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

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Sum of closed subspaces of a Hilbert space is closed

Let $M, N ⊂ H$ ($H$ Hilbert), be two closed linear subspaces. Assume that $\langle u, v\rangle = 0$ $∀u ∈ M$, $∀v ∈ N$. Prove that $M + N$ is closed. Take a sequence $(g_n)\in M+N$ such that $g_n\to ...
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1answer
52 views

Orthogonal projection on subspace

Let $\Omega$ be a measure space and let $h : \Omega → [0, +∞)$ be a measurable function. Let$$K = \{u ∈ L^2(\Omega);\ |u(x)| ≤ h(x)\ a.e. on\ \Omega\}.$$ Check that K is a non-empty closed convex ...
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42 views

Hilbert space Inequality

Let $K ⊂H$ be a nonempty closed convex set, $H$ Hilbert. Let $f ∈H$ and let $u=P_Kf$. Prove that $$||v − u||^2 ≤ ||v − f ||^2 − ||u − f ||^2, ∀v ∈ K$$ I've tried to use the parallelogram identity and ...
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1answer
40 views

Orthogonal complement of a subspace in a Hilbert space

In this question it is stated that if $W$ is a closed subspace of $V$, we may define $W^\perp$ to be the subspace of all vectors $v \in V$ such that $\langle v | w\rangle =0$ for all $w \in W$. ...
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38 views

Spectral definition of (fractional) Laplacian, need help understanding text

Let $\varphi_k$ and $\lambda_k$ be the eigenfunctions and eigenvalues of the Dirichlet Laplacian $-\Delta$ on some bounded domain $\Omega$. We know $\varphi_k$ are smooth and form an orthogonal basis ...
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2answers
29 views

Proving compactness of an operator

I'm having a hard time finding a solution for the following problem: Prove that the operator $ T \in \mathcal{L}(\ell_2) $ defined with the formula $$ T((x_1, x_2, \dots, )) = (0, x_1, x_2/2, x_3/3, ...
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1answer
27 views

Strict Bessel inequality in $\mathcal{l}^2$

I'm asked to give an example of an $x\in \mathcal{l}^2$ s.t. $$\sum_{j=1}^{\infty}|<e_j,x>|^2<||x||^2$$ Where $(e_j)$ is some orthonormal sequence. However, I think this question is more ...
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1answer
29 views

General Fourier coefficients and smoothness

Suppose $f\in L^2([0,1],\lambda)$. Are there assumptions on the smoothness of $f$ which translate into the particular behavior of Frourier coefficients. Namely, I have arbitrary complete orthonormal ...
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1answer
20 views

Equivalent finite subspaces of a hilbert space

I have to prove the following statement: Let $H$ be a Hilbertspace and $M,N$ closed subspaces. Then the following holds: If $M \sim N $ and $N$ is finite, then $M$ is finite. I think it should say ...
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2answers
61 views

Fock Space: NESS

Given the CAR-algebra with Hamiltonian dynamics: $$\tau^t[a^\#(\eta)]=a^\#(e^{itH}\eta)\quad(H:\mathcal{D}\to\mathcal{H})$$ (Caution that the Hamiltonian is usually unbounded.) Consider a KMS-state: ...
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1answer
40 views

Why does this completion of a Sobolev space contain constant functions? Please explain text.

Below, $\mathcal{C} = \Omega \times (0,\infty)$, $x$ refers to the variable in $\Omega$ and $y$ to the variable in $(0,\infty)$, and $\Omega$ is a bounded smooth domain. $tr_\Omega:H^1(C) \to ...
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2answers
40 views

Fock Space: Formal Adjoints

Problem Given a pre-Hilbert space $\mathcal{H}$. Consider unbounded operators: $$S,T:\mathcal{H}\to\mathcal{H}$$ Suppose they're formal adjoints: $$\langle ...
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0answers
38 views

What is the idea behind interpolation spaces?

I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific: Definition. Let $H$ be an ...
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59 views

When $1 \le p \le \infty, p\ne 2$, $L^p$ space is not a Hilbert space

It suffices to show that when $1 \le p \le \infty, p\ne 2$, $L^p$ norm does not arise from an inner product.(there is a hint saying that we can use the parallelogram law) I can proof a special case ...
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0answers
43 views

Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...
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1answer
42 views

Prove that if ,$||f||^2 = A\sum_{j}|<f, \phi_j>|^2 $ then $f = \sum_{j}<f, \phi_j> \phi_j$

Let $\phi_k$ be some sequence of real functions in an infinite Hilbert space $H$ such that there exists $A \in \mathbb{R}$ such that for all $f \in H$ ,$||f||^2 = A\sum_{j}|<f, \phi_j>|^2 $ ...
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1answer
22 views

For a given Hilbert space find a tight frame with bound A

For a given Hilbert space and $A>0$ find a tight frame with bound A. I know that an ortho-basis is a tight frame with $A=1$. Can I extend this to any $A>0$ by just scaling the ortho-basis?
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51 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
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1answer
49 views

Theorem 3.6-4 in Erwin Kreyszig's Introductory Functional Analysis With Applications

Here's the statement of Theorem 3.6-4 in Erwin Kreyszig's Introductory Functional Analysis With Applications: Let $H$ be a Hilbert space. Then (a) If $H$ is separable, then every orthonormal set in ...
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1answer
53 views

Invariant Subspace Problem: Non-Seperable Hilbert Space

I was reading an article, about the invariant subspace problem.. The statement of the problem is as follows: Given an $n$-dimensional Hilbert Space (or complex Banach Space) $\mathcal{H}^n$, does ...
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1answer
82 views

Example: operator injective, then the adjoint is NOT surjective

Let $T: V \rightarrow W$ be a bounded operator on normed spaces $V,W$. Now, there is a unique adjoint operator $T': W' \rightarrow V'$ defined by $T'(\alpha) = \alpha \circ T$. In finite dimensional ...
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1answer
58 views

How to use Triangle inequality to find the projection onto unit ball?

The projection onto the unit ball $$C:=\mathbb{B}(0,1)=\{x:||x||\leq1\}$$ is given by $$P_{C}(x)=\frac{x}{max\{||x||,1\}}, \quad\forall x\in X$$ where $X$ is Hilbert space. Now I can understand this ...
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64 views

How to prove a Banach normed vector space is NOT a Hilbert space?

We know that the Banach space $\big(\Bbb R^n,\|\cdot\|_2\big)$ is a Hilbert space with inner product $\langle x,y\rangle := \sum_{k=1}^n x_ky_k$. However, how to prove that $\big(\Bbb ...
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31 views

Construction of direct sum of Hilbert spaces

The following is a theorem of Takesaki's operator theory: I do not know why he puts $\bar H$ instead of $H$ for $n=-1,-2,\ldots\,{}$. Please help me. Thanks.
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1answer
54 views

Computing an orthogonal projection,

I'm trying to find a vector in $\mathbb{R}^4$ that is both orthogonal to the space $W$ spanned by $\{(1,2,0,1), (0,1,1,1)\}$ and happens to be "closest" to the vector $(3,3,3,3)$. From reading ...
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24 views

States: Liouvilleans

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider an invariant state: $\omega\circ\tau^t\equiv\omega$ Then the dynamics is unitarily implementable: ...
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63 views

An inequality for positive operators

Let $S$ and $T$ be positive operators on a Hilbert space $\mathcal{H}$. Suppose that $S \le T$. Since the square root function is operator monotone, it follows that $S^{1/2} \le T^{1/2}$. Does the ...
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1answer
46 views

Convergence of series in a Hilbert Space

I'm hoping for some help on the following question. I haven't gotten very far: Let $\{h_n\}_{n\geq 1}$ be a sequence of vectors in a Hilbert space $H$ with the property that $(h_n-h_m)\perp h_m$ ...
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1answer
99 views

If $p$ and $q$ are projections how can I prove that $\|p(1-q)(x)\|\le \|q(1-q)(x)\|$?

Let $p,q$ be orthogonal projection operators in a Hilbert space. Does the following equation hold? $$ \langle p(1-q)x,p(1-q)x\rangle = \langle p(1-q)x,x\rangle$$ It's clear to me that $ \langle ...
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1answer
40 views

Is this extension a Hilbert space?

Let $V$ be an inner product space over $\mathbb{F}$. Let $H$ be a complete subspace of of $V$ and $x\in V\setminus H$ Define $K= span(H\cup \{x\})$. Is $K$ a Hilbert space? How do I prove it?
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53 views

Existence of adjoint of the inverse

Let $H$ be a Hilbert space over $\mathbb{F}$ and $V$ be an inner product space over $\mathbb{F}$. Let $T:H\rightarrow V$ be a bounded linear bijection. If $V$ is a Hilbert space, then the open ...
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1answer
82 views

A operator is unitary if and only if it is a surjective isometry

I'm trying to prove the following result. Let U be an operator of a Hilbert space H, then $U$ is an unitary operator $\iff$ $U$ is an isometry and $R_u = H$ ($U$ is onto and isometry) I tried to use ...
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1answer
105 views

How to prove the sum of RKHS (Reproducing Kernel Hilbert Space)?

$k,k_1$ and $k_2$ are kernels on $\mathcal{X}\times\mathcal{X}$, and $k=k_1+k_2$, then we have the following properties for the RKHS (Reproducing Kernel Hilbert Space) $\mathcal{H}$, $\mathcal{H}_1$ ...
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1answer
81 views

Can Fourier transform be seen as a decomposition over a basis in a space of tempered distributions

Fourier series of a function that belongs to $L^2([0,T])$ can be seen as a decomposition of this function over an (orthonormal) basis in the Hilbert space $L^2([0,T])$. Fourier transform of a ...
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79 views

Proof of the Riesz-Schauder Theorem (for compact operators) using the Analytical Fredholm Theorem

First of all sorry for my bad English, I'm an Italian student, hope to let you understand! I'm having a little troubles with the proof of the Riesz-Schauder theorem for Compact Operators. Some infos ...
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5answers
538 views

Why do bases of infinite dimensional spaces need to be orthonormal?

I asked this question following a discussion in my Mathematical Methods course and didn't get a satisfactory answer. If we have an infinite dimensional Hilbert space, why do we need an orthonormal ...
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43 views

Coercivity of a sesquilinear form on a Hilbert space

Given two Hilbert Spaces $(V,||\cdot||)$ and $(H,|\cdot|)$ with the compact inclusion $V\hookrightarrow H$ and a sesquilinear form $a(\cdot,\cdot)$ on $V$ such that: $\bf (i)$ $Re\ a(u,u)\geq 0\ ...
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1answer
50 views

Summary: Spectrum vs. Numerical Range

Reference A proof of the statement below is split into: Normal Operators: Spectrum vs. Numerical Range Spectral Measures: Spectrum vs. Numerical Range Problem Given a Hilbert space ...
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34 views

How to prove this specific kernel is not in RKHS?

Consider $\mathcal{X}=\mathbb{R}$, and $k(x,y)=xy=[\frac{x}{\sqrt{2}},\frac{x}{\sqrt{2}}]\cdot [\frac{y}{\sqrt{2}},\frac{y}{\sqrt{2}}]^T$, where we thus can define two kinds of feature maps ...
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40 views

How to prove $\Phi_w(x)=\cos (w^T x+b)$ is outside the RKHS associated with the Gaussian kernel function$K(x,y)=\exp(-\frac{||x-y||^2}{2\sigma})$?

How to prove $\Phi_w (x)=\cos (w^T x+b)$ is outside the RKHS ( Reproducing Kernel Hilbert Space) associated with the Gaussian kernel function$K(x,y)=\exp(-\frac{||x-y||^2}{2\sigma})$?
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How to understand ' Let $\mathcal{H}$ be a Hilbert space of functions $f$ : $ \mathcal{X} \rightarrow R$, denoted on a non-empty set $\mathcal{X}$.'

I am a beginner. By asking this question, I means that, to construct a Hilbert space, should $\mathcal{X}$ satisfy some properties? Furthermore, in some papers especially on machine learning, ...
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1answer
69 views

Why $\|f\|_2^2$ can be written as $\int |f(x)|^2 p(x)dx$?

I am recently learning some papers on optimization in infinite-dimensional space, and I not familiar with function analysis. In some papers, I see $\|f\|_2^2$ is written as $\int |f(x)|^2 p(x)dx$ , ...
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2answers
51 views

What is the definition of closed subspace?

I am trying to understand what is intended with closed subspace, I took the following guess: A closed subspace $M$ of a Hilbert space $H$ is a subspace of $H$ s.t. any sequence $\{x_n\}$ of elements ...
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1answer
68 views

$L_2$ norm and RHKS norm in Hilbert spaces $\mathcal{H}$

According to this paper (just right below the Theorem 3 and above the section 3) Reproducing Kernel Hilbert Space(RKHS) $\mathcal{H}$ on $\mathcal{X}$ is a Hilbert space of functions from ...
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1answer
39 views

The inner product of the Cartesian Product space

I want to know how can one define the inner product in the Cartesian product of spaces, i.e. let $A,B$ two hilbert spaces. Let $a_1, a_2 \in A$ and $b_1, b_2 \in B$, how can one express the inner ...
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3answers
61 views

On the intuition behind the projection theorem.

I have recently proved the projection theorem in a Hilbert space setting. The statements were: If $M$ is a closed subspace of a Hilbert space $H$ and $x \in H$, then: There is a unique element ...
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1answer
27 views

Determining the orthogonal complement of $\{1 \}^\perp$ in $L^2[0,1]$

Consider the space $L^2[0,1]$ of complex valued square-integrable functions $f : [0,1] \to \mathbb{C}$. Let $\langle f, g \rangle = \int_0^1 f \bar{g}$ denote the standard $L^2$ inner product. For $M ...
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1answer
22 views

The Spectral Radius of a Product of Two Hilbert-Space Operators

I’m given a Hilbert space $ \mathcal{H} $ such that $ \dim(\mathcal{H}) > 1 $, and I’m supposed to construct two operators $ A $ and $ B $ on $ \mathcal{H} $ such that $ r(A B) \neq r(A) r(B) $. Is ...
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1answer
96 views

What is the orthogonal of an intersection?

Background I have been introduced to the notion of orthogonal complement of a subset of a (pre)hilbert space. Given $X$ a (pre)hilbert space and $A\subseteq X$, one defines $A^\perp:=\{x\in X:x\perp ...
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57 views

Construct a unitary operator U on H with prescribed spectrum

Given an infinite dimensional Hilbert space $H$. Let $|\lambda_k| = 1$ for $k = 1, ..., n$. Construct a unitary operator $U$ on $H$ such that $\sigma(U) = \{\lambda_k\}$ for $k=1,....,n.$ I can ...