq 0}$ be the orthonormal basis on $L^2(a,b)$ respected to $x$. If there exist a sequence $\psi_n(t)$ such ...
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0answers
13 views

Find the codimension of $\overline{\operatorname{span}}\{S^n(1,4,-1,0,0,\ldots):n=0,1,2,\ldots\}$ in $\ell_2$.

Find the codimension of $A=\overline{\operatorname{span}}\{S^n(1,4,-1,0,0,\ldots):n=0,1,2,\ldots\}$ in $l_2$ where $S$ is the shifting operator to the right: $Se_i=e_{i+1}$. I don't quote understand ...
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15 views

Is the distance attained?

Suppose that we consider the set $K:=\{ x \in \mathbb{R}^n: \sum_{j=1}^n |x_j|^p \leq 1 \}$ where $0<p<1$. In this case the set isn't convex. Indeed, if we pick for example $x=(1,0,0, \dots), ...
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1answer
36 views

Characterization Projection operator as distance minimizer

Let $H$ be a Hilbert space and $V$ be a subspace of $H$. How can I prove that for a map $P \colon H \rightarrow V$ the following are equivalent: $P^2=P$ and $P$ is linear $P(x) = ...
3
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1answer
76 views

Closure of an Operator in $l^2$

Let $l^2$ denote the Hilbert space of all complex sequences $\phi = (\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j|^2 < \infty$. Consider the linear subspace of $l^2$ defined by ...
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2answers
59 views

Adjoint of an Operator in $l^2$

Let $l^2$ be the Hilbert space of all complex sequences $\phi =(\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j |^2 < \infty$. Set $D= \{ \phi \in l^2 : \sum_{j=0}^{\infty} j ...
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1answer
22 views

A question concerning the triplet $V\subset H\subset V^*$

In Brezis' Functional Analysis book, p. 150, there is an exercise about the triplet $V\subset H\subset V^*$, where $(V,\|\cdot\|_{V})$ is a Banach space, $H$ is a Hilbert space with the scalar product ...
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1answer
34 views

Open sets in the unitary group $ U(\mathcal{H}) $ of a Hilbert space $ \mathcal{H} $.

Let $H$ be an infinite dimensional Hilbert space and let $(x_i)_1^\infty$ be an orthonormal basis for $H$. Consider $U(H)$ the unitary group of the continuous unitary operators on $H$. Equip $U(H)$ ...
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1answer
27 views

Find Riesz representation of $\phi=f({1\over 2})$

"Let $\rho$ be a space of complex polynomial and define $<f,g>={1\over 2\pi}\int_{0}^{2\pi}f(e^{it})\overline{g(e^{it})}dt$ for $f,g:\rho\to \Bbb{C}$. Let $\phi$ be a linear functional on ...
1
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1answer
24 views

If $U$ is unitary operator then spectrum $\sigma(U)$ is inside the unit circle- verification

In a Hilbert space, let $U$ be a continuous operator which it unitary. Prove $\sigma (U)\subseteq \Bbb{S}^1$. It is important for me to know how I am doing, and I didn't come by a clear explanation ...
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0answers
49 views

Proving compactness of an operator $(Kf)(t)=\int_{0}^{\infty}k(t+s)f(s)ds$

I was trying to prove the compactness of the following operator: $K:L_2([0,\infty))\to L_2([0,\infty))$ $(Kf)(t) = \int_{0}^{\infty}k(t+s)f(s)ds$, given that the function $k$ is continous, and ...
4
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0answers
56 views

Null Functional on $l^2$

Let $l^2$ be the hilbert space of all complex sequences $\psi= (\psi_n)_{n=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\psi_j |^2 < \infty$. Let $\phi= (\phi_n)_{n=0}^{\infty}$ be a sequence of ...
0
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2answers
57 views

problem on hilbert spaces

Let $X=C[0,1]$ with the inner product $\langle x,y\rangle=\int_0^1 x(t)\overline y(t)\,dt$ $\forall$ $x(t),y(t)\in C[0,1]$ $X_0 =\{x(t) \in X :\int_0^1 t^2x(t)\,dt=0\}$and $X_0^\bot$ be the ...
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0answers
40 views

Extend and restriction of operator on $B(H)$

Let ‎$‎‎H$ ‎be a ‎Hilbert ‎space ,‎‎‎‎‎‎$‎‎B(H)$ ‎be ‎bounded ‎operators ‎on ‎‎$‎‎H$ ‎and ‎‎$‎‎K(H)$ ‎be ‎compact ‎operators ‎on ‎‎$‎‎H$‎. Assume ‎that ‎‎$‎‎M$ ‎is a ‎close‎d subspace of ‎$‎‎H$ ‎and ...
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0answers
16 views

Compute the limit and show that uN converges weakly

full question I already know that the norm is 1, and that you can use the definition of weak convergence but that's where I get lost. Somebody told me I can use the Riesz representation theorem since ...