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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5 views

square root commutes with multiplication for positive elements in a $C^*$ algebra?

Let $A$ be a unital $C^*$ algebra. Let $a,b$ be positive elements. Through the functional calculus I can define $\sqrt{a}$ and $\sqrt{b}$. Now I don't necessarily see why ...
1
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0answers
7 views

For a densely defined symmetric operator $A$, is $A^2$ also densely defined?

Let $A : D(A) \to H$ be a possibly unbounded, densely defined symmetric operator on a Hilbert space $H$ ($A$ being symmetric means that $(\varphi, A\psi) = (A\varphi, \psi)$ for all $\varphi, \psi \in ...
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1answer
22 views

A factorization for operators

Let $a$ be an arbitrary operator in $B(H)$ and $b$ be a positive operator in $B(H)$. Assume $a$ and $b$ have the same null space and there exists an operator $u\in B(H)$ with $a=ub$. Q) Can we ...
3
votes
1answer
42 views

Conergence of linear functional

I want to determine wether the functional $\varphi_n:\ell^2\to \mathbb{R}$ defined by $$\varphi_n(x)=\frac{1}{n}\sum_{k=1}^n\sqrt k x_k\quad x=(x_1,x_2,\dots)$$ converges in norm, or in weak sense. ...
0
votes
1answer
21 views

Quotient maps in Banach spaces

I came across two definitions of quotient maps in Banach spaces. A bounded linear transformation $T:X\to Y$ is a quotient map if: A) $T(\textrm{int}(B_X))=\textrm{int}(B_Y)$ B) ...
2
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0answers
32 views

$\exp(x)$ as defined by a net

Motivation: So, I had an idle thought last week, and I thought I would ask it here before I forget about it. It is well known that we can define $$ e^x = \lim_{n \to \infty} \left(1 + \frac x{n} ...
-4
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0answers
24 views
3
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0answers
43 views

Do $L^p$ spaces measure something natural?

Obviously much of analysis on $\mathbb R^n$ considers $L^p$ spaces or other Banach spaces derived from them. The definition of $L^p(\mathbb R^n)$ looks very natural, but I've been bothered for some ...
0
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1answer
13 views

Existence of $\{a_n\}$ s.t $f(\vec{v})=\sum_{n \in \mathbb{N}} a_n v_n$ for continuous linear functions.

disclaimer: this is a homework assignment, I'm looking for a hint or a word of advice, not a full solution. Let $f: l^1 \to \mathbb{R}$ be a continuous, linear function. Prove that there exists a ...
0
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1answer
30 views

Unit ball of $L^1$, $L^\infty$ and $C(X)$ is not strictly convex

I need to show that the unit balls of $L^1(\mu)$, $L^\infty(\mu)$ and $C(X)$ are not strictly convex. I have already shown that if $1<p<\infty$ then the unit ball of $L^p(\mu)$ is strictly ...
4
votes
1answer
54 views

Prove that $f\in L^2$ and $\lim_{n\rightarrow\infty} \int_A f_n = \int_Af$

Let $A$ be a bounded, measurable susbset of $\mathbb{R}$. Prove that if $(f_n) \subset L^2 (A)$ converges uniformly to $f$ on $A$, then $f\in L^2(A)$ and $\lim_{n\rightarrow\infty} \int_A f_n = ...
0
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0answers
13 views

Showing any bounded sequence in Holder space $C^{1/2}$ has a convergent subsequence in Holder space $C^{1/3}.$

Prove that any bounded sequence in $C^{1/2}([0,1])$ admits a convergent subsequence in $C^{1/3}([0,1]),$ where we say that $f \in C^{\alpha}([0,1])$ if $f$ is Holder continuous of order $\alpha.$ The ...
2
votes
0answers
29 views

Identity operator on a dense subset cannot be extended to a continuous function from X to Y

I'm practicing some old exams for my Functional Analysis test and i'm stuck on the following question: Let $X$ be any Banach space with $Y \subset X$, $Y \neq X$ and $Y$ dense in $X$. Show that the ...
1
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1answer
26 views

The dual of the space of all the bounded functions

I 'd like to know what is the dual space of the space of all the bounded functions on the set $X$, where $X$ can be any set. Also, I don't assume that the function $f$ is measurable relative to any ...
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0answers
42 views

Function help needed ASAP [on hold]

How can I find the value of (x) for which the function of a curve f(x) is not defined??? Eg. F(x)= 3x^2 - 4x -24 =0???
2
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2answers
42 views

A characterization of tempered distributions

The Schwartz space on $\mathbb{R}^n$ is the function space $$ S \left(\mathbf{R}^n\right) = \left \{ f \in C^\infty(\mathbf{R}^n) : \|f\|_{\alpha,\beta} < \infty,\, \forall \alpha, ...
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0answers
38 views
+50

If $u \in W^{1,p}(I) \cap C_c(I)$ then $u \in W_{0}^{1,p}(I) $

I want to show the following statement ($1 \leq p < \infty$), for an open interval $I$: If $u \in W^{1,p}(I) \cap C_c(I)$ then $u \in W_{0}^{1,p}(I) $. $W^{1,p}(I) $ is the Sobolev Space, ...
0
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0answers
24 views

I don't understand how the adjoint operator is used in a book that I'm reading

I'm reading Stochastic Differential Equations in Infinite Dimensions and don't understand what the authors do in Chapter 2.3.1. Let me introduce the necessary objects: Let $K$ and $H$ be real ...
0
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0answers
19 views

Is there a o.g basis of $H^1(\Omega)$ that is o.n in $L^2(\Omega)$?

Let $\Omega$ be a bounded smooth domain, and define $H^1(\Omega)$ with the usual norm involving the function and its gradient. I am wanting to know if $H^1(\Omega)$ possesses a basis $b_j$ such that ...
2
votes
1answer
15 views

Summation of L1-function values finite

For $f \in L^1(\mathbb{R})$, we can proof, that $\sum_{n\in \mathbb{Z}} |f(t+na)|<\infty$ ALMOST EVERYWHERE for $t\in [0,a]$ ($\star$ proof below) My Question: If we assume $f\in ...
2
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1answer
22 views

The quotient norm on $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$.

I try to show that the norm on the quotient space $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$, where $x = (x_n)_{n \in \mathbb{N}} \in \ell^{\infty} ...
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0answers
12 views

the imageof the closed unitball of a Hilbert space under a compact linear map between Hilbert spaces, is compact

Could you plz give me a hand in this problem? Thanks in advanced let T: H_1→H_2 be a compact linear map between Hilbert spaces H_1 and H_2. then the image of the closed unit ball of H_1 under T is ...
1
vote
1answer
9 views

Convergence of unitary products on a Hilbert space

First: I'm sorry for the basic question--I can move it to Math SE if necessary... Let $X$ be a Hilbert space and suppose $\{U_k\}_k$ is a sequence of unitary operators on $X$. Let $\|\cdot\|$ be the ...
8
votes
0answers
113 views

Fast convergence in $L^1$ implies convergence almost everywhere

This is a proof-verification request. Claim: Let $(X,\mathscr M,\mu)$ be a measure space. Let $f_n$ ($n\in\mathbb N$) and $f$ be measurable, integrable, real-valued functions such that ...
2
votes
1answer
45 views

Proof of Functions from R to R

I need hints in how to prove this. Given $X,Y\subset \Bbb R$, and let $F=${$f:\Bbb R \rightarrow \Bbb R| f(X)=0$} and $G=${$g:\Bbb R \rightarrow \Bbb R| g(Y)=0$} prove that: $Hom(\Bbb ...
0
votes
2answers
57 views

The product of two distributions is not a distribution

Suppose $u(x)$ is the Heaviside function, which takes value $1$ when $x\ge0$ and takes value $0$ when $x<0$. Then the derivative of $u$ is a delta function $\delta_0$, or a distribution. Now my ...
1
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2answers
37 views

Minimizing an integral — Hilbert space

Find the real values of $a, b$ which minimize $$\int_1^{\infty} \left| \frac{1}{x^2} - a \frac{1}{x^3} - b\frac{1}{x^4}\right|^2 \; dx.$$ Hint : Work in an appropriate Hilbert space. Here is ...
0
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1answer
25 views

Find widest subset on which Fourier series can be integrated and derived term by term

As part of one problem I need to find the widest subset of $\mathbb{R}$ on which the obtained Fourier series can be integrated and derived term by term. I found that it has something to do with ...
0
votes
1answer
19 views

How do calculate not emty set levels of function $f$?

Let $f$ be defined as follows: $$f:\mathbb{R}^{2}\to\mathbb{R}:(x,y)\mapsto\begin{cases}\frac{xy^{2}}{x^{2}+y^{4}}&\text{if } (x,y)\neq (0,0)\\ 0&\text{if } (x,y)=(0,0)\end{cases}$$ I put ...
2
votes
1answer
48 views

Open set in $\ell^2$

Let $a=(a_n)_n\subset(0,\infty)$ be a sequence and $S^{(a)}:=\{(x_n)_n\in\ell^2:\lvert x_n\rvert\ <a_n \forall n \}$. I want to prove that $S^{(a)}$ is open in $\ell^2$ iff $\inf_{n\in\mathbb{N}} ...
0
votes
0answers
22 views

I want to calulate the range of an operator $S$ which maps $L^{2}$ into $H^{1}(\Omega)$?

I have an equation $s(\lambda,\mu)=l(\mu)$, where s(.,.) is a symetric positive definite bilinear form in $L^{2}(\Omega)$, and $l(.)$ is in $H^{1}(\Omega)$. I want to show that the range of the ...
0
votes
1answer
19 views

Convex set in a vector space gives a norm

Given an $\mathbb{R}$ or $\mathbb{C}$ vector space $X$ and a function $p:X\rightarrow[0,\infty)$ with $p(x)=0$ iff $x=0$ and $p(\alpha x)=|\alpha|p(x)$ for all $x,\alpha$, I want to show that $p$ is a ...
0
votes
1answer
23 views

Removable singularities in Sobolev spaces

I'm going to state and prove a theorem (as we did during a lecture), which basically is contained in Differentiable Functions on Bad Domains - Vladimir G. Maz'ya, Sergei V. Poborchi, and I'll add my ...
3
votes
0answers
29 views

Closed map $T:X \to Y$ has closed graph?

Let $T:X\to Y$ be a linear operator between two normed vector spaces. My question is: If $T$ is a closed map (sends closed sets to closed), then is the graph of $T$ a closed set of $X \times Y$? ...
0
votes
0answers
39 views

Convergence of sequence of vectors in $C^*$-algebra

Let $B$ is $C^*$-algebra and $x_i \in B$ - linear independent vector system, $\alpha_i \in \mathbb{C}$ such that: $$\|x_i\| = 1$$ $$\lim_{N \to \infty} \|\sum_{k=1}^N \alpha_k x_k\| = \lim_{N \to ...
0
votes
1answer
23 views

Banach fixed-point theorem for a recursive functional equation

I was asked to prove that the functional equation $$ x(t) = \begin{cases} \frac{1}{2} x(3t) + \frac{1}{2},& 0 \leq t \leq \frac{1}{3}\\\ f(t), & \frac{1}{3} < t\leq \frac{2}{3}\\\ ...
3
votes
1answer
41 views

Universal $C^*$-algebra of countable family of self-adjoint operators have boundedly complete standard Schauder basis

Let $A = C^*(T_1,T_2,T_3,... | T_i=T_i^*, ||T_i|| \leqslant 1)$ - universal enveloped $C^*$-algebra of countable family of self-adjoint operators. $A$ have standard Schauder basis, which contains all ...
0
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1answer
34 views

Can you recover a distribution from mollification?

Let $f\in \mathcal S'(\mathbb R)$ be a Schwartz distribution. Given $\rho \in C^\infty_c(\mathbb R)$ define the convolution as the function $$x\mapsto (f\ast\rho)(x):=\langle f, \rho (\cdot ...
3
votes
4answers
75 views

Solving a functional equation ( $ f(x-y) = f(x)/f(y)$ )

Consider the functional equation $$f(x-y)=f(x)/f(y)$$ If $f'(0)= p$ and $f'(5)=q$, then what is the value of $f'(-5)$ ? My attempt. Using the equation written above I was able to determine the ...
0
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1answer
788 views

Show that for any partition $P$ of $[a,b]$, $U(f,P) - L(f,P) \leq C(b-a)mesh(P)$.

Suppose $f:[a,b]\to \mathbb{R}$ is Lipschitz, i.e $|f(x)-f(y)| \leq C|x-y|$ for all $x,y$ in $[a,b]$ and thus $f$ is continuous. Show that for any partition $P$ of $[a,b]$, $U(f,P) - L(f,P) \leq ...
5
votes
1answer
100 views

Connection of Fourier's work with Fredholm's

Im trying to formulate for myself in what sense Fredholms work on the Dirichlet problem is connected to Fouriers work on the heat equation. Fourier idea seems to have fundamental problems with ...
9
votes
1answer
2k views

Proof of separability of $L^p$ spaces

The following is a proof in Brezis book. It shows the separability of $L^{p}$ spaces: I have a few questions regarding the proof: It says 'it is easy to construct a function $f_{2} \in ...
0
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0answers
32 views

Is there a way to derive/prove that $ \langle Ux, t \rangle = \langle x, U^{-1}t \rangle$ is true for unitary transformations?

Why is it that $ \langle Ux, t \rangle = \langle x, U^{-1}t \rangle$ is true for unitary transformations? I have a argument (not quite a proof I guess) of why that is true for finite vectors but I ...
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votes
0answers
16 views

How do I find not empty level sets of function?

Let $f$ be defined as follows: $$f:\mathbb{R}^{2}\to\mathbb{R}:(x,y)\mapsto\begin{cases}\frac{xy^{2}}{x^{2}+y^{4}}&\text{if } (x,y)\neq (0,0)\\ 0&\text{if } (x,y)=(0,0)\end{cases}$$
1
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0answers
27 views

If $Q$ is a trace class operator on $U$, then each bounded, linear operator from $U$ to $H$ is a Hilbert-Schmidt operator from $Q^{1/2}U$ to $H$

Let $U$ and $H$ be Hilbert spaces $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ with finite trace $U_0:=Q^{1/2}U$ $L$ be a bounded, linear operator from $U_0$ to $H$ I ...
3
votes
1answer
58 views

Is the derivative of differentiable function $f:\mathbb{R}\to\mathbb{R}$ measurable on $\mathbb{R}$?

Suppose we have a bounded differentiable function $f:\mathbb{R}\to[a,b]\subset\mathbb{R}$. Hence $f$ is continuous and measurable (in terms of standart Lebesgue measure) on $\mathbb{R}$. I want $f$ ...
1
vote
2answers
42 views

finite spectrum eigenvalue

Let $T:X \to X$ be a linear bounded operator where X is Banach space ,and $\sigma (T)$ is a finite set.Then does the spectrum consist of eigenvalues only? Any hint or counterexample is ok. thanks in ...
3
votes
1answer
27 views

Why is $C(\beta \mathbb{R})/C_0(\mathbb{R})\cong C(\beta \mathbb{R}\setminus \mathbb{R})$ as $C^*$-algebras?

Let $\beta \mathbb{R}$ be the Stone-Čech compactification of $\mathbb{R}$ (with euclidean topology) and $C_0(\mathbb{R})$ the $C^*$-algebra of continuous complex-valued functions vanishing at ...
0
votes
1answer
39 views

Exercise 1.65 of Megginson's “An Introduction to Banach Space Theory”.

Unfortunately I do not succeed in completing the following exercise: Let $X$ be a Banach space and let $T : X \to \ell^{1} (\mathbb{N})$ be a linear operator. For each $n \in \mathbb{N}$, let ...
2
votes
1answer
23 views

Weak/Weak* topologies compared to topologies generated by semi-norms from dense subset

The weak topology on normed linear space $X$ can be defined as being induced by semi-norms $\|\cdot\|_{x'}$, $x'\in X'$ with $\|x\|_{x'}=|x'(x)|$. Similarly the weak* topology is induced by ...