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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Showing that $d_\infty(x,y)$ is finite for all $x,y\in\mathscr L^2$

I want to show that $d_\infty(x,y)$ is finite for all $x,y\in\mathscr L^2$, however I am not to sure how to go about doing so. We have, $$d_\infty(x,y)=\sup_i|x_i-y_i|$$ and $\mathscr L^2$ the ...
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0answers
16 views

Operators whose spectrum has a finite number of connected component

Assume that $H$ is a separable Hilbert space. Let $Q$ be the set of all operators$T \in B(H)$ such that the spectrum of $T$ has a finite number of connected component. Is $Q$ a subvector space ...
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2answers
26 views

Metric spaces, manipulating the absolute value function.

I have the following problem involving the set $Y$ of infinite sequences that absolutely converge such that, $$\sum_{i=0}^\infty x_i^2 \lt\infty$$ where $x_i$ is the $i$-th term in the infinite ...
2
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1answer
21 views

Compact operator

Suppose $A : H \to H$, where $H$ is a Hilbert space, is bounded. Also, $A$ is a diagonal operator with diagonal $\{a_n\}$. Show: If $A$ is compact, then $a_n \to 0$ as $n \to \infty$. Should I prove ...
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14 views

Question about a proof in Clapp's paper, “On the number of positive symmetric solutions of a nonautonomous semilinear elliptic problem”

I have this: And I want to understand this proof: I don't understand the choice of $f(p)$ why it is sufficient to prove that $\lim_{p\rightarrow2^*} f(p)=f(2^*)$ since we see that ...
1
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1answer
38 views

Are infinite-dimensional singletons measurable?

Consider the wiener measure space $C[a,b]$ of all real-valued continuous functions on $[a,b]$ with the wiener measure $\mu$ on the $\sigma$-algebra $\mathcal{A}$ of Carathéodory measurable sets in ...
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1answer
15 views

The relationship between function space embeddings and their respective inequalities

Let $L^{p,\infty}$ be the weak $L^p$ space consisting of measurable functions $f$ satisfying \begin{equation*} ||f||_{p,\infty}:=\sup_{\rho}\rho\lambda (|f|>\rho)^{\frac{1}{p}}<\infty . ...
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1answer
10 views

Approximation by $\mbox{Im }(t-z)^{-1}$ with $\mbox{Im } z > \epsilon$

It is a standard fact of harmonic analysis that the span of the functions $$g_z(t) = \mbox{Im } (t-z)^{-1},$$ ranging over all $z \in \mathbb{C}$ with $\mbox{Im } z > 0$, is dense in ...
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0answers
14 views

Hyperbolic equations with time dependent coefficients associated with the time derivatives

I'm concerned with evolution equations (of second order) and am hoping for some literature hints regarding a special situation. The equations I'm working with basically look like (complemented with ...
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0answers
15 views

Equivalence of the Sobolev norm $\| f\|_{W^{k,\infty}}$ and $\| f\|_{H^{k,\infty}}$.

I want to know that when $k=0,1,2,...$, the Sobolev norm $$ \| f\|_{W^{k,p}} :=( \sum_{|\alpha|\le k} \|\partial^\alpha f\|_{L^p(\mathbb R^n)}^p )^{1/p} $$ is equivalent to the norm $$ \|f\|_{H^{k,p}} ...
3
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22 views

Least expensive way to “walk” with a convex potential

Let $V(x)$ be a non-decreasing, convex function on $\mathbb{R}^+$. Let $A_{L,k}$ be the space whose elements are sequences of positive integers $r= (r_i)_{i=1}^{N}$ such that $\sum\limits^N_{i=1} r_i ...
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0answers
10 views

A pde that cannot solves by Lax-Milgram theorem

Consider the following pde: $-u''(x)+au'(x)+bu(x)=f(x) \qquad\text{in}\; (0,1)\\$ $u'(0)=\alpha\\$ $u'(1)+u(1)=\beta$ How could I prove that it has a nontrivial solution? The bilinear form ...
3
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1answer
25 views

Manipulating the maximum function, metric spaces.

I am trying to show that the supremum metric, $d_{\infty}$, is indeed a metric on $\mathbb R^2$. I have shown that the first two properties of a metric space hold, but am having trouble showing the ...
1
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0answers
13 views

Inequality in a Dirichlet BVP

For the Drichlet boundary value problem $Lu = -u''+p(x)u'+q(x)u = f(x), \; \; \; x \in I=[a,b]$. with $u(a)=u(b)=0$. Then for $v \in H^2(I) \cap H_0^1(I)$ show that $\left\lvert \right\rvert v ...
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55 views

What is the implication that $\| \cdot \|_2$ and $\| \cdot \|_\infty$ are equivalent norms on $\mathbb{R^2}$

Given $\mathbb{X}$ = $\mathbb{R^2}$, consider $\| \cdot \|_2$ and $\| \cdot \|_\infty$ We can show that $\| x \|_\infty \leq \| x \|_2 \leq \sqrt2 \| x \|_\infty$ Hence $\| \cdot \|_2$ and $\| ...
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0answers
24 views

Additional assumptions on function to ensure uniform convergence

Given a sequence $u=(u_n)_{n\geq1}$ converging to $1$, I would like to prove uniform convergence of the sequence of functions $f_n$ defined by $f_n(x)=f(u_n x)$ for $x\in\mathbb{R}_+$ to the function ...
2
votes
1answer
24 views

Big an open ball inside small open ball in metric space

When i was reading a book "Elements of the Theory of Functions and Functional Analysis " ([A. N. Kolmogorov, S. V. Fomin) I encountered with very interesting(for me) problem. Problem: Creat a ...
6
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1answer
51 views

Is this set of random variables a Hilbert space?

Consider a sequence of i.i.d. random variables $\left\{ {{\varepsilon _t}} \right\}_{t = 1}^\infty $ with $E\left( {{\varepsilon _t}} \right) = 0$ and $E\left( {\varepsilon _t^2} \right) = {\sigma ...
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1answer
36 views

Is there an error in the solution for this exercise?

I have this exercise: H is a complex hilbert space. And T is a compact operator on H. Show that if H is not separable, then 0 is an eigenvalue of T. Hint: Use lemma 1, and theorem 2. The ...
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1answer
20 views

Finite Rank Operator: Continuity

I keep forgetting it, so... Given Banach spaces $X$ and $Y$. Then it is wrong: $$\dim\mathcal{R}F<\infty:\quad F\in\mathcal{L}(X,Y)\implies F\in\mathcal{C}(X,Y)$$ Can I construct such?
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1answer
31 views

How to compute high order differential?

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 ...
2
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1answer
40 views

Calculus stay to Real Analysis as $x$ stay to Functional Analysis

Hi guys i had a look to book which treat the subject of Calculus (of course...) Analysis and Functional Analysis. Is that correct to state that Calculus is more focused on "computing" while ...
6
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1answer
39 views

structure theorem for Banach spaces

The following is a theorem in the Banach Algebra Techniques in Operator Theory by Douglas: Here are my questions: Could one come up with some reference (or proof) regarding the remark right ...
2
votes
1answer
78 views

Difficult to read about different subjects simultaneously, should I leave one for now? [on hold]

I learn math by reading books. Usually I read 3 books (about 3 different subjects) simultaneously and switch focus every couple of days. The books i'm studying right now are Rudin's functional ...
9
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2answers
75 views

Existence of a function

I need some help: I am thinking about this problem. Any advice would be appreciated. Let's fix $\epsilon>0$. Does there exists some $f\in C^0([0,\pi])$ such that: ...
2
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0answers
24 views

derivative of a linear operator

I am a little confused by the proof in the picture. Doesn't the calculation show $dB(u)h = B(h_1,u_2)+ B(u_1, h_2)$?
2
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1answer
30 views

Symmetric function of two normal distribution implies bilinear

This question is related to my previous question which was partially answered my @MichaelHardy. Let $X$ and $Y$ be two independent standard normal random variables. Now, suppose that ...
3
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0answers
35 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 ...
4
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1answer
52 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
36 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 ...
2
votes
1answer
33 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 ...
4
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1answer
23 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 ...
2
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0answers
40 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 ...
4
votes
1answer
50 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 ...
1
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1answer
37 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
38 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
votes
2answers
40 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 ...
0
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1answer
26 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
22 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
36 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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0answers
47 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
vote
1answer
26 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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0answers
24 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
82 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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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?
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0answers
27 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
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0answers
13 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 ...
2
votes
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 ...
2
votes
0answers
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 ...