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

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On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem. I would like to thank for all kind help and construction. Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
9
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0answers
350 views

Is this question solvable? $2$ non-linear equations and the proof that the solution is unique (with asymmetric bounty option)

As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
9
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0answers
474 views

An infinite series expansion in terms of the polylogarithm function

We have the complex valued function: $$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$ We wish to recover the coefficients $a_{n}$. The only thing I though would ...
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4answers
955 views

An idempotent operator is compact if and only if it is of finite rank

Would you help me to solve this problem. Show that an idempotent operator on hilbert space is compact if and only if it has finite rank.
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2answers
4k views

Weak Convergence implies boundedness and componentwise convergence

Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in ...
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2answers
523 views

Haar's base for $L^2[0,1]$

$\newcommand{\span}{\operatorname{span}}$ Define $e_{0,0}\equiv 1$, and for all $n\in \mathbb{N}$ $$e_{n,k}=\begin{cases} 2^{n/2} &\text{if } \frac{k-1}{2^n}\leq x\lt \frac{k-\frac{1}{2}}{2^n}\\ ...
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2answers
625 views

Is compactness a stronger form of continuity?

Let $H$ be a Hilbert space. We say that a linear operator $T \colon H \to H$ is compact if it maps bounded sets to precompact ones, that is, if for every bounded sequence $(a_n)$ in $H$, $(Ta_n)$ has ...
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1answer
2k views

Equivalent inner products on a Hilbert space

Take a Hilbert space $(\mathcal H,(\cdot,\cdot)_{\mathcal H})$ and two equivalent inner products $(\cdot,\cdot)_1$ and $(\cdot,\cdot)_2$ on $\mathcal H$, i.e. such that there are $a,b \in \mathbb R$ ...
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1answer
1k views

Is the right shift operator bounded?

I was reading my lecture notes for functional analysis when I came across the following statement: Let $(e_{n})$ be a total orthonormal sequence in a separable Hilbert space H. The right shift ...
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2answers
170 views

Prove of inequality under a Hilbert space.

Let $x\neq y$ when $x,y\in H$ and H is a Hilbert space which satisfy $\|x\|=\|y\|=r$. Show that $\|\frac{x+y}{2}\|<r$. Actually in my question r=1 but as far as i could understand there is a way ...
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1answer
134 views

Sufficient Condition for $f\in L^{1}(\mathbb{R}^{d})$ to belong to $L^{2}(\mathbb{R}^{d})$ in terms of its Fourier coefficients

Question. Let $\left\{\varphi_{j}\right\}$ be a complete orthonormal system for $L^{2}(\mathbb{R}^{d})$ such that each $\varphi_{j}\in C_{b}(\mathbb{R}^{d})$ (the space of continuous, bounded ...
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1answer
160 views

Is $\mathcal{C}([0,1])$ homeomorphic to a Hilbert space?

Let $\mathcal{C}([0,1])$ the Banach space of continuous functions from $[0,1]$ to $\mathbb{C}$. The norm on $\mathcal{C}([0,1])$ is $f \mapsto \| f\|_{\infty}= \sup_{x \in [0,1]} |f(x)|$. Is it ...
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3answers
341 views

For what sequences of real numbers $\left\{ k_{n}\right\}$ is the set of functions $\left\{ e^{ik_{n}x}\right\}$ a basis?

It is well known that the set of functions $\left\{ e^{^{inx}}\right\}$, for integer $n$, is an othonormal basis for the space of square integrable real functions in the interval $[-\pi,\pi]$. Now ...
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1answer
134 views

Vector space that can be made into a Banach space but not a Hilbert space

Are there any (real or complex) vector spaces which can be made into a Banach space given a suitable norm, but cannot be given a norm that makes it a Hilbert space? I know that the parallelogram law ...
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1answer
629 views

Physical (Quantum Mechanical) Significance of completeness of Hilbert Spaces.

I'm not sure if the question is very 'mathematical',but I'm asking any way. I have a basic knowledge of quantum mechanics and I'm studying Hilbert spaces. I was wondering what is the physical ...
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1answer
78 views

A sequence that converges weakly but not in the Cesàro sense

Let $H$ be a Hilbert space over $\mathbb{C}$ with inner product $\langle\cdot,\cdot\rangle$, and let $\{x_n\}_{n=1}^\infty\subseteq H$, $x\in H$. I'm using the following definitions: ...
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1answer
102 views

What's the intuition behind the direct integral of a family of Hilbert spaces?

In order to understand better the mathematically rigorous version of Dirac's formalism in Quantum Mechanics I've been reading about direct integrals of Hilbert spaces, projector-valued measures and so ...
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1answer
248 views

Is my statistician friend right/wrong on metric spaces and norms?

I was talking to a statistician friend of mine who said that instead of minimizing this function $\sum_{i,j}W_{ij}d_{ij}^2(X)$ over $X$ it would be better to solve an analogous minimization problem ...
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4answers
4k views

A linear operator on a finite dimensional Hilbert space is continuous

How do I show that a linear function from a Hilbert space $H$ to itself is continuous if $H$ is finite dimensional? Also, what would be an example of a linear function from a Hilbert space to itself ...
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3answers
601 views

A complete orthonormal system contained in a dense sub-space.

Let H be a separable complex Hilbert space. Let A be a dense sub-space of H. Is it possible to find a complete orthonormal system for H that is contained in A?
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3answers
578 views

If a linear operator has an adjoint operator, it is bounded

This is a question I'm struggling with for a while: Let $H$ be a Hilber space. Let $T,S: H\rightarrow H$ be linear operators (not neccessarily bounded) such that for every $x,y\in H$: $\langle ...
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1answer
949 views

Does there exist a real Hilbert space with countably infinite dimension as a vector space over $\mathbb{R}$?

Essentially what the title says - where to me a Hilbert space is a complete (Hermitian) inner product space, am I safe to assume every such real Hilbert space is of uncountable dimension over ...
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146 views

The spectrum of a self-adjoint operator on $\mathcal l^2$

Let $S$ be the unilateral shift operator on $\mathcal l^2$ (which shifts one place to the right) and $S^*$ its adjoint, the backward shift (which shifts one place to the left). I've been trying to ...
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2answers
3k views

Example that in a normed space, weak convergence does not implies strong convergence.

The book "Introductory Functional Analysis with Applications" (Kreyszig) presents the following definitions. A sequence $(x_n)$ in a normed space $X$ is said to be strongly convergent if there is ...
7
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3answers
362 views

Sequences are Cauchy depending on the norm?

Assume that we have two Banach spaces $B_1, B_2$ with their underlying sets being identical. Is it possible that a Cauchy sequence in one of the spaces would fail to be Cauchy in the other? I am ...
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1answer
300 views

Existence of the Pettis integral

This is related to a question of MO: A question on the integral of Hilbert valued functions. I'm sure it's easy, but I cannot think right now, so I thought I'd ask. Let $f:[0,1]\rightarrow H$ be a ...
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1answer
239 views

Energy estimate of the differential equation $\dot{x}=Ax$

Conside the differential equation $$\dot{x}=Ax,\qquad x(t):{\bf R}\to{\mathcal H}$$ where $\mathcal{H}$ is a Hilbert space and $A$ is a bounded linear operator. With the initial condition, one can ...
7
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2answers
144 views

Is every Hilbert space a Banach algebra?

Let $H$ be a Hilbert space. Could we say that, always there is a multiplication on $H$, that makes it into a Banach algebra? If not, under which conditions does it exist?
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1answer
141 views

“Refinement” of the existence of faithful representations of C*-algebras?

Conway, in a course in operator theory, brings the statement 1. below as a theorem and statement 2. below as an exercise. Still, he states that 2. refines 1., but I can't see it. Every C*-algebra ...
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295 views

Counterexample for the stability of orthogonal projections

Let $V$ be a seperable Banach space, which is dense and continuously embedded in a Hilbert Space $H$. Let $(V_m)$ be a Galerkin scheme (See definition below) for $V$. Using the embedding we can ...
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1answer
1k views

A few questions about the Hilbert triple/Gelfand triple

I am attempting to fully understand Hilbert triples by reading Brezis' Function Analysis book. Consider $V \subset H \subset V^*$, where $V$ is Banach and $H$ is Hilbert. $V$ is dense in $H$. ...
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1answer
76 views

A question concerning Mazur's Lemma

I have a problem with application of Mazur's Lemma. Just consider $B(H)$ when $H=\ell_2$. Then, $B(H)$ is a normed vector space. Then, take operators $$X_n:={\rm ...
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2answers
1k views

Contexts For Bessel's Inequality?

Bessel's inequality appears to be about orthonormal sequences. But (in the context of inner product spaces), I've thought of this inequality as being a demonstration that the hypotenuse of triangles ...
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1answer
329 views

Non-examples for the Kato-Rellich Theorem

The Kato-Rellich Theorem is a classical result stating that if $A,B$ are unbounded operators on a Hilbert space with $A$ self-adjoint, $B$ symmetric, $\mathcal D (A)\subset \mathcal D(B)$ and $$ ...
7
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1answer
113 views

Tight Probability on Hilbert space

I am considering the following problem. Let $(X_j)$ be i.i.d. $N(0,1)$ random variables and $H$ a Hilbert space with orthonormal basis $(e_j)$. Let $$X:=\sum_j \frac{X_j e_j}{j}$$ And for any ...
7
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1answer
278 views

Reproducing Kernel Hilbert Space is dense?

Let $E=C[0,1]$, space of all real-valued continuous functions on $[0,1]$, $\mathcal{E}$ be its Borel $\sigma$-algebra and $\mu$ a Gaussian measure on $E$. Let $E^*$ be a space of all continuous ...
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2answers
677 views

Does this statement about Hilbert spaces make any sense?

I have found this tweet about git and don't know what to make of it. git gets easier once you get the basic idea that branches are homeomorphic endofunctors ...
7
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1answer
370 views

Approximating a Hilbert-Schmidt operator

Let $H$ be a separable Hilbert space. Recall that a bounded operator $A : H \to H$ is said to be Hilbert-Schmidt if $$\|A\|_{HS}^2 := \sum_{i=1}^\infty \|A e_i\|^2 < \infty$$ where ...
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2answers
280 views

Absolute Continuity of Finite Borel Measure Characterized by Orthonormal Basis

I've been working through Fundamentals of Stochastic Filtering (Bain, Crisan) and am a little perplexed by the following (initially) seemingly straightforward exercise and its given solution. We are ...
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1answer
1k views

Isomorphic Hilbert spaces

As part of a broader proof , I need to show that every two separable Hilbert spaces (that contains a dense countable set) are isomorphic (the linear mapping from one space to the other is injective ...
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1answer
1k views

Hilbert Schmidt integral operator

Hilbert-Schmidt Integral operators are usually defined from $H=L_2[a,b]$ into $H=L_2[a,b]$ as $$(Tf)(x) = \int_a^b K(x,y)f(y) dy,$$ provided that $K(x,y)$ is a Hilbert Schmidt kernel, namely ...
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2answers
517 views

On the isometry between bounded linear operators and the dual of nuclear linear operators

Let $H$ be a separable Hilbert space. Let $(e_i)_i$ be an orthonormal basis. For any bounded linear map $T$ we write, whenever possible $$\operatorname{tr} T := \sum_{i}^{\infty} \langle T e_i, e_i ...
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1answer
256 views

Holomorphy of a function with values in a Hilbert space

Denote by $\mathbb C^\infty $ the Hilbert space $\ell^2(\mathbb C)$. Fix $1\leq N,M \leq \infty$, and let $U$ be an open subset of $\mathbb C^N $. Following Mujica's book "complex analysis in Banach ...
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1answer
257 views

Does this sequence of operators converge in norm or strongly?

Let $H$ be a Hilbert space and $\mathcal{L}(H)$ the set of all bounded linear operators $L:H\to H$, equiped with the usual norm $\|\cdot\|_{\mathcal{L}}$. Let $T:D(T)\subset H\to H$ be a ...
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5answers
495 views

What does it mean for two functions to be orthogonal? [duplicate]

When two finite dimensional vectors are orthogonal, i.e. perpendicular, their dot product is exactly zero, e.g. $$\mathbf{a}\cdot\mathbf{b}=a_1b_1+\cdots+a_nb_n=0.\tag{1}$$ When I studied functional ...
6
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2answers
2k views

Relationship of Fourier series and Hilbert spaces?

I just read in a textbook that a Hilbert space can be defined or represented by an appropriate Fourier series. How might that be? Is it because a Fourier series is an infinite series that adequately ...
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2answers
1k views

Prove that this integral operator is compact

Let $X,Y=L^2(0,1)$, $k\in C^0([0,1]^2)$. Define $$ K:X\to Y,\,\,\,\,\,Kf(x):=\int_0^1k(x,y)f(y)dy\,\,\,\,\forall\, f\in L^2(0,1). $$ I have to show that $K$ is compact. My idea is to prove that $K$ ...
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2answers
351 views

For a multiplication operator $M_f$ on $L^2$ with $f\geq 0$, is $SM_fS^{*}$ positive?

I have the following problem. Let $\Omega \subset R^n$ have finite measure, let $H = L^2(\Omega)$ and let $S: H \to H$ be a bounded linear operator. Then it is well known that $P = SS^*$ is a positive ...
6
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1answer
2k views

Compactness of Multiplication Operator on $L^2$

Suppose we have an bounded linear operator A that operates from $L^2([a,b]) \mapsto L^2([a,b])$. Now suppose that $A(f)(t) = tf(t)$. Is A compact? Edit: I know $A = A^*$ but I'm not really sure ...
6
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2answers
116 views

Does orthogonal decomposition characterize direct sums in Hilbert space?

Let $H$ be a Hilbert space with inner product $(\cdot, \cdot)$. I know that if $M$ is a closed subspace of $H$, then $H$ can be written as the direct sum $M \oplus M^\perp$, where $M^\perp$ stands ...