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

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3
votes
1answer
34 views

Do the holomorphic or meromorphic functions on a domain $D \subseteq \mathbb{C}$ form a Hilbert space $\mathcal{H}$?

In a physics paper I am reading that the meromorphic functions on $\mathbb{C}$ with $f(x) = f(\overline{x})$ form a Hilbert space. $$ \mathcal{H} = \{ f(x) : f(x) = f(\overline{x}) \}$$ Even let $f(...
3
votes
1answer
67 views

Spectrum of right shift operator in weighted $l2$ sequence space

Let $l_2(a)$ be a hilbert space defined with following inner product: $\langle x_n,y_n\rangle = \sum a^k x_k y_k$. (It's a weighted sequence space with the weights $\omega_i = a^i$). It's elements ...
1
vote
1answer
31 views

Relation betwen dimension of Hilbert space and cardinality of its dense subset

Suppose $H$ is infinite dimensional Hilbert space. Let $A$ be a dense subset of $H$. How to prove that $\mathrm{dim}_{\mathrm{orth.}}\ H \le \mathrm{card}(A)$ ? When we have equality ? I need only ...
1
vote
2answers
49 views

Consequence of the polarization identity?

Here is a proof which I do not fully understand. Theorem : Let $H$ be a Hilbert space. A continuous linear map $T : H \rightarrow H$ is self-adjoint (hermitian) if and only if $$\big\langle T(x), ...
0
votes
1answer
31 views

Linear Algebra proof involving traces and rank one projections

How does one prove that if $T$ is a non-negative linear operator over a finitely generated Hilbert space that satisfies $tr(T^2)=tr(T)=1$, then $T$ must be a rank one projection? I am sorry to give no ...
0
votes
1answer
36 views

Isometry between $L^{\bot}$ and $H/L$ where $H$ is hilbert and $L$ is a closed subsapce

Show $L^{\bot}$ and $H/L$(with $||\cdot||_{H/L}$) where $H$ is hilbert and $L$ is a closed subsapce are isometric, given $||\cdot||_H$ is defined by the inner product. By now I have considered $T:L^{\...
0
votes
1answer
42 views

Bilinear form is bounded

$\newcommand{\ints}{\mathbb{Z}}$ $\newcommand{\norm}[1]{ \lVert #1 \rVert }$ Given a bilinear form $a(x,y) = \sum_{n\in\ints}\sum_{m\in\ints}v_m x_{n-m} y_n$. $x \in \ell^2$, $y \in \ell^2$, $v \in \...
1
vote
1answer
46 views

Hilbert space sequence

Let $$ L^2:=\left\{f:[0,1] \to \mathbb R \; \middle| \; \int_0^1 |f|^2 < \infty\right\}, $$ $$ \langle f,g \rangle:= \int_0^1 f g, $$ and $$ \|f\| := \sqrt{\langle f, f \rangle}. $$ Prove that $\|...
4
votes
2answers
61 views

Almost orthogonal collection of $L^{2}$ functions

Suppose I had an infinite sequence of smooth compactly supported functions $\{f_{n}\}_{n \geq 1}$ such that for each $n$, $f_{n}$ is mutually orthogonal to all but a fixed absolute constant $C$ of the ...
1
vote
1answer
42 views

Haar functions form a complete orthonormal system

I want to show that the Haar functions in $L^2([0,1])$ forms an orthonormal basis: Let $$f = 1_{[0, 1/2)} - 1_{[1/2,0)} \ \ \mbox{,} \ \ f_{j,k}(t) = 2^{j/2}f(2^jt - k).$$ Let $\mathscr{A} = \{(j.k) :...
2
votes
1answer
24 views

Unitary Operator on Hilberspace to show that Fourierbasis is a maximal Orthogonal Set

I have looked at the proof Proving that the Fourier Basis is complete for C(R/$2*\pi$ , C) with $L^2$ norm but am having trouble understanding the argumentation about the Hilberspace. I think the ...
2
votes
1answer
59 views

Approximation property for Banach space and $l^{p}$

Let's consider a compact operator $T: X \rightarrow l^{p}, 1 \leq p < \infty$. I would like to check, whether it's possible to approximate $T$ by the operators of a finite rank with respect to an ...
1
vote
0answers
24 views

An operator which moves on the boundary

Let $H$ be a non-separable Hilbert space and $E$ be an orthonormal basis in $H$. Let $E_0$ be a countable subset of $E$ and $p$ be the projection onto the space generated by $E_0$. Let $\{\zeta_n\}...
1
vote
1answer
24 views

Please help with a general solution of a functional equation involving projections

I saw the following claimed : Let's say we have the functional equation $f(R+S) = f(R) + f(S)$ where R and S are projections in a vector space, and f is a real valued function. Then its general ...
2
votes
1answer
40 views

A system of equations

Let $H$ be a non-separable Hilbert space. Assume $E$ is an orthonormal basis in $H$. Let $E_0=\{e_n\}$ be a countable subset of $E$ and let $\{\zeta_n\}$ be a bounded sequence in $H$. Let $E_1$ be a ...
2
votes
0answers
21 views

Fréchet differentiability of Nemyckij operator defined on $L^2$

I have been told the following. Suppose $\Omega\subseteq\mathbb{R}^n$ is a bounded borel set, $f$ is Carathéodory function on $\Omega\times\mathbb{R}=\{(x,s):x\in\Omega,s\in\mathbb{R}\}$, $f_s$, ...
1
vote
0answers
21 views

What is Hoeffding's inequality in Hilbert space?

Suppose I have random variables $X_1, X_2,...,X_n \in \mathcal{H}$, where $ \mathcal{H}$ is some Hilbert space. How can I bound the following term - $ P(\| \sum_{i = 1}^n X_i - E[X_i] \|_{\mathcal{...
1
vote
0answers
25 views

If $T$ is quasinilpotent operator, then $‎‎\parallel‎ T^n ‎‎\parallel‎^{\frac{1}{n}}‎\rightarrow‎ 0$

Can someone prove that if $T$ is quasinilpotent operator, then $‎‎\parallel‎ T^n ‎‎\parallel‎^{\frac{1}{n}}‎\rightarrow‎ 0$, where $T$ is a bounded linear operator on a separable Hilbert space $H$.
0
votes
0answers
16 views

operators with finite trace form an ideal in B(H,H).

Let H be a Hilbert space and S$\in$ B(H,H). Define |S| to be the square root of S*S. the trace of nonpositive operators is given by tr(S) = tr(|S|). how can I show that the set of operators with ...
1
vote
1answer
28 views

compactness in $\ell^2$

How can I show T is compact when T is defined as $$ \text{T :}\,\ell^2 \to\ell^2\,\text{by Tx=y where} \,y_j=\alpha_jx_j\text{and}\,\alpha_j\to0\,\text{as}\,n\to\infty$$
0
votes
1answer
54 views

Is this “trick” allowed with an arbitrary inner products?

Say I have two sequences, $\{x_n\} \rightarrow x$ and $\{y_n\} \rightarrow y$, in a Hilbert space. Is this the following allowed?: $$\langle x_n,y\rangle - \langle x_n,y_n\rangle = \overline{\langle y,...
1
vote
2answers
38 views

Integral of an operator

In quantum mechanics we know that if $q$ corresponds to a complete set of parameters characterizing a quantum system, then the state vectors $|q\rangle$ satisfy the following identity: $$\int |q\...
-3
votes
1answer
64 views

What is the closed linear span of the Rademacher system in $L^2$? [closed]

The question is in the title: What is the closed linear span of the Rademacher system in $L^2$?. The definition of the Rademacher system can be found here.
0
votes
0answers
45 views

Vector valued functions VS Tensor Products - Anisotropic Banach Space

I am learning about anisotropic function spaces. To be simple consider functions $f(x,y)$ on the two-dimensional torus, ie functions on two circles. Suppose they are Sobolev functions $H^s$ in one ...
3
votes
1answer
81 views

Hilbert Space multiplication Operator, shift operator

I have this problem and am not sure how to even approach it.. Hilbert space $l^2(\mathbb{Z})$ with orthonormal basis$ $$(e_n)$ and Hamiltonian operator $He_n=i(e_{n+1}-e_{n-1})$ a)I need to use ...
4
votes
1answer
39 views

Can the system of shifts of an $L^2(\mathbb{R})$ function be an ONB?

In Wavelet theory, one constructs wavelet bases via translations a dialations of an $L^2$ function... Is it possible for some set of translations alone to form an Orthonormal Basis? That is: Does ...
0
votes
1answer
39 views

An operator satisfying in a sequence of equations

Assume that $H$ is a non-separable Hilbert space. Let $\{\eta_n\}$ be an arbitrary sequence in $H$. Let $\{\zeta_n\}$ be a sequence in $H$ which forms a linearly independent set. Does there exist ...
-3
votes
1answer
73 views

Curvature tensor for a particular Hilbert manifold

My question involves an infinite dimensional Hilbert manifold with a Riemannian metric. My question is: What is the form of the curvature tensor for a infinite dimensional Hilbert manifold with ...
1
vote
1answer
12 views

Determination of Rodrigues' Polynomials norm

Consider the $L^2([-1,1])$ function space with the follwoing inner product: $$\langle u(x),v(x)\rangle =\int_{-1}^{1} u(x)\bar{v}(x) dx$$ Rodrigues's Polynomials in $[-1,1]$ are defined as: $$R_n(...
1
vote
0answers
29 views

Correct this solution to finding $||A||$ of $(Af)(x) = g(x)f(x)$.

I need some help with a question I tried to solve it, but I am just not quite sure if my answer is correct. (I have got the feeling it can be - much - better). Suppose we have a complex Hilbert space ...
0
votes
1answer
26 views

Neighborhood of a Hilbert Space

Let $\mathbf 0$ be the sequence of real numbers with all the components equal to $0$ and, for each $n \in \Bbb N$, let $δ_n$ be the sequence of real numbers whose n-th component equals $1$ and all ...
1
vote
0answers
13 views

Prove Harr wavelet orthonormal basis

Define $$h = 1_{[0, 1/2)} - 1_{[1/2,0)} \ \ \mbox{and} \ \ h_{(j,k)}(x) = 2^{j/2}h(2^jx - k).$$ Let $A = \{(j.k) : j \geq 0, k = 0, 1, 2, ..., 2^j -1\}.$ I want to show that $\{1_{[0,1]}\} \cup \{h_{(...
0
votes
1answer
39 views

Proof involving Hilbert Cube

Let $\mathbf 0$ be the sequence of real numbers with all the components equal to $0$ and, for each $n \in \Bbb N$, let $δ_n$ be the sequence of real numbers whose n-th component equals $1$ and all ...
0
votes
0answers
20 views

A translation invariant sigma algebra in $B(H)$

Assume that $H$ is a non-separable Hilbert space. Let $s_0$ be the family of all basic neighborhoods in the strong operator topology. We denote $M_s$ by the sigma algebra generated by $s_0$. ...
1
vote
1answer
69 views

Show that every Lebesgue integrable function can be approximated in norm and almost everywhere by a sequence of continuous functions.

Show that every Lebesgue integrable function can be approximated in norm and almost everywhere by a sequence of continuous functions. I'm not sure where to even start with this. The question doesn't ...
0
votes
1answer
34 views

If $A$ is a positive operator and $B$ is a bounded operator, show that $B^*AB$ is positive.

If $A$ is a positive operator and $B$ is a bounded operator, show that $B^*AB$ is positive. Where both $A$ and $B$ are operators in a Hilbert space $H$. I know that if $A$ is a positive operator, ...
3
votes
1answer
110 views

Why is this set compact in $L^2(\mathbb{N})$?

Suppose $L^{2}(\mathbb{N})$ is the Hilbert space of sequences $(a_{n})_{n \in \mathbb N}$ which satisfy $\sum |a_{n}|^{2}$ with $(a,b) = \sum a_{n} \bar{b_{n}}.$ Prove the set of sequences $\{a_{n}\}...
0
votes
0answers
18 views

Approximate to integral of continuous function

I'm wandering given summation converges to integral.Original problem is this. Let $f$ is real valued continuous function on $\mathbb R $ such that $f (x)=f (x+1)$ for all $x$ in $\mathbb R$. ...
0
votes
1answer
43 views

Infinite dimensional hilbert space & compactness

I got a trouble for solving my home work. I have to prove the following If $H$ is infinite dimesional hilbert space, then specify a closed unit ball in $H$ is compact or not. First I though since $H ...
0
votes
1answer
51 views

Closure of a differential operator

Consider $A:\mathcal{D}(A)\subset L^{2}[0,1]\to L^{2}[0,1]$ given by $$A:=-\frac{d^{2}}{dx^{2}},\qquad\mathcal{D}(A):=C^{2}_{0}(0,1)$$ Now, I assume that the closure of $A$ is its extension defined ...
1
vote
1answer
34 views

The special case of the Riemann lebesgue lemma

I'm trying to prove the following Let $A$ be a measurable subset of $[0,2\pi]$ $$\lim_{n\to \infty} \int_A e^{inx} \, dx=0$$ There is a hint "this is the special case of the Riemann Lebesgue ...
2
votes
2answers
48 views

Condition for orthonormal set to be basis of Hilbert space

Let $H$ be a hilbert space. And let$ B$ be a basis of $H$. I think a orthonormal set$ S$ to be a basis iff $|S| =|B|$. (But I'm not sure about this) Am I right? If this is wrong, is the same argument ...
2
votes
2answers
47 views

Show $\|A\|=\sup_{x\neq 0} \langle Ax,x\rangle/\langle x,x\rangle$ for a positive operator $A$.

I have a positive operator $A$ on the Hilbert space $\mathcal{H}$. I must prove that $\|A\|=\sup_{x \ne 0}\frac{(Ax,x)}{(x,x)}$. I am only able to get one inequality: Assume $x$ is nonzero: $$\frac{...
2
votes
1answer
83 views

Best approximation theorem. Hilbert space

$Y$ is a closed subspace of a Hilbert space $H$ and $x \in H$. To prove that $x \in $ orthogonal complement of Y if and only if $||x-y|| \geq ||x||$ for all $y \in Y$. Necessary condition is easy to ...
-2
votes
2answers
43 views

A problem on conditional expectation and inner product in a Hilbert space [closed]

Let $X$ and $Y$ be two random elements in a Hilbert space. Then I have seen in a paper to use the following $$E[\langle X,Y \rangle] = E[\langle E[X|Y], Y\rangle].$$ Here $E[X|Y]$ is also an element ...
2
votes
1answer
39 views

weak convergence in hilbert space and exchanging of limits

Question: Let $\{x_n\}$ be a sequence of elements of a Hilbert space $X$ which weakly converge to $x\in X$. Assume also that $\limsup\|x_n\|\leq\|x\|$ Show that $\|x_n-x\|\to0$. Proposed Solution: ...
1
vote
2answers
46 views

Example of a densly defined positive self adjoint unbounded operator.

I know example of a densely defined positive self-adjont unbounded operator with discrete spectrum. What is the example of a self-adjoint positive unbound operator with continuous spectrum?
1
vote
2answers
42 views

Show that the set of functions in $L^2[0,1]$ with a zero integral on $[0,1]$ is a closed vector subspace of $L^2[0,1]$.

Let $H = L^2([0, 1])$ and let $K \subset H$ be defined as $K = \{f \in H \, : \, \int_{[0,1]} f \, \mathrm{d}m = 0\}$. Show that $K$ is a closed vector subspace of $H$. Find the element of $K$ that is ...
1
vote
1answer
23 views

Proving $0.5\exp(-|x-y|)$ is reproducing kernel for $W^{1,2}(\mathbb{R})$

Prove that $K(x,y)=0.5\exp(-|x-y|)$ is a reproducing kernel for $W^{1,2}(\mathbb{R})$, i.e. that $K(x,y)\in W^{1,2}(\mathbb{R})$ and for the continuous representative $\hat{f}$ of $f\in W^{1,2}(\...
1
vote
0answers
48 views

Prove the following are equivalent.

Let $E$ and $F$ be Hilbert spaces. For, $T \in B(E,F)$, show that the following are equivalent: $(i)$ $T$ is compact $(ii)$ $T^*$ is compact $(iii)$ There exists a sequence of finite rank operators ...