Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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9 views

looking for some exercise to test understanding of covector

I am trying to understand the concept of "covector" so that I can work with it for problems I come across in PDE. My knowledge on differential geometry is very little. And my topological background ...
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1answer
46 views

Some questions about an exercise about $C^\infty \subset L^\infty$

Let $$ L^\infty (\mathbb R) = \{f : \mathbb R \to \mathbb C\mid \text{essential sup of } f < \infty \text{ and } f \text{ Borel measurable} \}$$ and $$ C^\infty (\mathbb R ) = \{ f: \mathbb R ...
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2answers
27 views

Trace evaluation via complex analysis

We are given $U$, $V$ unitary matrices of size $N \times N$ whose spectral decomposition is known (in my specific problem, $N=4$, and $U$, $V$ are matrices with real coefficients but we can keep it ...
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1answer
30 views

Show that T is compact if $T^*T$ is compact

Let $H$ be a Hilbert space and $T: H\to H$ be a bounded linear operator. $T^*$ is the Hilbert adjoint operator. Show that $T$ is compact if $T^*T$ is compact. I am stuck with this proof. Any help ...
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12 views

“Occupation time” nonlinear functional measurable?

My question is for which functions $f$ the following nonlinear functional $f\rightarrow\int \mathbf{1}_B(f(x))dx$ is Borel measurable; $B\in\mathcal{B}(\mathbb{R})$ and $\mathbf{1}_B(.)$ is a ...
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35 views

isomorphism between function space and complex matrices

How would you show that $\mathcal{L}(X) \cong \mathbb{C}^{n \times n}$, where $X= \mathbb{C}^{n}$. Note that $\mathcal{L}(X)$ denotes the space of linear bounded functions on $X$. Is this a specific ...
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1answer
42 views

Proof of the integral operator in $L^2(\mathbb{R})$ being self-adjoint “by hand”

Suppose we have an integral operator $A$ such that $$Af(x) = \frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-\frac{(x-y)^2}{2}}f(x) \, dy$$ This operator is bounded and $\|A\|=1$ (see Norm of the ...
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20 views

Can we say $\| f\|_{\dot H^s(\mathbb R^n)} \le \| f \|_{\dot H^q(\mathbb R^n)}$ if $s\le q$?

If $s\le q$, then can we say that $$ \| f\|_{\dot H^s(\mathbb R^n)} \le \| f \|_{\dot H^q(\mathbb R^n)} $$ holds? Here the homogeneous Sobolev seminorm $\|f\|_{\dot H^s(\mathbb R^n)} = ...
3
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0answers
36 views

The Weak topology on an infinite-dimensional space is not metrizable

Let $X $ be an infinite-dimensional normed space I want to prove that weak topology on $X$ is not metrizable, this is my solution Assume that there is a metric $d$ on $X$ such that induced weak ...
2
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2answers
37 views

Badly formed question? $\|x\|=1=\|y\|$ and $\|x+y\|=\|x\|+\|y\|$, there is a line segment in the unit sphere

Show that if a normed linear space $X$ contains linearly independent vectors $x$ and $y$ such that $\|x\|=1$ and $\|y\|=1$ with $\|x+y\|=\|x\|+\|y\|$, then there is a line segment contained in the ...
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1answer
25 views

Strange inequality in the proof of differentiability of Fourier series

I am looking at a proof and I found a strange inequality. Let $n\in \mathbb{Z}^d$ then it is stated that $\sum_{j=1}^d{(2\pi)^{2k}n_j^{2k}}>>\parallel n\parallel_2^{2k}$ due to the inequality ...
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0answers
26 views

Is the limit of absolutely uniformly convergent series of functions a uniformly continuous function?

Let $f_n$ be a series of continuous and bounded functions on $\mathbb R$ such that $\sum|f_n|$ is uniformly convergent. Does $\sum |f_n|$ define a uniformly continuous function, and if so, how to show ...
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1answer
19 views

Do all continuous piecewise affine functions belongs to the class ($A_1$) of Muckenhoupt functions?

Given $\Omega\subset\mathbb R^N$ is open, we say $\omega$: $\Omega\to [0,+\infty)$ belongs to Muckenhoupt class $A_1$ if there exists some $C>0$ such that $$ \frac{1}{|{B}|}\int_{B(x,r)} ...
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1answer
34 views

Quotient set and inverse image.

(What i'm about to say is a subpart of a proof of a theorem). Let $M, N$ be a closed subspace and a finite dimensional subspace, respectively, of a normed linear space $X$. We define the natural map ...
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45 views

What are some functions $f \in L^\infty(\Omega)$

In my text book all it said about $L^\infty(\Omega)$ space is that it is the space of all measurable functions that are bounded almost everywhere. No example given. I can't see how any member of this ...
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1answer
25 views

Is there a name for embedding of Lebesgue spaces $L^q(a,b) ⊂ L^p(a,b)$

Let $1 ≤ p ≤ q ≤ ∞$ then $L^q(a,b) ⊂ L^p(a,b)$ where $L^q(a,b), L^p(a,b)$ are Lebesgue spaces Is there a name for this relationship? Is this the Sobolev embedding theorem?
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23 views

How Kriging, Bochner theorem and Positive definite (PD) function are related?

This question referes to the link: https://en.wikipedia.org/wiki/Kriging I can understand the relation between Bochner's theorem and PD function. But could not properly understand and connect all ...
4
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64 views

Random variables that span copies of $\ell_p$

Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation ...
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39 views

What is unit ball in the weak star topology of a Banach space? [on hold]

Let X is Banach space with dual $X^{*}$. What is unit ball of $X^{*}$in the weak star topology?
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26 views

Normal Operators: Superalgebra (II)

Problem highlighted at the end! Application Reduction to only one operator!! Reference This builds up on: Superalgebra (I) Convention All operators possibly unbounded!! Structures Given a ...
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0answers
12 views

Is the space of all adapted processes with Càdlàg paths a Banach space?

Consider first the definition of a stochastic integral for simple predictable processes. $$I:\mathbb{S}\rightarrow\mathbb{D},\ H\mapsto I_X(H):=H_0X_0+\sum_{i=1}^nH_i(X_{T_{i+1}}-X_{T_i})$$ The ...
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1answer
45 views

Dimension of $\left(\lambda |\psi\rangle \langle\psi| +(1-\lambda)\frac{\mathrm{I}}{2}\right)^{\otimes N}$

I have the $N$-fold tensor product of a convex combination of a pure state, i.e. $|\psi\rangle\langle\psi|$ with $|\psi\rangle$ a unit vector in a complex Hilbert space of dimension two, and the ...
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23 views

The dual space of weighted compact supported function?

Let $\Omega\subset \mathbb R^N$ be open bounded. It is well know that the dual space of $C_c(\Omega)$, i.e., compacted supported continuous function, can be identified by finite Radon measure ...
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26 views

Is $\mathbb R^N$ an $C$-distinguished topological space?

I am reading a paper which has some complicated construction on a Hausdorff topological space called $C$-distinguished topological space. The paper says that a $C$-distinguished topological space $X$ ...
6
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4answers
102 views

Definition of Equivalent Norms

Two norms $F,G$ are equivalent when there are constants $a,b$ such that $aF \le G \le bF$. I'm reading about this idea, and so far I've seen that equivalence of norms implies that the underlying ...
1
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1answer
20 views

Second differential of the norm in an infinite dimensional Hilbert space

Let $f: E \to \mathbb{R}$ sending $x \mapsto \|x\|$ and make some simple hypothesis $E$ is a Hilbert Space Let's say that the norm $\|\cdot\|$ is derived from a scalar product So we can easily ...
3
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1answer
32 views

Sobolev spaces and the domain of fractional Laplacian

I'm reading this paper on arxiv link. So far OK. Now this I don't understand. Take $s=\frac 12$. They say that by density the operator $(-\Delta)^s$ is defined on $\mathbb{H}^s(\Omega)$. ...
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1answer
37 views

Existence stochastic integral

I am trying to understand the prove of the existence of the stochastic integral for a local martinglale null at $0$ and continuous, $M\in \mathcal{M}^c_{0,\text{loc}}$, a predictable process $H\in ...
3
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22 views

The Hardy-Littlewood-Sobolev Inequality

Let $f:\mathbb R^n \to \mathbb C$, $n\ge 2$. I saw the line that the inequality $$ \left\| |x|^{-1} * |f|^2 \right\|_{L^\infty} \le C\|f\|_{L^{\frac{2n}{n-1},2}}^2 $$ with some constant $C>0$. Here ...
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38 views

Can a positive definite kernel expanded as the product form with an arbitrary orthonormal system?

Notations mostly follow https://en.wikipedia.org/wiki/Mercer%27s_theorem. Mercer's theorem uses the eigenfunctions $\{e_j\}$ of the integral operator as the expansion function. I wonder if we could ...
2
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0answers
30 views

Hyperplane in a complex vector space

This is my first question on MSE, I'm sorry if there already exists similar questions, I couldn't manage to find it. My friend, who studies Physics, asked me about the meaning of "functional" so I ...
2
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0answers
35 views

Expected value of multidinesional symmetric function is zero

Does anybody know a simple proof of this statement or reference to such proof? Statement Let $h: R^n \to R$ be a bounded function, symmetric in its arguments, i.e. for any permutation $\pi$ of set ...
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1answer
28 views

Use Riesz theorem to show functional bounded

I have the linear functional: $ F(v) = \int_\Gamma v \mathbf{g}\cdot\mathbf{n} d\Gamma$ where $\Gamma$ is a (smooth) part of the boundary of a domain $\Omega$, $\mathbf{g}$ is given (assumed smooth) ...
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3answers
35 views

Convergence on the dual of a Banach space

I have a simple question : What it means $$||v_n||_{(W^{1,p}_0)^*}\rightarrow 0$$ Where $(W^{1,p}_0)^*$ is the dual space of $W^{1,p}_0$ I know that $v_n\rightarrow 0$ in $(W^{1,p}_0)^*$ mease ...
2
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1answer
36 views

Specific question on $l^p$ spaces and its dual in weak * topology

I am covering now Lp spaces in my summer real analysis course and this problem from Folland related to the dual of Lp stumped me hard, it is problem 19 chapter 6 reads as follows: We define $ ...
4
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2answers
35 views

Parallelogram law in $L_1$ space

Exercise 5.5 from Capinski's and Kopp's book "Measure, Integral and Probability" asks to show that it is impossible to define an inner product on the space $L^1([0,1])$. In order to get this result we ...
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0answers
22 views

Generate complete functions set in hilbert space

It is just a curiosity, but is there a general method (or a class of methods) that allows to derive orthonormal complete function set for a given hilbert space? (Except Gram Shmidt algorithm and ...
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1answer
29 views

Quasi ideal sequence in $B(H)$

According to comments by Hamza I revise the question. Let $H$ be an infinite dimensional separable Hilbert space. Is there an increasing sequence of subvector spaces $V_{1} \subsetneq V_{2} ...
2
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1answer
44 views

Why is this a bounded operator?

Let $\mathcal{H}$ be the Hilbert space $l^2(\mathbb{N})\otimes l^2(\mathbb{Z})$. I want to prove that the operator $T$ defined by $$T:=\sum_{k=1}^{\infty}{\sqrt{1-q^{2k}}e_{k-1,k}\otimes 1}$$ is a ...
4
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2answers
177 views

Compact support vs. vanishing at infinity?

Consider the two sets $$ C_0 = \{ f: \mathbb R \to \mathbb C \mid f \text{ is continuous and } \lim_{|x|\to \infty} f(x) = 0\}$$ $$ C_c = \{ f: \mathbb R \to \mathbb C \mid f \text{ is continuous ...
3
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1answer
34 views

$\ell^p$ as a direct summand of $L^p$

I've been struggling with the following problem from a previous year's quals, and I don't know where to look it up (or even if it's supposed to be too obvious to write down). How do we embed $\ell^p$ ...
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1answer
27 views

Dense Domain: Preimage

Given Banach spaces $X$ and $Y$. Regard a bounded operator: $$A\in\mathcal{B}(X,Y)\implies A\in\mathcal{C}(X,Y)$$ Then for dense sets: $$W\leq Y:\quad \overline{W}=Y\implies\overline{A^{-1}W}=X$$ ...
3
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3answers
67 views

Multiplication operator on $L^1$

Let $\phi :X \rightarrow \mathbb{C}$ be measurable with respect to the measure space $(X,\mu)$. Suppose that $\phi f \in L^1(\mu)$ whenever $f \in L^1(\mu)$. Define $M_{\phi}(f)=\phi f$, for $f \in ...
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0answers
22 views

How to solve differential equations for linear operators?

I want to solve the differential equation $$ BA = \frac{\partial}{\partial t} A $$ for $A$. Here $A : H_1 \mapsto H_2$ and $B : H_2 \mapsto H_2$ are operators and $H_1, H_2$ are some Hilbert spaces. ...
3
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1answer
27 views

The dual function of composite functions

Given $X$ $Y$ are two finite dimensional Hilbert space. Let $K$: $X\to Y$ be linear and $F$: $Y\to \mathbb R^+$ is convex. Let us use $F^\ast$ to denote the dual (conjugate) function of $F$. Recall $$ ...
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1answer
34 views

Problem in showing that a sequence is a Cauchy sequence on a space with the integral metric.

I'm having difficulty following what is going on and understanding parts in the following example. It is quite similar to an example I posted before (Changing of the limits of integration with the ...
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2answers
24 views

denseness of polynomials in bounded borel measurable functions

Let $K\subseteq \mathbb{R}$ be compact, consider $B(K)$ the set of all bounded borel measurable functions $f:K\to \mathbb{C}$ and endow $B(K)$ with the uniform norm, so you obtain a Banach space. My ...
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0answers
19 views

Abelian Algebras: Generator

Given a Hilbert space $\mathcal{H}$. Consider normals: $$N:\mathcal{D}(N)\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ Denote their algebra: ...
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1answer
28 views

Analytic version of Hahn-Banach using geometric version

When studying the Hahn-Banach theorem, one can demonstrate the geometric version from scratch and use it to prove the analytic version, as is outlined in Hahn-Banach theorem: 2 versions. To do so, it ...
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1answer
18 views

Why is $ab=ba=a^\ast b=ab^\ast=0$ (orthogonal elements in a $C^\ast$-algebra)?

Let $a,b$ be elements in a $C^\ast$-algebra $A$, such that $$a^\ast ab^\ast b=b^\ast ba^\ast a=0$$ $$a^\ast abb^\ast=bb^\ast a^\ast a=0$$ $$aa^\ast b^\ast b=b^\ast b aa^\ast =0$$ $$aa^\ast ...