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

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1answer
19 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 ...
2
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1answer
35 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 ...
2
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4answers
41 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 ...
1
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1answer
37 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
14 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 ...
0
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0answers
19 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 ...
1
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0answers
22 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 ...
1
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1answer
31 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.. ...
1
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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 ...
0
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1answer
20 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 ...
3
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1answer
41 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 ...
0
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0answers
26 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 ...
2
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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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1answer
37 views

left shift operator and compact operator [closed]

Let ‎$‎‎S$ ‎be the ‎left ‎shift ‎on ‎‎$‎‎\ell^2$ ‎i.e ‎‎$‎‎S(x_1,x_2,x_3,...)=(x_2,x_3,...)$‎‎. ‎ Assume that ‎$‎‎T$ ‎is a ‎compact ‎operator ‎such ‎that ‎‎$‎‎TS=ST$.‎ ‎ Q:‎$‎‎T$ ‎should ‎be ‎zero ...
0
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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 ...
1
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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 ...
-1
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0answers
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), ...
0
votes
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) = ...
2
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0answers
43 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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votes
2answers
45 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 ...
0
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1answer
21 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 ...
1
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1answer
33 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 $U(H)$ ...
1
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1answer
26 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
22 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 ...
3
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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
54 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
56 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 ...
1
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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 ...
0
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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 ...
3
votes
1answer
52 views

Bounded Operators on a finite-dimensional Hilbert space - Linear combination of at most two unitaries and from a partial isometry to a unitary

Good day, In the lecture of functional analysis the proof of two statements were skipped as a task for us but I'm not sure how I approach these questions. a) Show that every partial isometry $V \in ...
0
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1answer
33 views

Semigroups on Complex Hilbert Spaces

Let $H$ be a separable complex Hilbert space, let $(e_i)_{i\in\mathbb{N}}$ be a complex othonormal vasis, and let $(\lambda_i)_{i\in\mathbb{N}}$ be a sequence of complex numbers s.t. $\sup_{i\in ...
1
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1answer
23 views

$L_1+L_2$ is close if $L_1\bot L_2$ are close sub-spaces of a Hilbert space $H$

$L_1+L_2$ is close if $L_1\bot L_2$ are sub-spaces of a Hilbert space $H$. While I do understand why it is true, I can't be completely sure how deduction is done here. I do know that if $\langle ...
3
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1answer
37 views

The weak topology on $H$ is the weak* topology on $H^*$ pulled back via $\Phi$

I'm reading the following in Analysis Now by Pedersen: The map, $H$ a Hilbert space $$\Phi:H\to H^*: x\mapsto(\cdot\mid x)=[y\mapsto (y,x)]$$ is a conjugate linear isometry. Then define the weak ...
0
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1answer
16 views

Show $A^*A$ is self adjoint for $A$ closed and description of the form domain.

I have been stuck with the next problem Suppose $A$ is a closed operator defined on $\mathfrak{D}(A)\subset \mathfrak{H}$, where $\mathfrak{H}$ is a Hilbert space. Show that $B=A*A$ is self ...
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0answers
20 views

Books to learn tensor product on hilbert spaces

I have just started to work on Quantum Computing. I have began to read a paper which deals with tensor product on hilbert spaces. I have a had a course in functional analysis. So I don't have an ...
3
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1answer
21 views

Orthonormal Bases in a Hilbert Space and Vector Subspaces

Let $H$ be a Hilbert space and $S$ a vector subspace of $H$ which is dense in H. Does there exist an orthonormal basis $(u_{\alpha})_{\alpha \in A}$ of $H$ such that $\{ u_{\alpha} : \alpha \in A ...
2
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0answers
30 views

Smallest closed subspace of $A$ in pre-Hilbert spaces [duplicate]

Let be $A\subset H$ a subset of $H$ Hilbert space. I know that $A^{\perp\perp}$ is the smallest closed subspace of $H$, such that $A\subset A^{\perp\perp}$. But if $H$ is a inner product space (or ...
1
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2answers
64 views

$L^{2}[-\pi,\pi]$ is unitarily isomorphic to $l^2(\Bbb C)$

So I have countable orthonormal basis of $L^2[-\pi,\pi]$ as $\{e^{inx}\}_{n \in \Bbb Z}$ and countable orthonormal basis of $l^2(\Bbb C)$ as $\{a_n\}_{n \in \Bbb Z}$ such that ...
1
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0answers
51 views

Lie Algebra SU(2)

Given a two dimensional Hilbert-space, $\mathcal{H}$, and a vector $\eta \in \mathcal{H}$, of this space, if $\eta$ transforms in SU(2) like this, $$\eta \rightarrow e^{(-i\alpha ...
2
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1answer
25 views

Countable Complete Orthonormal Set implies countable dense subet

Let $\mathcal H$ be a Hilbert Space, let $B = \{u_j\}_{j=1}^{\infty}$ be a countable orthonormal basis. So we know that if a set is a complete orthonormal basis, the set of all finite linear ...
1
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1answer
99 views

Show $\Bigl\{\sqrt{2\over {\pi}}\sin (nx)\Bigr\}_{n=1}^{\infty}$ is an orthogonal basis of $L_2[0,\pi]$

Show $\Bigl\{\sqrt{2\over {\pi}}\sin (nx)\Bigr\}_{n=1}^{\infty}$ is an orthogonal basis of $L_2[0,\pi]$. What I need is a verification and guidance. I managed to show that the set is orthogonal. My ...
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0answers
22 views

An operator is linear and bounded on a hilbert space

an operator linear and bounded on a hilbert space Let H be the Hilbert space L^2(R), and assume that the continuous function g satisfies 0
0
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0answers
52 views

unique function on a hilbert space

unique function on a hilbert space How do you show that with Ω=(-1,1) there exists a unique function u such that the equations in the picture is correct?
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0answers
25 views

Use Cauchy-Schwarz inequality to prove that $\langle\,,\rangle : \mathscr H \times \mathscr H \to \Bbb C$ is continuous.

Let $(a,b) \in \mathscr H \times \mathscr H$ be fixed. So we have to prove that for a given $\epsilon \gt 0$, we can find $\delta_1 \gt 0$ and $\delta_2 \gt 0$ such that $\lvert \langle x,y\rangle - ...
0
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0answers
33 views

Prove that $f(x,y)$ defines an inner product [duplicate]

Let $(E,\left\lVert . \right\rVert)$ be a normed vector space defined on $\mathbb{R}$ . We suppose that the norm satisfies the Parallelogram law. Prove that: $$f(x,y)=(1/4)[(\left\lVert x+y ...
0
votes
1answer
18 views

Is the distance of an element $a$ from a subspace $M$ always $||a-P_M a||$?

The distance of an element $a$ from a subspace $M$ is $||a-P_Ma||$? ($P_Ma$ is the orthogonal projection of $a$ on $M$). During the course of studying about Hilbert Spaces and The Operators Theory, I ...
3
votes
1answer
43 views

Dense subsets of functional spaces

In books on Malliavin calculus and stochastic PDE, I found the following result is frequently used. I state it here in the simplest form. Given a separable Hilbert space $\left(H, \langle \cdot, ...
1
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0answers
32 views

Multiplication of Matrices in a Hilbert Space

So I was having a discussion with a friend as follows: Let $\mathcal H$ be a Hilbert space. Let $\mathcal H^{\otimes n} = \mathcal H \otimes \mathcal H \otimes \cdots \otimes \mathcal H$. $\mathcal ...
1
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2answers
38 views

Example of an operator that is not subnormal

In some recent questions the term subnormal operator has appeared. A bounded operator $A$ acting on a Hilbert space $H$ is called subnormal if there exists a Hilbert space $K$ containing $H$ as a ...