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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Proving subsets of $l^{\infty}$ are compact

Recently I started reading up on some set theory and metric spaces. I just read about compact subsets and I thought I understood it but in the exercises I'm having difficulty with the following ...
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5 views

Some questions about Krein-Rutman Theorem

I would like to figure out the Krein-Rutman Theorem. And I'm following the notes: ftp://ftp.ma.utexas.edu/pub/papers/llave/.grad/5999_chap1-1.pdf However, I got some questions. Defintion. Let X ...
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1answer
18 views

Can one define a functional on a Hilbert space based on its action on a Hilbert basis?

I know that the actions of a functional on a vector space can be uniquely described by the value the functional takes on each element of a (Hamel) basis. My question is, in a Hilbert space, would ...
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2answers
120 views

One of these two operators is not invertible

I have a Hilbert space $H$ and a bounded self-adjoint operator $T$ on it with $||T||=1$. I've been trying to show that at least one of $I+T, I-T$ are not invertible, but I haven't been able to make ...
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0answers
10 views

Infinite matrix defining a bounded operator on $l^2$

I think I only need some help to clear up the terminology and make sure I understand correctly: I've shown that for a sequence $\{f_{k}\}_{k=1}^{\infty}$ in a Hilbert space $H$, there exists $C>0$ ...
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2answers
31 views

Proving functional's continuity

Prove that functional $f:C[a,b]\to \mathbb{R},\ f(x)=\int_a^bx^2(t)dt$ is continuous. Any ideas on how to approach this problem?
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2answers
22 views

Proving mapping is contraction

Prove that mapping $B:C[0,\tau]\to C[0,\tau]$. $$(Bx)(t)=\left( \int_0^\tau \sin x(s)ds\right) t, \ t\in [0,\tau], \ \tau >0$$ is contraction mapping if $\tau^2<1$. I want to show that for all ...
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0answers
7 views

Existence of state on a C*-algebra satisfying $|\tau(ab)|=\|ab\|$

Let $a,b$ be elements of a unital C*-algebra $A$ with $0\leq a,b\leq 1$ (e.g., $a,b$ are projections). Is it the case there is a state $\tau$ on $A$ such that $|\tau(ab)|=\|ab\|$? If $ab$ is normal ...
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23 views

Wot topology on $B(H)$ is not metrizable

Let $H$ be a infinite dimensional Hilbert space and $B(H)$ be the space of bounded and linear operators on $H$. I know that weak operator topology (wot) and strong operator topology (sot) are ...
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0answers
5 views

Sufficient conditions on integration kernel for continuity of the integral operator

Suppose that we have a measure $d\mu(v)=e^{-|v|^2}dv$ on $\Bbb R^d$. We define a linear operator $$T[f](u)=\int_{\Bbb R^d} |u-v|^\beta d\mu(v).$$ I want to establish conditons on $\beta\in\Bbb R$ so ...
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16 views

Unitary G-module

I'm not sure if I understand this sentence correctly: "By a unitary G-module we will mean a Hilbert space W on which G acts by means of a strongly continuous unitary representation". $G$ is a ...
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1answer
14 views

Strong convergence of product of operators on a Banach space

If $\{T_n\},\{S_n\}$ are two sequences of bounded operators on a Banach space $X$, such that $\{T_n\}$ converges weakly to $T$, and $\{S_n\}$ converges strongly to $S$, does it follow that $T_nS_n\to ...
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1answer
16 views

Convergence in the weak operator topology implies uniform boundedness in the norm topology?

If $\{T_n\}$ is a sequence of bounded operators on the Banach space $X$ which converge in the weak operator topology, could someone help me see why it is uniformly bounded in the norm topology? I ...
2
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1answer
42 views

Show that this path is differentiable but not rectifiable

My path is defined as follows: $\gamma:[1,1]\rightarrow \mathbb R, \space \gamma(t):= \begin{cases} \ (0,0) & \text{if $t$=0} \\[2ex] t,t^2 \cos (\frac{\pi}{t^2}), & \text{if $t$ $\in$ ...
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1answer
37 views

What are the hypotheses in Levi's monotone convergence theorem?

Today I read monotone convergence theorem , dominated convergence theorem and fatou's lemma And I need some help We know the dominated convergence theorem in Measure theory In its proof we ...
3
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1answer
32 views

Convergence in Fréchet spaces and the topology

actually i have two questions concerning Fréchet spaces (i am not familiar with these spaces so i need some help). I have a vector space $V$ with a distance $$d(x,y) = \sum_{n\in \mathbb{N}} ...
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11 views

show there does not exist a best Approximation element in $E$ [on hold]

Let $c_{0}$ to be the space of sequence which converge to $0$,with $l^{\infty}$ norm,and $$ E=\bigg\{x=(x_{n})\in c_{0}\bigg| \sum_{n=1}^{\infty}\frac{x_{n}}{2^{n}}=0 \bigg\} $$ We know that $E$ is ...
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1answer
27 views

Is this sufficient for $f'' \in L^2$?

Let $f \in L^2(0,2\pi)$ be taken such that $f$ and $f'$ are absolutely continuous on $[0,2\pi]$ with $f(0) = f(2\pi)$ and $f'(0)= f'(2\pi).$ Is this sufficient to conclude from this that $f'' \in ...
2
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1answer
14 views

Normal Operators: Construction

Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(T)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$A:=N(1+N^*N)^{-1}\in\mathcal{B}(\mathcal{H})$$ ...
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0answers
30 views

A question regarding Parseval's identity.

In most books/websites, Proposition 2 (see below) is either stated for a Hilbert space or proved via Riesz-Fischer. Does the follow approach (which seems to work in an inner product space) fall down ...
2
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1answer
34 views

Identity of the Operator Norm applied to the differential $df$ for convex $U \subset \mathbb{R}^n$ and $f \in C^1(U, \mathbb{R}^k)$

In my Analysis II Script they often use 'special' norms such as the Operator norm to make proofs more 'elegant' or just shorter. There is also the following statement (without a proof) which I can't ...
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0answers
41 views

Showing that the sequence of functions is not Cauchy

I need to show that $ g_n(x)=x^{1/(2n-1)} $ is not a Cauchy sequence in $C[-1,1] $ w.r.t. supremum norm. I tried to find the maximum of the difference of $g_n$ and $g_m$ by just differentiating but ...
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1answer
46 views

A rather abstract strongly continuous semigroup

Define $X$ as the Hilbert space $L^{2}(0,\infty)$ and let the operators $T(t):X\to X$, $t\ge 0$ be defined by $(T(t)f)(\zeta):=f(t+\zeta)$ I want to show that $(T(t))_{t\ge 0}$ is a ...
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0answers
13 views

C* Algebra, f(x,z)

Let $A$ be a $C^*$ algebra, $x\in A$ and $||x|| < 1$. Let $f(x,z) = (1-x x^*)^{-\frac{1}{2}}(1+zx)$, $|z|=1, z\in \mathbb{C}$, $\mathbb{C}$ is the complex field. How to prove: $$ f(x,z)^* f(x,z) ...
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0answers
20 views

When is it possible to bound a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ with $\big|\ f(x_1,x_2,\ldots,x_n)\ \big| \le {\prod}_{i=1}^n h_i(x_i)$

Is there any result that specifies when a multivariate function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ can be bounded (either locally or globally) by a product of some functions $h_i:\mathbb{R} ...
3
votes
1answer
31 views

I need help understanding the proof of Lemma 2.4-1 from Kreyszig's Functional Analysis.

Lemma: Let $\{x_1, \ldots, x_n \}$ be a linearly independent set of vectors in a normed space $X$ (of any dimension). Then there is a number $c > 0$ such that for every choice of scalars $\alpha_1, ...
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0answers
42 views

How to prove the space $H$ is Banach?

$H$={$f$:$f$ and its derivative are absolutely continuous and squared integrable in $\mathbb{R}$}. The norm is ...
4
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1answer
50 views

Categorical Banach space theory

Consider the category $\mathsf{NormVect}_1$ of normed vector spaces with short linear maps$^{\dagger}$ and the full subcategory $\mathsf{Ban}_1$ of Banach spaces with short linear maps. Both ...
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20 views

Weak convergence in $D^{1,p}(\mathbb{R}^n)$ and in $\mathcal{M}(\mathbb{R}^n)$

What it means: $$u_n\rightharpoonup u ~\text{weakly in }~ D^{1,p}(\mathbb{R}^n)\\ |\nabla(u_n-u)|^p\rightharpoonup\mu ~\text{weakly in}~ \mathcal{M}(\mathbb{R}^n)\\|u_n-u|^{p^{*}}\rightharpoonup\nu ...
1
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1answer
26 views

Time derivative of logistic function [on hold]

I was wondering whether there is a possible solution to this. If we have function $$ y_t = \frac{x_t}{1+x_t}. $$ given that $x_t>0$ we can represent it as a logistic function $$ y_t = ...
3
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1answer
33 views

How can I prove that $f$ is inner product function

We know the polarization identity in inner product space : $$\langle x,y\rangle= \frac{1}{4} (\|x+y\|^2-\|x-y\|^2) + \frac{i}{4} (\|x+iy\|^2-\|x-iy\|^2) $$ But the question is if we have ...
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1answer
24 views

Obtain the triangle inequality from the Minkowski inequality

I want to prove: $$\|f-h\|_p \leq \|f-g\|_p + \|g-h\|_p$$ Minkowski inequality: $$\|f+g\|_p \leq \|f\|_p + \|g\|_p$$ Is $\|f-g\|_p \leq \|f\|_p + \|g\|_p$? It looks like we have: $$\left(\int_a^b ...
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0answers
21 views

$c_{00}$ is a dense subset of $c_0$

I would like to show that $c_{00}$ is a dense subset of $c_0$. I am not sure if I am overly simplifying the argument or even making the right argument for that matter. proof: Suppose that $x \in ...
2
votes
1answer
20 views

For fn(z)= 1/nz, If we make fn(0)= 1, does that make the family of functions bounded?

I have a problem that requires me to use a theorem requiring a bounded family of functions. The family provided that I am supposed to use this theorem for is $f_n (z) = \frac 1 {nz}$ when $z \neq 0$ ...
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0answers
22 views

Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$ where $P$ is a strictly hyperbolic linear operator. For ...
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0answers
23 views

What is the Fourier basis for all 2 dimensional functions?

Let us say we have a set of all 2-dimensional functions (E.g. 1 time and 1 space dimension). What is the (Schauder) basis for this set?
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0answers
15 views

First order elliptic pseudodifferential operator and Sobolev space

Let $P$ be an elliptic pseudodifferential operator of order 1 on a compact manifold $M$. Also let the spectrum of $P$ lie inside $(0, \infty)$. I am trying to see how one could prove that $P : H^1(M) ...
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2answers
33 views

Swapping series and linear operators

If $T$ is a continuous linear transformation between normed spaces. Under what conditions of $T$ and $(a_n)_n$ we have $T(\sum_{n=0}^\infty a_n)=\sum_{n=0}^\infty T(a_n)$?
3
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1answer
20 views

Lower semicontinuous energy functional on compact space of Lipschitz functions

Let $\Omega \subset \mathbb{R}^{n}$ be a bounded open subset containing $0$ and let $L>0$ be some positive constant. Consider the space $A_{0}=\{f \in C^{\infty}(\overline{\Omega}) \mid f \text{ ...
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1answer
26 views

how can we get Pythagoras from the parallelogram law

When using the definition and properties of the inner product, we get the parallelogram law: $||x+y||^2= \langle x+y, x+y\rangle= \langle x, x\rangle + \langle x, y\rangle +\langle y, x\rangle ...
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1answer
22 views

Evaluating norm of the operator

I have to calculate norm of the operator $\varphi : l_{1} \rightarrow \mathbb{C}$, where $$ \varphi( (x_n)_{n=1}^{\infty} ) = \sum_{n=1}^{\infty} (-4)^{-n} x_{2n}.$$ My attempt was as follow: Let ...
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0answers
14 views

Definition of equi-absolute continuity

Could someone provide (or point me to) a definition of equi-absolute continuity for functions defined on an open bounded subset $\Omega \subseteq\mathbb{R}^n$? I only managed to find a definition for ...
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0answers
23 views

Estimating a sum

i want to show the following: Assume that $\sum_{m\in\mathbb{N}}{|i+\lambda_m|^{-p}}<\infty$ (where $(\lambda_m)_m$ is a sequence of real numbers). I want to show that then also holds for each ...
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0answers
34 views

Question about convergence in $L^2$ (revisited)

Yesterday I asked the folowing question: Question about convergence in $L^2$ which was answered negatively with a counterexample. Here, I wonder if one can find the right set to look at: Assume we ...
4
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1answer
34 views

Is the following statement true on $L^0$ spaces?

Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $X,Y\in L^0(\Omega;\mathbb{R})$ two random variables taking values in $\mathbb{R}$. Is it true that: $$\int_{A} f(X(\omega)) P(d\omega) = ...
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2answers
35 views

Inequality error possibly. How are two inequalities equal?

Notation: $\underline{x}\in \Bbb R^n,||\cdot||_p =\left(\sum \limits_{i=1}^n |\cdot|^p\right)^{\frac1p}$ $$||\underline{x}||_p\left( \sum \limits_{i=1}^n |x_i + ...
0
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0answers
18 views

Elliptic regulartiy for nonlinear elliptic equations

Here is the question: Let us consider the Schrodinger type equation $$ \left\{ \begin{aligned} &-\Delta u + u = |u|^{p-2}u \quad \text{in } \mathbb{R}^N \\ &u\in H^1(\mathbb{R}^N) \qquad ...
1
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1answer
39 views

Determine if the set of all $f \in C(\mathbb{R})$ such that $f(1/2)$ is a rational number is a subspace of $C(\mathbb{R})$

Let $C(\mathbb{R})$ denote the vector space over $\mathbb{R}$ of all continuous functions on $\mathbb{R}$. Determine if the set of all $f \in C(\mathbb{R})$ such that $f(1/2)$ is a rational number is ...
1
vote
2answers
32 views

Hölder inequality conditions for $L_p$ spaces?

The Hölder inequality is the statement that if $f,g$ are measurable functions then $$ \|fg \|_1 \le \|f\|_p \|g\|_q$$ if $p,q$ are such that ${1\over p}+ {1 \over q} =1$. But it's not clear to me ...
0
votes
1answer
17 views

About the self-adjoint extension of an operator.

Let $B$ be a selfadjoint extension of an operator $A$ on a Hilbert space $H$. Let $\varphi \in \ker(A^\ast-z_0)$. Then i want to show that $\varphi + (z- z_0)(B-z)^{-1} \varphi \in \ker(A^\ast-z)$. I ...