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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4
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2answers
37 views

Find all $\alpha$ such that $n^\alpha\chi_{[n,n+1]}$ converges weakly to 0 in $L^p$.

Edit: $1 < p < \infty$ Let $f_n(x) = n^\alpha \chi_{[n,n+1]}.$ Then $$ \begin{align} \left|\int_{-\infty}^{\infty} n^\alpha \chi_{[n,n+1]}g(x)dx\right| &\le n^\alpha \lvert\lvert\chi_{[n,n+...
6
votes
1answer
50 views

Why is the Minimum in the Min-Max Principle for Self-Adjoint Operators attained?

Let's consider a self-adjoint operator $A$ (not necessarily bounded) on a Hilbert space which is bounded from below, with domain $D$ and whose resolvent is compact. Then, the spectrum consists solely ...
1
vote
1answer
22 views

$G$ is dense in $X^*$ in weak* sense if and only if $G$ is total set

I have some question on functional analysis. Recently, I'm reading an article of Coifman and Weiss, "Extensions of hardy spaces and their use in analysis". They proved some important theorem to me by ...
0
votes
0answers
10 views

Numerical analysis of bifurcation problems - notational confusion

I'm going through the following set of review notes about numerical analysis of bifurcation problems and was wondering if someone could explain to me what is going on on the page 50. First of all, I'...
2
votes
1answer
26 views

Duality Pairing and self-adjoint operators notation

Let $X$ be a Hilbert space and let $X'$ be the dual space of $X$ with respect to the duality pairing $\langle\cdot,\cdot\rangle$. Let $A: X \mapsto X'$ be a bounded linear operator. We assume that $A$...
1
vote
1answer
28 views

Show that $(\mathcal C^0([a,b]),\|\cdot \|)$ is complete.

Let $\|f\|=\sup_{[a,b]}|f|$ where $f\in \mathcal C^0([a,b])$. I have to show that $(\mathcal C^0([a,b]),\|\cdot \|)$ is complete. I did as follow. Let $x\in ]a,b[$. Let $(f_n)$ a Cauchy sequence. ...
0
votes
1answer
57 views

Has the distributional Laplacian $\Delta f:C_c^\infty(\Omega)'\to C_c^\infty(\Omega)'$ a unique extension in $H_0^1(\Omega)'$?

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D:=C_c^\infty(\Omega)$ and $$H=\overline{\mathcal D}^{\langle\;\cdot\;,\;\cdot\;\rangle_H}\tag 1$$ with $$\langle\phi,\psi\rangle_H:...
-1
votes
0answers
31 views

Fabian. Functional analysis and infinite dimensional geometry

Where can I find solutions to the problems of Fabian. Functional analysis and infinite dimensional geometry, 3 - Weak Topologies
1
vote
3answers
51 views

A canonical example for spaces that aren't $1^{\text{st}}$ countable

To get the feeling for metric spaces (or $1^{\text{st}}$ countable spaces, in general), I always find that visualizing them as $\Bbb R^3$ gives a good intuition of what is going on (locally, of course)...
1
vote
0answers
34 views

Line in Modified Bessel Function of the First Kind

Plotting the modified Bessel function of the first kind $I_\nu(x)$ as a function of two real variables, it looks like to one side of $x=\frac{2}{3}\nu$ the function falls rapidly to zero and on the ...
0
votes
1answer
51 views

Questions about the proof of Poisson's formula for half-space in Page 38 of Evans' book

I have questions about the proof of Theorem 14 (poisson's formula for half-space) in Page 38. Let $K(x,y)$ be the Poisson's kernel for $\mathbb R^n_+$: $$K(x,y)=\frac{2x_n}{na(n)}\frac{1}{|x-y|^n}dy\...
3
votes
2answers
76 views

The convergence of unordered sum

$X$ is an arbitrary normed vector space. $A=\{x_i\in X\mid i\in J\}$ is an indexed set. $J$ contains the indices and is uncountably infinite. Let $\mathcal F=\{F\mid F\subseteq J, F \text{ is finite}\}...
0
votes
0answers
23 views

Distribution Convergence in PDE theory

I'm trying to follow an Example (5.1) from a PDE book (Vasy). I was having trouble following the proof of the following question (modified): Define a bump function: ${\delta_j}^{-n}\phi(\frac{x}{\...
1
vote
0answers
26 views

Minimal prime ideals of the ring of continuous functions

Let $X$ be a topological space. Are there any conditions on $X$ which guarantee that that the minimal prime ideals of $C(X)$, the ring of real-valued continuous functions on $X$, have a nice ...
1
vote
1answer
22 views

Find the spectrum of an operator related to Fourier series

As an exercise, I was told to find the spectrum of the bounded operator $K\in B(L^2[-\pi,\pi])$ defined by $$K\varphi (t)=t\int_{-\pi}^\pi\varphi (x)\cos (x)dx+\cos t\int_{-\pi}^\pi x\varphi(x)dx.$$ ...
0
votes
0answers
17 views

When is orthogonal projection compact? [duplicate]

Let $M$ be a closed subspace of a Hilbert space $H$. Let $P$ be the orthogonal projection on $M$. I was told to find the eigenvalues and eigenvectors of $P$ and moreover say when it is compact. Since ...
3
votes
1answer
18 views

Find eigenvalues and eigenvectors of infinite symmetric matrix of powers of two

Let $a_n=2^{-n}$. What are the eigenvalues and eigenvectors of the $\ell^2$ operator represented by the infinite matrix below? $$A=\begin{pmatrix} a_1 & a_2 & a_3 & \dots \\ a_2 & a_3 &...
1
vote
1answer
16 views

Eigenvalue of every eigenvector is an eigenvalue of element of o.n eigenvector basis

I need to prove (or disprove) that given a bounded operator $A$ on a Hilbert space with orthonormal basis of eigenvectors $\left\{ e_i \right\}$, if $Av=\lambda v$ for $v\neq 0$ then $\lambda $ is an ...
6
votes
4answers
889 views

Is diameter of a set a measure?

Suppose the diameter of a nonempty set $A$ is defined as $$\sigma(A) := \sup_{x,y \in A} d(x,y)$$ where $d(x,y)$ is a metric. Is $\sigma(.)$ a 'measurement'? I.e., how do I prove the countable ...
0
votes
1answer
42 views

definition of Fourier transform questions

I'm completely stumped by the problem below because I haven't attended the lectures which used only Riemann integration and am not sure what the author is getting at. Let $f\in C^1(\mathbb R)\cap ...
1
vote
1answer
55 views

Limit of sequence of linear functionals

Determine whether the following sequence of functionals over $l_2$ $$\phi_n(x) = \frac{1}{n}\sum_{k=1}^n \sqrt k\ x_k\ \ \ \ \ \ \ \ \ \ \ x = (x_1, x_2, x_3,...)$$ converges in the weak sense (i.e. ...
1
vote
1answer
22 views

Simple norm inequality

Trying to follow the comments to this question I am struggling very much to understand how to simplify $\|Ax\|_2=\sup_{\|x\|_2=1}\sqrt{\sum_i(\sum_ja_{ij}x_j)^2}$ to arrive at an $x$-free bound. Can ...
5
votes
1answer
69 views

Functional Analysis by Reed and Simon, chapter 3 exercise 15

Let $H$ be a Hilbert space with an orthonormal basis $\{ x_n \}_{n=1}^\infty$ and let $\{ y_n \}_{n=1}^\infty$ be a sequence of elements in $H$. Show that following two statements are equivalent $$ \...
1
vote
0answers
24 views

Prove multiplication by sequence is a compact operator

Let $c_0(\mathbb N)$ be the space of sequence in $\mathbb C$ whose limit is zero, equipped with the $\ell^\infty$ norm. Let $u_n$ be a sequence in $\mathbb C$ and define the operator $A$ taking a ...
1
vote
0answers
19 views

$ (k\otimes h^\ast)^\ast=h\otimes k^\ast$?

Let $H,K$ be Hilbert spaces with $h\in H,k\in K$. Let $k\otimes h^\ast(g)= \left\langle g,h \right\rangle k$. I'm supposed to prove $ (k\otimes h^\ast)^\ast=h\otimes k^\ast$, but I don't see how this ...
0
votes
1answer
39 views

Proving an operator is compact exercise

Suppose $(a_{ij})_{i,j\in \mathbb N}$ satisfy $\sum_{i,j}|a_{ij}|^2<\infty$ and define $A:\ell ^2 \rightarrow \ell ^2$ by $(Ax)_i)=\sum _j a_{ij}x_j$. I need to prove $A$ is compact. Unfortunately,...
0
votes
0answers
26 views

Interior of a solid self dual cone

Let $X$ be a real Hilbert space with a solid self dual positive cone $K$, that is, $\mathrm{int}(K)$ is non-empty and $K^{*}=K$. If $X$ is finite dimensional, I know that the $\mathrm{int}(K)$ = $\...
0
votes
0answers
21 views

Image of unit ball precompact implies bounded

I need to prove that if the image of the unit ball $X_1$ of a Banach space $X$ along an operator $A:X\rightarrow Y$ is precompact, $A$ is bounded. Is my solution correct? $\|A\|=\sup_{\|x\|=1}\|Ax\|...
0
votes
0answers
17 views

How to proof $M(H,K)X=U([M(\lambda_{i},\mu_{j})]_{ij}\circ (U^{*}XV))V^{*}$

Let $M(x,y)$ be positive real function on $(0,\infty)\times (0,\infty)$ satisfies $M(x,y)=M(y,x)$ $M(\alpha x,\alpha y)=\alpha M(x,y)$ for all $\alpha>0$ $M(x,y)$ is non-decreasing in $x$ and $y$...
6
votes
2answers
62 views

Correspondence between maximal ideals and multiplicative functionals of a non unital, commutative Banach algebra.

Let $\mathcal{A}$ be a non (necessarily) unital commutative Banach algebra, and let $$ M_{\mathcal{A}} = \{ \phi:\mathcal{A} \to \mathbb{C} : \phi \mbox{ is multiplicative and not trivial}\} $$ and $$...
2
votes
1answer
49 views

Why are two characters of a commutative Banach algebra with the same kernel equal?

Let $A$ be a commutative Banach algebra. Let $\chi_1$ and $\chi_2$ be characters of $A$. I am having some difficulty seeing why the following statement is true: If $\ker \chi_1 = \ker\chi_2$, then ...
0
votes
0answers
39 views

Implicit function theorem for Banach spaces

I was wondering if someone could give a bit of broad advice regarding working with Implicit Function Theorem (IFT) and, I guess, the Catastrophe theory. This is something completely new to me. ...
4
votes
1answer
54 views

Solving the ODE $y^{\prime\prime}(x)-y(x)=g(x)$ using the Fourier transform, without missing solutions

I'm supposed to solve the ODE $y^{\prime\prime}(x)-y(x)=g(x)$ using the Fourier transform and then explain if I got the most general solution. First of all, I don't know what "solve" means here ...
2
votes
1answer
69 views

Determining whether equality $ \|T v\| = \|T\| \cdot \|v\|$ is possible

As an exercise, I'm supposed to determine whether for the operators $A_\lambda:g(t)\mapsto \sqrt\lambda g(\lambda t)$ on $C[0,1],\lambda\in (0,1)$ and $T_f:g\mapsto g(t)f(t)$ on $L^2$ it is possible ...
7
votes
3answers
96 views

Trace norm of a triangular matrix with only ones above the diagonal

For $n\in\mathbb N^*$, we consider the triangular matrix $$ T_n = \begin{pmatrix} 1 & \cdots & 1 \\ & \ddots & \vdots \\ 0 & & 1 \end{pmatrix} \in M_{n,n}(\mathbb R) \,. $$ ...
1
vote
1answer
34 views

A neat characterization of measurable functions $f:\mathbb R\rightarrow \mathbb C$ for which $\lim_R\int_{-R}^R|f|dx<\infty$

Is there a neat characterization of measurable functions $f:\mathbb R\rightarrow \mathbb C$ for which the limit of Riemann integrals satisfies $\lim_R\int_{-R}^R|f|dx<\infty$ in terms of elements ...
0
votes
0answers
32 views

Exercise on the operator $g(t)\mapsto \sqrt \lambda g(\lambda t)$

I have the linear operator $A_\lambda :g(t)\mapsto \sqrt \lambda g(\lambda t)$ on $C[0,1]$ with $\lambda\in (0,1)$ (which extends to $L^2[0,1]$). I need to find its adjoint and then prove that while $...
1
vote
0answers
16 views

Fibers of unbounded linear functional are dense

I'm supposed to prove that if $f$ is a discontinuous linear functional $H\rightarrow \mathbb C$, each of its fibers $f^{-1} \left\{ \alpha \right\} $ is dense. I already know the kernel, i.e $f^{-1} \...
0
votes
0answers
29 views

What's the difference between the topology defined by a seminorm and the topology defined by the norm it induces?

I was just wondering whether there's some big difference between the topology generated by a seminorm and the norm it induces. For instance, Suppose $X$ is normed and $A$ is a subspace. $X/A$ is semi-...
0
votes
1answer
40 views

Showing space is complete and functional is bounded

I've got two questions. (1) Define $ \|f\| := |f(0)| + \max \limits_{0 \le t \le 1} |f'(t)|$ to be a norm on $C^1([0,1])$. (I have shown it's indeed a norm). Show that $(C^1([0,1]), \|\cdot\|)$ is ...
0
votes
2answers
31 views

Density of complex polynomial in the space of entire holomorphic functions.

Let $\int_\mathbb C f(w)e^{\frac{-|w|^2}{2}} p(w) dA(w)=0$ for all complex polynomial $p(w)$. Then show that $f=0$. Anyone could please help me for this. Thanks in advance.
2
votes
1answer
28 views

Question about the proof of the fact that minimum is attained for a l.s.c. convex function over a convex compact set.

I quote here the proof of a result given in Haim Brezis Functional Analysis, Sobolev Spaces and partial differential Equations: I haven't been able to conclude where exactly is this hypothesis used: ...
1
vote
1answer
66 views

Corollary 2.1 in Ekeland and Temam on lower semicontinuity

Why in Corollary 2.1 on page 10 (see the picture) from Ekeland and Temam book Convex Analysis and Variational Problems there is equality in (2.11), i.e why $$\forall u\in V,\quad \overline F(u)=\...
2
votes
1answer
24 views

Is it true that a rational transfer function without RHP pole must be square integratable?

I am stuck with a problem: For a rational transfer function without RHP pole, for example $$H(s)=\frac{(-z_0 + s)(-z_1+s)\cdots(-z_n+s)}{(-p_0+s)(-p_1+s)\cdots(-p_m+s)}$$ where $Re\{p_i\}<0$ and $...
0
votes
2answers
50 views

If $p,q$ are distributions with $\partial_ip=\partial_iq$, then $p=q$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ $\mathcal D(\Omega):=C_c^\infty(\Omega)$ Let $$\frac{\partial p}{\partial x_i}(\phi):=-p\left(\frac{\partial \phi}{\partial x_i}\right)\;\;\;\...
2
votes
1answer
43 views

If $\|e_1+e_2+\cdots+e_n\|\leq C$ for all $n$ then $(e_i)_{i=1}^n$ is uniformly equivalent to the basis of $\ell_\infty^n$?

Conjecture. Let $n\in\mathbb{N}$, and let $(e_i)_{i=1}^n$ be a normalized unconditional basis for an $n$-dimensional Banach space $E_n$ with Schauder basis constant $K\in[1,\infty)$. If $\|e_1+e_2+\...
4
votes
2answers
179 views

explain why one can write $\hat{f}(\xi)=\lim_{n\to\infty}\frac{1}{\sqrt{2\pi}}\int_{-n}^{n}e^{-i\xi x}f(x)dx$ when $f\in L^2(\mathbb{R})$

Let $f\in L^1(\mathbb{R})$ where the measure is taken to be the Lebesgue measure. The Fourier transform of $f$ is the function $\hat{f}$ defined as $$\hat{f}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{...
0
votes
0answers
28 views

The $L^2$ convergence of semi-$p$-lapace equation

This question is similar to the one I post early here. But this one might be more reasonable I think... Let $g\in L^\infty(\Omega)$ be given, where $\Omega\subset \mathbb R^2$ is open bounded with ...
1
vote
0answers
28 views

The convergence of $p$-laplace equation

Let $g\in L^\infty(\Omega)$ be given, where $\Omega\subset \mathbb R^2$ is open bounded with smooth boundary. Define, for $1<p\leq 2$, $$ u_p:=\operatorname{argmin}\left\{\int_\Omega|u-g|^pdx+\...
1
vote
2answers
40 views

Spectrum of the Resolvent of a Self-Adjoint Operator

Let $\mathcal{H}$ be a Hilbert space, and $A$ a self-adjoint operator with domain $D_{A} \subseteq \mathcal{H}$. Assume that $\lambda_0 \in \rho(A)$, where $\rho(A)$ is the resolvent set of $A$. For ...