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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1answer
34 views

Weak compactness of a set of translates in $C_0(\mathbb{R})$

Let $f \in C_0(\mathbb{R})$. Consider the set of translates of $f$ $$ A = \{ f_t : t \in \mathbb{R} \}$$ where $f_t(x)=f(x+t), x\in \mathbb{R}$. I want to show that $A$ is relatively compact in the ...
2
votes
2answers
34 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 ...
2
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0answers
21 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 ...
4
votes
1answer
312 views

Properties of the square norm in Banach spaces

Let $X$ be a Banach space with its dual $X^*$. Consider the mapping $f: X\rightarrow \mathbb{R}$ given by $$ f(x)=\frac{1}{2}\|x\|^2. $$ We have know that when $X$ is a real Hilbert space ($X=X^*$) ...
3
votes
1answer
30 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 $\mathbb{H}^s(\Omega)$. ...
1
vote
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 ...
1
vote
4answers
92 views

Are there an infinite number of open balls in an open set in a metric space?

Let's start off by recalling the definition of an open set in a metric space: A set $A$ in a metric space $(X,d)$ is open if for each point $x\in A$ there is a number $r\gt0$ such that $B_r(x)\subset ...
1
vote
2answers
111 views

Compact Operators: Trace

Given a Hilbert space $\mathcal{H}$. Consider a bounded operator: $$A:\mathcal{H}\to\mathcal{H}:\quad\|A\|<\infty$$ Regard ONB's: ...
-1
votes
0answers
22 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 ...
0
votes
1answer
24 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 ...
0
votes
1answer
17 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)} ...
1
vote
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 ...
2
votes
1answer
54 views

What's the difference between a Banach Algebra and a C*-algebra?

I'm currently looking at going into a PhD program in mathematics and need to decide on a specialization. In meeting with my advisor he pushed me into looking at C*-algebras based on my interests. ...
6
votes
2answers
194 views

Universal separable Banach algebras

The well-known Banach–Mazur theorem says that $C([0, 1])$ is a universal separable Banach space, in the sense that if $X$ is any separable Banach space then there is a map $f : X \to C([0, 1])$ which ...
0
votes
0answers
43 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 ...
1
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0answers
19 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?
2
votes
1answer
43 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 ...
0
votes
1answer
27 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) ...
3
votes
2answers
117 views
+400

Spectral theorem for representations proof.

Let $H$ be a separable Hilbert space, and $U$ a unitary representation of $\mathbb{Z}^d$ on $H$. Let $\chi_m$ be the characters of the Torus $T^d$, and $m$ the Haar measure on $T^d$. I would like to ...
4
votes
0answers
64 views
+50

What plays the role of the identity for the generalized convolution associated to the Fourier-Bessel transform?

In traditional Fourier theory, the Dirac delta plays the role of an "identity" for the $L^1$ algebra with respect to the usual convolution. The convolution is traditionally built out of group ...
6
votes
4answers
92 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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3answers
60 views

How to find a function in $ L_{2}(0,\infty)$ but not $L_{q}(0,\infty)$, $q\neq 2 $

How to find $x\in L_{2}(0,\infty)$ \ $L_{q}(0,\infty)$ , $q\neq 2 $ i tried $\frac{1}{t.lnt}$ with various degrees on $t$ and $ln(t)$ Could you please help me with this question
1
vote
1answer
362 views

Equality in the Cauchy-Bunyakowsky-Schwarz inequality for a semi-inner product

I am stuck with an exercise that I found in a textbook by Conway. First, I would like to clarify what is meant by a semi-inner product. Definition. Suppose that $\mathscr X$ is a vector space over ...
1
vote
0answers
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
votes
0answers
60 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 ...
0
votes
0answers
22 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 ...
0
votes
0answers
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?
3
votes
1answer
66 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: ...
14
votes
2answers
443 views

Can a dynamic programming problem be transformed into a linear algebra problem?

Here is a simple standard economic problem: Let Robin Crusoe have an endowment $w_0$ of lembas bread. She is immortal and discounts at a rate of $\beta$ per period. Each period (from $t=0$ on) she ...
0
votes
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 ...
1
vote
0answers
71 views
+50

Two ODEs, why is one solution the solution of the other?

Consider the ODE: find $u:[0,T] \to \mathbb{R}^n$ s.t. $$u'(t) = F(t,u(t))$$ $$u(0) = u_0$$ given $F:[0,T]\times \mathbb{R}^n \to \mathbb{R}^n$ Caratheodory, and we know that if it has a solution, it ...
2
votes
0answers
21 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 ...
-1
votes
1answer
49 views

Need help proving the equivalence of two norms?

Hey I could use a lot of help with this problem please! Let $(X, \langle\,\cdot,\cdot\,\rangle)$ be a Hilbert space over $\mathbb{R}$. Then, let $A\colon X \to X$ be a linear operator. Suppose that ...
0
votes
2answers
37 views

Ordering : Ranges

Given a Hilbert space $\mathcal{H}$. Denote selfadjoints: $$\mathcal{S}(\mathcal{H}):=\{A\in\mathcal{B}(\mathcal{H}):A=A^*\}$$ Note that one has: $$\Delta ...
-1
votes
0answers
50 views

Ordering: Compactness

Given a Hilbert space $\mathcal{H}$. Denote compacts: $$\mathcal{K}(\mathcal{H}):=\{K\in\mathcal{B}(\mathcal{H}):K\text{ compact}\}$$ And selfadjoints: ...
1
vote
0answers
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 $X$ ...
1
vote
0answers
43 views

Want to prove an inequality of two norms in a Hilbert space

So here is my problem, Let $D:=[-d,d]\times[-d,d]$ and $C_0^{\infty}$(D) be the set of all smooth functions with compact support in $D$ which are zero on the boundary of $D$. Moreover we have the ...
1
vote
1answer
35 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
votes
0answers
20 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 ...
2
votes
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 ...
0
votes
0answers
36 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 ...
0
votes
1answer
39 views

How is Lipschitz continuity for Fréchet derivatives defined?

Let $(X,||\cdot||_X)$ be a Banach space and $X^*$ it's dual of linear functionals $X\to\mathbb{R}$. The Fréchet derivative $\nabla f(x)$ of a function $f:X\to\mathbb{R}$ at $x$ is an element of $X^*$. ...
2
votes
0answers
29 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 ...
1
vote
2answers
84 views

uniqueness of positive operator

Let $A,B$ be commuting positive operators on a Hilbert space such that $\langle(A-B)(A+B)x,x\rangle=0$ for all $x$ in the Hilbert space. Prove that $A=B$. My attempt: The above implies that $A=B$ on ...
0
votes
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
votes
1answer
42 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 ...
2
votes
1answer
35 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 $ ...
0
votes
1answer
95 views

inequality between entries of the vector and $l_2$ norm of the vector

Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. I am wondering for which vectors the following would be true: $$ \|a\|_2^2\geq c \sum_{i\ne j,i,j}a_ia_j, \quad i,j=1, \ldots, n. $$ Here $c>0$ is ...
1
vote
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} ...
4
votes
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 ...