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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3
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1answer
43 views

Example of a self-adjoint bounded operator on a Hilbert space with empty point spectrum

Good day, I wanted to find a self-adjoint bounded operator on a Hilbert space with empty point spectrum i.e. $$ T = T^* ~\text{but}~ \sigma_p(T)= \emptyset $$ Some definitions and results of the ...
2
votes
1answer
66 views

Prove that $\int_{0}^\infty (e^{As})^T e^{As}\,ds$ is convergent

Consider A to be a matrix that has all eigenvalues $\lambda$ with negative real part, that is, $Re(\lambda) < 0$. a)Show that the integral $\int_{0}^\infty (e^{As})^T e^{As}\,ds$ is convergent, ...
0
votes
0answers
13 views

Covariance functional of a generalized real-valued Brownian motion

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $[0,\infty)$ and $$\langle ...
0
votes
0answers
21 views

When are Fourier spaces included in each other?

Given the Fourier spaces $V(N_1, T_1)$ and $V(N_2,T_2)$, what necessary and sufficient conditions are required in order to have $V(N_1, T_1)\subset V(N_2,T_2)$? I know that if $V(N_1, T_1)$ is ...
3
votes
1answer
29 views

‎Jointly ‎continuous of product in $B(H)$

‎Let ‎$‎B(H)‎$ ‎be the set of‎ ‎bounded ‎operators ‎on a Hilbert space ‎‎$‎‎H$.‎ ‎ I ‎know that ‎$u_{‎\alpha‎}‎\longrightarrow u‎$‎‎‎ ‎in ‎S.O.T ‎if ‎and ‎only ‎if‎ ‎$u_{‎\alpha‎}(x)‎\longrightarrow ...
1
vote
1answer
20 views

Prove order of distribution $\Lambda_{1/x}$ is 1

The distribution is defined as: $$\Lambda_{1/x}(\varphi)=\lim_{\varepsilon\rightarrow0+}\int_{\mathbb{R}\backslash(-\varepsilon,\varepsilon)}\frac{\varphi(x)}{x}\ \mathrm{d}x$$ I tried integrating ...
1
vote
1answer
20 views

Largest invariant subspace

If $A$ is a $n\times n$ matrix with complex entries, denote by $inv(A)$ the dimension of a largest dimensional non-trivial invariant subspace of $A$. What is: $$\inf\{inv(A): ...
1
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0answers
28 views

Are the set of probability functions with compact support in a fixed closed ball complete under the Wasserstein norm?

Let $B_R$ be a closed ball of radius $R$ in the space $\mathbb{R}^d$. As the title suggests I have this feeling that the set of functions $$S:= \left\lbrace f:\mathbb{R}^d \to \mathbb{R} ...
2
votes
1answer
20 views

Small perturbations and eigenvalues

Suppose $A$ is a $n\times n$ matrix. Given $\epsilon>0$, can one find a rank one matrix $B$ with euclidean norm at most $\epsilon$ such that $A+B$ has $n$ distinct complex eigenvalues? Given a ...
-1
votes
1answer
37 views

can you prove this theorem An introduction to wavelet Analysis? [closed]

Definition. The sequence $\{f_n(x)\}$, $n\in \mathbb{N}$ defined on an interval $I$ converges in mean-square to the function $f(x)$ on $I$ if $\lim_{n\to \infty} \int_I {|f_n(x)-f(x)|}^2\, dx =0$. We ...
0
votes
2answers
33 views

Reflexive Banach spaces, compactness

Let $X$ be a reflexive Banach space. Then, consider a linear and compact operator $T \colon X \to X$. Prove that if: $\text{inf} \{ \|Tx\| : x \in X\quad \text{s.t.}\quad \|x\| = 1 \} > 0$, ...
-5
votes
0answers
48 views

can you prove this theorem? [closed]

Definition.The sequence $\{f_n(x)\}_{n\in \mathbb N}$ defined on an interval $I$ converges in mean-square to the function $f(x)$ on $I$ if $\lim_{n\to \infty} \int_I |f_n(x)-f(x)|^2 \, dx =0$ We write ...
1
vote
1answer
21 views

If $X$ is maximal ideal then it consists of non-invertible elements?

I'm reading through a paper where I came across the following theorem Let $A$ be a commutative complex Banach algebra with unit element $e$. Theorem: A subspace $X\subset A$ of codimension $1$ is ...
1
vote
1answer
28 views

Lax Milgram Lemma. Prove coercivity

Consider the following problem: \begin{cases} (\mu u'-au)'=f \\ u(0)=u(1)=0 \end{cases} The domain being the interval $\Gamma$=(0,1) Where $\mu$ is a positive function in $C^1$($\Gamma$) and a is a ...
0
votes
0answers
14 views

Number of copies of irreducible unitary representation in $L^2(G)$ for compact group $G$?

Peter-Weyl Theorem is concerned with expressing $L^2(G)$ as closure of direct sum of subspace generated by irreducible unitary representation. Every irreducible representation on a compact group is ...
0
votes
0answers
13 views

limit of function depending on parameters

This may be a very vague question, but I'm looking for a vague answer. Suppose I have a function $f(x_i; t_a)$, where I think of $x_i$ as the arguments of the function, and $t_a$ as parameters (the ...
2
votes
1answer
30 views

Is the square root function norm continuous?

Let $\{a_n\}$ be a sequence of positive operators in $B(H)$. What about the following implication, True or false? $$||a_n-a||\to 0\Longrightarrow ||a_n^{\frac{1}{2}}-a^{\frac{1}{2}}||\to0$$
0
votes
0answers
26 views

A remark on the polar decomposition

Let us focus on $L^1(H)$, the space of trace class operators on a Hilbert space $H$. Assume $\{x_n\}$ converges to $x$ (in the trace class norm). Let $x_n=u_n|x_n|$ and $x=u|x|$ be the polar ...
1
vote
1answer
35 views

Check whether the set is closed/has an interior point?

Let $H= \{ (x_n) \in l^2: \sum_n \frac {x_n}{n}=1 \}$ Then check whether H is $1$ Closed $2$ Has an interior point? I think that $H$ is closed and it does not have any interior point but i ...
0
votes
1answer
30 views

Double dual of the space of bounded operators on Hilbert space [duplicate]

Every Banach space $X$ is canonically, isometrically embedded in its bidual $X^{**}$. But it is not always $1$-complemented in the bidual: for example, there is no projection from $\ell_\infty$ onto ...
2
votes
1answer
24 views

Generating Infinite Set with Function Composition

I imagined myself today being infinitely small, standing on the inside of a closed and perfectly mirrored surface and holding a laser. Could this surface be shaped in some way where I could turn on ...
1
vote
3answers
38 views

Subset of $(l^{2},d_{2})$ is open

Show that $A = \{\phantom{i}\{x_{n}\} \in l^{2} \hspace{2mm}:\hspace{2mm} |x_{n}| < 1, \forall \phantom{i}n \in \mathbb{N}\phantom{i} \}$ is open in $(l^{2},d_{2})$. The $d_{2}$ metric is: $$ ...
0
votes
0answers
23 views

support of an operator on a Hilbert space [closed]

let $x\colon\mathcal{H}\to\mathcal{H}$ be a self-adjoint operator, the support $s(x)$ of $x$ is defined as the smallest projection $e\in B(\mathcal{H})$ such that $ex=xe=x$. let $A=\int\lambda \, ...
1
vote
0answers
24 views

Spherical Bessel expansion of Green function

Any reference/advice would be good. I can use eigenfunction to solve the Green function for $$\Delta u(x) + k^2 u(x) = \delta(x - y)$$ boundary condition given as $u = 0$ on $\partial B(1)$, unit ...
1
vote
2answers
38 views

What is the correct method of finding the leading order behavior of a function in a given limit?

I am kind of confused about finding the correct leading order behavior of a function. Example: If I want to understand the behavior of the following function $$f(x)=\coth(x)-\frac{1}{x}$$ I can ...
3
votes
1answer
40 views

Question about weak convergence, $\lbrace f(x_{n}) \rbrace$ converges for all $n$, then $x_{n} \rightharpoonup x$

I found the following question in my textbook Let $E$ a reflexive space and $\lbrace x_{n} \rbrace \subset E$ a sequence such that $\lbrace f(x_{n}) \rbrace$ converges for all $ f \in E^{*} $, Show ...
3
votes
1answer
48 views

Computing the limit of an integral

Consider the following integral $$ \int_{-\infty}^{\infty}f(t) K(\frac{a-t}{h})dt $$ where (1) $h>0$, $a \in \mathbb{R}$ (2) $f:\mathbb{R}\rightarrow[0,\infty)$ is such that ...
1
vote
0answers
43 views

Eigenfunctions of integral operator

I am faced with the problem of calculating the eigenfunctions for an operator of the form: $(Kf)(x) = \int_{-\infty}^{\infty} K(x-\alpha y) f(y)dy $ Does anyone know for which functions (or types of ...
14
votes
1answer
270 views

Mathematical meaning of certain integrals in physics

While studying on texts of physics I notice that differentiation under the integral sign is usually introduced without any comment on the conditions permitting to do so. In that case, I take care of ...
0
votes
0answers
46 views

Continuous non-differentable functions

I'm looking for some examples of everywhere continuous functions which are nowhere differentable. I found already Takagi curve and Weierstrass function. Can you point out some online courses or pdf ...
2
votes
2answers
50 views

Prove that there is no norm for to make this mapping continuous

I am dealing with an exercise which is as follows: Show that there is no norm such that the set of all the mappings $T_a$ which map every element $f\in C(\mathbb{R}, \mathbb{R})$ (where the latter is ...
1
vote
2answers
30 views

Orthonormal projection contracts inner product?

I wonder if an orthonormal projection $P^2=P$ in a Hilbert space $\mathcal{H}$, contracts its inner product i.e. $\langle PW,V \rangle \leq \langle W,V \rangle $ for every pair of elements $W,V \in ...
2
votes
1answer
22 views

Image of a dense set through unbounded operator

Let $T$ be a densely defined, closed operator on a Hilbert space $H$ such that $T^*T$ remains densely defined. Obviously, $\sigma(I+T^*T)\subset [1,\infty)$, which in particular implies this operator ...
1
vote
0answers
33 views

Finding the general function that satisfies this property

I've been playing around with some introductory examples in my functional analysis course, and I came across. Given $X = \mathbb{R} \setminus \{0,1,2\},$ I want to find all functions $f: X ...
2
votes
1answer
37 views

$C ([1,2] \times [0,1] \to \mathbb R)$ dense in $C ( [1,2] \rightarrow L^{2} ([0,1] \to \mathbb R))$?

Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense ...
2
votes
1answer
37 views

Continuous functions with values in separable Banach space dense in $L^{2}$?

Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense ...
0
votes
2answers
34 views

Spectrum of operator $T((x_n)_{n\in\mathbb{Z}})=\left(\frac{1}{n^2+1}(x_n-x_{-n})\right)_{n\in\mathbb{Z}}$

The eigenvalues should satisfy: $$T(x_n)=\lambda x_n$$ $$\frac{1}{n^2+1}(x_n-x_{-n})=\lambda x_n$$ $$\left[(n^2+1)\lambda+1\right]x_n=x_{-n}$$ I suppose that this should mean that ...
2
votes
1answer
57 views

Pythagorean theorem does require the Cauchy-Schwarz inequality?

In Wikipedia I found a proof of Schwarz inequality making use of the Pythagorean Theorem. Anyway I'm wondering... isn't that a petitio principii? Isn't the Pythagorean Theorem in Hilbert spaces proved ...
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votes
0answers
145 views

Show that there is no normed space X s.t. its dual X' is equivalent to $((C^1[0,1];\mathbb{R}), ||.||_{\infty})$.

Show that there is no normed space X s.t. its dual X' is equivalent to $((C^1[0,1];\mathbb{R}), ||.||_{\infty})$. Normed spaces are equivalent if there are bounded linear maps which are inverses of ...
2
votes
1answer
35 views

Show that if we had a complete metric space $X$ with no isolated points, then every singleton $\{x\}$ is nowhere dense

My attempt: The closure of the singleton is again the singleton itself Since there are no isolated points, then clearly $\{ x \}$ does not contain any non-empty open set hence the interior of the ...
0
votes
1answer
43 views

If we had a complete metric space with no isolated points, then singular points are nowhere dense

Let $X$ be a complete metric space. I am trying ot prove whether or not each point in $x$ is nowhere dense if $X$ has no isolated points. idea: the closure of a point is itself, and the interior of a ...
2
votes
1answer
41 views

Square Root of the shift operator indexed by $\mathbb{Z}$

My question is very similar to this question, but instead of indexing by $\mathbb N$ I am indexing by $\mathbb Z$. Consider the shift operator $T : \ell^1(\mathbb Z) \to \ell^1(\mathbb Z) $ given by ...
0
votes
1answer
35 views

How do we get the inequality?

Proposition: If $A \in \mathbb{R}^{n \times n}$ a symmetric matrix then $||A||= \sup \{ ||Ax||_2: ||x||_2=1\}= \sup \{ |\langle x, Ax \rangle|: ||x||_2=1\}$. Proof: It suffices to show that $||A|| ...
0
votes
0answers
17 views

For a convex function, why does lower semicontinuity imply weak lower semicontinuity?

I have come across the statement that, for a convex function, all notions of lower semicontinuity are equivalent. That is: weak lower, sequential lower, and weak sequential lower semicontinuity are ...
0
votes
0answers
21 views

Local banach algebra without zero divisors

I need to construct example of banach algebra with unique nontrivial maximal ideal and without zero divisors. I think it is must be a subalgebra of $\mathbb{C}[[z]]$, but I could not build anything.
2
votes
0answers
35 views

Conditions under which an Convolution operator is normal.

I have a possibly complex valued convolution operator given by $ \int_{\mathbb{R}}K(x-y)f(y)dy$ I know that the operator is self-adjoint if $K(x)=\overline{K(-x)}$ holds. But are there softer ...
1
vote
1answer
33 views

problem on finding norm

let $f$: $l^2$ $\to$ $\Bbb R$ be defined by $$ f(x_1,x_2,x_3,......) = \sum_{n=1}^\infty \frac{x_n}{2^\frac{n}{2}} \ \forall x=(x_1,x_2,....) \in l^2$$ then, what is the value of $\left\lVert ...
0
votes
1answer
26 views

Properties about reflexive space

I'm studying fuctional analysis and specifically reflexive spaces. My textbook has a introductory level, so don't cover so many things. My questions are: 1) If $X$ and $Y$ are isomorphics and $X$ is ...
1
vote
1answer
47 views

Properties of mollification

We have this theorem For any $1\le p<\infty$ and $f\in L^p(\mathbb{R}^k)$, then $\|f*\phi_\delta - f\|_p\to 0$ as $\delta\to0$, where $\phi$ is any nonnegative measurable function on ...
0
votes
0answers
26 views

On the sequence of orthonormal basic

I have a question : Let $0 \leq a \leq b \leq +\infty $, supposing that ${\phi_n(x,t)}_{n \geq 0}$ be the orthonormal basis on $L^2(a,b)$ respected to $x$. If there exist a sequence $\psi_n(t)$ such ...