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

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3answers
160 views

show that $l^2$ is a Hilbert space

Let $l^2$ be the space of square summable sequences with the inner product $\langle x,y\rangle=\sum_\limits{i=1}^\infty x_iy_i$. (a) show that $l^2$ is H Hilbert space. To show that it's a ...
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0answers
26 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
20 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
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 ...
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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
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 - ...
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4answers
44 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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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 ...
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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 ...
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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 ...
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1answer
32 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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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 ...
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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 ...
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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
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 ...
3
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1answer
688 views

Hilbert cube is compact

Let $\{u_n\}_{n\in \mathbb N}$ be an orthonormal set in $H$ (Hilbert space). How prove that the set $\displaystyle Q=\{x\in H :\ x=\sum_{i=1}^{\infty}{c_nu_n}, \ \mbox{where} |c_n|\leq\frac{1}{n} \}$ ...
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 ...
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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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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 ...
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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

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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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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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 ...
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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) = ...
44
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3answers
5k views

Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)

I am trying to understand the differences between $$ \begin{array}{|l|l|l|} \textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline \text{metric}& \text{metric ...
3
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1answer
77 views

Normal Operators: Superalgebra (I)

Given a Hilbert space $\mathcal{H}$. Consider normal operators: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Denote their calculus: ...
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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
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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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), ...
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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 ...
2
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1answer
802 views

Sesquilinear forms on Hilbert spaces

Definition of the problem Let $\mathcal{H}$ be a Hilbert space, and let $B:\mathcal{H}\times\mathcal{H}\rightarrow\mathbb{K}$ be a sesquilinear form. Prove that TFAE: $(i)$ $B$ is continuous. ...
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 ...
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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 $U(H)$ ...
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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 ...
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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 ...
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 ...
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1answer
477 views

distributivity of tensor product and direct sum for Hilbert spaces

Before I ask my actual question about direct sums and tensor products of Hilbert spaces, let's first talk about direct sums and tensor products of vector spaces. We might define direct sums of ...
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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 ...
3
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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 ...
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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 ...
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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 ...
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 ...
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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 ...
3
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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, ...
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1answer
176 views

Proving the closed unit ball of a Hilbert space is weakly sequentially compact

I bumped into this statement in Hofer-Zehnder in the middle of proving a Hamiltonian field always has a periodic orbit over a level set of the hamiltonian if that set is a regular compact and strictly ...
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 ...
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 ...