0
votes
1answer
64 views

Weak convergence of scaled elements implies norm convergence

Let $u_{k}\in l^2{\mathbb{(Z)}}$ be a sequence such that for every sequence $n_{k} \in \mathbb{Z}$ the sequence $n_{k}u_{k}\rightharpoonup 0$. Prove that $ u_{k} \rightarrow 0$ in $l^{q}(\mathbb{Z}) , ...
2
votes
1answer
24 views

Find the values of $p$ for which the given sequence converges in $l^p$ norm or weakly

We consider $E_p=(c_{00}, ||\cdot||_{p})$ (where $c_{00}$ are the sequences that are zero except in a finite number of values and $1\leq p \leq \infty$) and the sequence:$$(x_n)_{n\in ...
2
votes
1answer
36 views

If a compact operator satisfies $T^nx\to0$ weakly for all $x$, then $\|T^n\|\to0$

Let $H$ be a real Hilbert space, $T:H\to H$ be a compact operator. Suppose that for every $x\in H$, sequence $(T^n x)_{n\in \mathbb{N}}$ converges weakly to $0$. How to prove that $ ...
1
vote
1answer
46 views

Convergence of square root operators

Let $Q_n$ and $Q$ be compact positive and symmetric operators. Let $A_n = {Q_n}^{\frac12}$ and $A=Q^{\frac12}$. Given $Q_n$ converges to $Q$ w.r.t. operator norm. Does $A_n$ converges to $A$? Thanks. ...
1
vote
1answer
32 views

Hilbert Space: Weak Convergence implies Strong Convergence

This probably might be a duplicate - let me know if so. I read the following in Graf's notes on quantum mechanics - can you give me a hint for the proof. In Hilbert spaces weak convergence in a way ...
5
votes
1answer
37 views

Show that $C^1([0,1])$ is not reflexive

Aim of this exercise is proving that $(C^1([0,1]),\|\cdot\|_{C^1})$ is not reflexive. We know that, if $(f_h)_h\subset C^1([0,1])$ is a sequence that weakly converges to $f\in C^1([0,1])$ (that is ...
1
vote
1answer
47 views

If holomorphic $\{f_n\}\to f$ uniformly on compact subsets of $U$, then do $f_n$ and $f$ eventually have the same number of zeros?

Let $U$ be an open subset of $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $U$ such that $f_n\to f$ uniformly on any compact subset $K$ in $U$. Suppose $f$ is not constant, ...
2
votes
2answers
50 views

For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives.

Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset ...
2
votes
1answer
43 views

completely bounded maps - convergence

Let $x$ be a completely bounded map between operator spaces $W \subset \mathbf{B}(\mathcal{H})$ and $V \subset \mathbf{B}(\mathcal{K})$, where $\mathcal{H}$ and $\mathcal{K}$ are Hilbert spaces, and ...
0
votes
0answers
35 views

Proving the existence of limit of an integral

Let $g:\mathbb{R}^d\to\mathbb{R}$ a smooth function and $B:\mathbb{R}^d\to\mathbb{R}^d$ a Lipschitz continuous vector field. I have to study the limit of the following integral ...
0
votes
0answers
33 views

Convergence of norms

I have this space $H_{0,p}^1=\lbrace u\in AC([0,+\infty),\mathbb{R}),u(0)=u(+\infty)=0, \sqrt{p} u'\in L^2(0,+\infty)\rbrace $ endowed with the norm $||u||^2=\int_0^{+\infty} p(t) u'^2(t) dt$ ...
1
vote
1answer
29 views

Norms and convergence in $\mathcal{C}^{\infty}(O)$

Let $O \in \mathbb{R}^d$ be open, $K \subseteq O$ compact and $n \in \mathbb{N}$. For $f \in \mathcal{C}^{\infty}(O)=\mathcal{E}(O)$ we define $$\|f\|_{n,K}^{(1)}:=\sup_{|\alpha| \leq n} \|D^{\alpha} ...
1
vote
1answer
25 views

weak convergence of probability measures and unbounded functions with bounded expectation

Assume that $\mu^n$ are probability measures on $R$ that convergence weakly(-*) to $\mu$, i.e for all $f \in C_b (R)$ (bounded and continuous), we have that $\int f(x) \mu^n(dx) \rightarrow \int f(x) ...
1
vote
1answer
22 views

Bijective transformation of $L^2$-weak convergence sequence again weak converging?

Let $f_n$ converge weakly in $L^2(x)([0,1])$ to $f$, with $|f_n(x)|\leq C$ for almost all $x\in]0,1]$ and all $n$. Let $H:R\rightarrow R$ be strong monotone increasing and continuous with $H(0)=0$. ...
1
vote
1answer
47 views

Does weak convergence in $L^2$ implies convergence almost everywhere along subsequence?

If I know $\int_{[0,1]} f_{n}(x) g(x) dx \rightarrow \int_{[0,1]} f(x) g(x) dx$ as $n \rightarrow \infty$ for all $g \in L^2([0,1])$ (weak convergence in $L^2$) and $|f_n(x)|_{L^2} <C$ (uniformly ...
2
votes
0answers
31 views

Weakly closed convex set and norm convergent

Let $X$ be a normed space and let $\{x_n\}$ be a sequence in $X$ such that $x_n\to x$ weakly. Show that there is a sequence $\{y_n\}$ such that $y_n\in \operatorname{co}\{x_1,...,x_n\}$ and ...
0
votes
1answer
40 views

How to deal with discontinuous points when proving that step functions are dense in $PC[a,b]$

This question is a follow-up to my previous question: How does one prove that a space is dense in another under some norm? I figured out a way to solve (part of) the exercise. Given some function ...
2
votes
2answers
37 views

Closed subspaces of $L^2(0,1)$

I would like to prove that the almost-everywhere constant functions, and the functions whose integral is 0 are closed subspaces of $L^2(0,1)$. It's readily seen that they are subspaces. I'm finding ...
0
votes
1answer
28 views

Equivalence of weak-* convergence in Banach spaces.

Let $X$ be a Banach space and $f,(f_n)_{n \in \mathbb N} \in X^*$. $f_n \xrightarrow{w^*} f$ if and only if a) $\sup_{n \in \mathbb N} \|f_n\| < \infty$ and b) $\exists A \subset X: ...
0
votes
1answer
26 views

a non-decreasing sequence of functions with bounded L^p-norm is a Cauchy sequence in L^p space

Let $\{f_k\}_{k=1}^\infty$ be a sequence in $L^p(\mathbb{R})$ for $1\leq p<\infty$. Suppose $f_1\leq f_2\leq\cdots$ and $\sup ||f_k||_p<\infty$. Prove that $\{f_k\}_{k=1}^\infty$ converges in ...
5
votes
0answers
111 views

Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper. In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...
0
votes
2answers
24 views

Let $u_n \to u$ in $L^1(\Omega)$. Does $u_n^p \to u^p$ in $L^1(\Omega)$ if we know $u_n^p \in L^1(\Omega)$?

Suppose $u_n \to u$ in $L^1(\Omega)$ where $\Omega$ is a bounded domain. Suppose that $u_n^p \in L^1(\Omega)$ (actually $L^\infty(\Omega)$ for each $n$). Fix $p \in [1,\infty)$. So $u_n(x) \to u(x)$ ...
0
votes
1answer
37 views

Theorems on continuous embedding

Let $X,Y$ be two normed spaces such that $X\hookrightarrow Y$. Show the following: 1) If $A\subset X$ is closed in $Y$ then $A$ is closed in $X$; 2) If $X, Y$ are Banach spaces if $x_n$ tends ...
1
vote
1answer
34 views

bounded sequence in $L^p(\mathbb{R}^n)$ that converges a.e.

Let $1<p<\infty$. Let $\{f_k\}$ be a sequence in $L^p(\mathbb{R}^n)$. Suppose $f_k\to f$ a.e. and there exists $C>0$ such that $||f_k||_p\leq C$ for all $k$. Prove that for all $g\in ...
1
vote
1answer
37 views

Convergence of a sequence of unit vectors in a Hilbert space

Let $E$ be a vector space (over $\mathbb{R}$) with a positive definite hermitian form and let $\{x_{n}\}$ ($x_{n} \not= 0$) be a sequence converging to $x$ in the $L^{2}$ norm ...
0
votes
1answer
34 views

Show that if a series of functions converges uniformly then it also converges in $L^2$ and pointwise sense. [closed]

Consider any series of functions on any finite interval. Show that if the series converges uniformly, then it also converges in the $L^2$ sense and pointwise. Need help with this problem thank you.
5
votes
1answer
84 views

Strong convergence of an “averaging” operator

Let $X$ be an Hilbert space and $S:X \rightarrow X$ be a bounded linear operator with $||S||=1 $ Define $$T_n= \frac{1}{n} \sum_{r=0}^{n-1} S^r$$ I want to show it converges strongly to some ...
1
vote
1answer
58 views

If $a_n \to a$ in $L^2$ and $F:\mathbb{R} \to \mathbb{R}$ is continuous, does $\int F(a_{n_j}) \to \int F(a)$?

Suppose $a_n \to a$ in $L^2(\Omega)$. Let $F:\mathbb{R} \to \mathbb{R}$ be continuous with $F(0) = 0$. We have that $F(b) \in L^1(\Omega)$ if $b \in L^2(\Omega)$ and $|F'(x)| \leq C_1 + C_2|x|$. I ...
1
vote
1answer
26 views

Convergence of $\chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}}$ for $u_n \to u$?

Let $u_n \to u$ in $L^2(\Omega)$ and let $I$ be an bounded interval. Does it follow that $$\chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}}$$ at least for a subsequence of $u_{n_j}$ ...
1
vote
1answer
19 views

convergence in measure implies the composition of the sequence of functions and a continuous function also converges in measure

Let $D$ be a measureble set in $\mathbb{R}^n$. Suppose $\mu(D)<\infty$. Let $\phi: D\times \mathbb{R}\to \mathbb{R}$ be a continuous function such that for almost every $x\in D$, ...
1
vote
0answers
35 views

Extension of a Pseudodifferential Operator

Let $M$ be a smooth manifold with countable atlas, and define the distributions $\mathscr{D}'(M)$ as the dual space to the smooth densities with compact support, and $\mathcal{E}'(M)$ as the dual ...
0
votes
3answers
40 views

Does $L^2$ strong convergence and bounded $L^\infty$ imply convergence in $L^\infty$ along subsequences?

Take a sequence of functions $f_n \in L^2([0,1])$ such that $f_n \rightarrow f$ in $L^2([0,1])$ (convergence in $L^2$ norm) and $|f_n|_{L^\infty},|f|_{L^\infty} <R$ (uniformly bounded in ...
1
vote
1answer
39 views

Counterexamples for L1-convergence not implying L2-convergence and vice versa

I am trying to find counterexamples for the following statements. Let $\{f_n\}$ be a sequence in $L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$, and let $f$ also be in $L^1(\mathbb{R}^d) \cap ...
1
vote
1answer
31 views

Is strong convergence?

Let $F_n(f) = f(\frac{1}{n}) - f(-\frac{1}{n}) \in \big ( C^{(1)} [-1, 1] \big ) ^*$. For every $f \in C^{(1)} [-1, 1]$ we have pointwise limit: $$ \lim_{n \to \infty} F_n (f) = \lim_{n \to \infty} ...
2
votes
1answer
23 views

Denseness: $\overline{\mathcal{l}^2_0}=\mathcal{l}^2$

How to prove that the finite sequences are indeed dense within the space?
2
votes
2answers
66 views

What is $L^p$-convergence useful for?

Why do people care about $L^p$-convergence $f_n \rightarrow f$? Are there any interesting application of $L^p$-convergence? For example, if $p=\infty$, then the limit $f$ of the sequence $f_n$ of ...
1
vote
1answer
40 views

Completeness: $\mathcal{l}^2(S)$

Surely, for countable index sets this is just the diagonal trick: $\#S<\infty$ However for arbitrary index sets how do I prove that the limit will actually have only countable non vanishing ...
2
votes
1answer
92 views

Pointwise a.e. convergence and weak convergence in Lp

I'm trying to prove the following theorem: Let $\{f_n\}\subset L^p(\Omega)$, $f_n \rightharpoonup f$ in $L^p(\Omega)$ ($\Omega\subset\mathbb{R}^n$ is open and bounded, $1\leq p \leq \infty$) and $f_n ...
0
votes
0answers
37 views

Convergent sequences in normed vector spaces

Consider a normed vector space $X$. I have a couple of questions regarding convergent sequences and subspaces of $X$(topological subspaces, not necessarily linear subspaces). Let $W$ be a topological ...
2
votes
1answer
47 views

How to show that the sequence $(x^{(n)})$ weakly convergent in $l_p$, $1\le p\lt \infty$

How to show that the sequence $(x^{(n)})$ weakly convergent in $l_p$, $1\le p\lt \infty$. where $(x^{(n)})=(\underbrace{0,0,..0}_{n-1},1/n,1/(n+1),...,1/(2n),0,0,...)$ for $n\in\mathbb{N}$
1
vote
1answer
18 views

Proof of Strong Operator Convergence Theorem

Recall the theorem : $T_n \in B(X,Y)$ where $X,\ Y$ are Banachs, is strongly convergent iff (a) $ \parallel T_n \parallel $ is bounded (b) $T_nx$ is Cauchy where $x$ is in total subset ...
1
vote
1answer
31 views

How to give a criterion for strong convergence of in $L_p[0,1]$ for this example

let $x_n=\alpha_n e^{-nt}$ for $n\in \mathbb{N}$ and $1<p<\infty$. How to give a criterion for strong convergence of in $L_p[0,1]$ for this example: $x_n\rightarrow 0$ (strong convergence). ...
4
votes
1answer
57 views

Dense subspace of the space of measures on the torus $\mathbb{T}$.

Every measure $\mu$ on the torus $\mathbb{T}$ is the weak-$\ast$ limit of a sequence of absolutely continuous measures on $\mathbb{T}$ with $C^{\infty}$ densities. I'd like to see a proof of this ...
2
votes
1answer
74 views

When does $ f'_{n}(x) \to g(x) =1$ imply $f'(x) =1 $

I considered the following: $f_{n}(x) \in C^1(0,1)$ (class of continuously differentiable functions) and $f_{n} \to f:(0,1) \to \bf{R} $ with $f'_{n} \to g =1$. Does this imply that $f \in C^1(0,1)$ ...
1
vote
0answers
18 views

Limit of function of an operator

Let $A_n$ be a sequence of bounded, self-adjoint operators on Hilbert space $\mathcal{H}$. Let us assume that for some vector $\psi\in\mathcal{H}$, $$\lim_{n\rightarrow\infty}A_n\psi = \alpha ...
0
votes
1answer
40 views

Are these $f_n$ equicontinuous?

Let $f_n$ be a sequence of real-valued functions defined on $\mathbb{R}$ satisfying $f_n \to f$ uniformly in the compact subsets of $\mathbb{R}$ $f_n^{-1}$ is bi-Lipschitz $1 \leq (f_n^{-1})'(x) ...
0
votes
1answer
34 views

Converse of existing question on L^p convergence

My question is about this: Convergence in $L^p$ of $f_n$ implies convergence in $L^1$ of $|f_n|^p$ and $f_n^p$ It was shown that the author's question was indeed true by the use of MVT. Is the ...
0
votes
0answers
26 views

Prove operator convergence

I have to proove that given $X$ a normed space and $Y$ a Banach space, if the sequence of bounden linear operators from $X$ to $Y$ $\{A_n\} \rightarrow A$ and the sequence $\{x_n\} \rightarrow x$ then ...
3
votes
1answer
98 views

Convergence of characteristic functions on hypercube

I have a question regarding the following partition of a hypercube $H_{R}(x)$ centered at $x$ with sides of length $R$ in $\mathbb{R}^{n}$: Consider this hypercube $O = H_{R}(x) = ...
2
votes
1answer
46 views

Convergence of function in $L^1$ space

Let $\Omega \subset \mathbb{R}^{n}$ be bounded and open. Assume that $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a Caratheodory function which means $a(x,.,.)$ is ...