1
vote
1answer
38 views

Convergence in $l^p$ space?

We know that $\lim_{k \to \infty} a_{n}^{k} = b_n$. All $(a_{n}^{k}) \in l^p$ space. How to show that $(\lim_{k \to \infty} a_{1}^{k},\lim_{k \to \infty} a_{2}^{k},...,\lim_{k \to \infty} ...
0
votes
1answer
65 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
26 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
29 views

Show that $S = \{f \in L^1(\mathbb{R}) \mid \int_{\mathbb{R}}f dm = 0\}$ is closed in $L^1(\mathbb{R})$.

This should be a relatively easy question, but I can't seem to figure it out. I want to show that $S = \{f \in L^1(\mathbb{R}) \mid \int_{\mathbb{R}}f dm = 0\}$ is closed in $L^1(\mathbb{R})$. As ...
2
votes
2answers
41 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 ...
2
votes
2answers
59 views

Properties of a sequence of iid rv's

I cannot do part a), and Im fairly sure that b) and c) will follow from it. If possible could I please have a solution to part a) and hints if you feel necessary to parts b) and c).
0
votes
2answers
26 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 $u_n(x) \to u(x)$ ...
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 ...
2
votes
2answers
67 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 ...
3
votes
1answer
70 views

can $L^p$ norm convergence and pointwise monotonic imply pointwise convergence?

Let $(f_n)_{n=1}^\infty$ be a sequence of measurable function such that $\lim_{n\to\infty}||f_n-f||_p=0$. If for any $x\in \Omega$, $\{f_{n}(x)\}_{n=1}^\infty$ is a monotonic sequence, can we deduce ...
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
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). ...
0
votes
1answer
30 views

If $u_n$ is bounded and pointwise convergent, then $u_n$ convges in $W_{2,p}$.

I'm reading this paper about solving semilinear elliptic pde's through iterated approximations. The line i'm trying to understand is "Then, since $u_k = Tu_{k-1}$ and since $\{u_k\}$ is a bounded, ...
0
votes
1answer
35 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 ...
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 ...
2
votes
1answer
78 views

Two questions on Banach-valued spaces of integrable functions

Let $V$ be a Banach space with dual $V'$ and suppose that $V$ is included into the Hilbert space $H$ so that the inclusion is continuous and dense. Then after identification $H = H'%$ we have $V ...
3
votes
0answers
32 views

Difference between almost everywhere convergence of whole Fourier series and a subseries of $L^2$ functions

Let $f \in L^2(\mathbb{R})$ and $F_{N}f$ denote the $N$-th partial sum of its Fourier series. Then $||F_{N}f -f||_{L^{2}} \rightarrow 0$ as $N \rightarrow \infty$. But this implies there exists a ...
1
vote
0answers
32 views

A useful criterion in dispersive PDE.

I would like to prove the following theorem: Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let $f:I\mapsto Y$ ...
2
votes
1answer
152 views

Convergent sequence in Lp has a subsequence bounded by another Lp function

For $E$ a measurable set and $1\leq p<\infty $, assume $f_n\to f$ in $L^p(E)$. Show that there is a subsequence $\{f_{n_k}\}$ and a function $g\in L^p(E)$ such that $|f_{n_k}|\leq g$ almost ...
2
votes
1answer
60 views

Integral convergence involving Lp and Sobolev spaces

Quick question, any contribution or hint would be appreciated: How does it follow that: $$\lim\limits_{k \rightarrow \infty}\int_{\Omega}a(u_{k})\frac{\partial u_{k}}{\partial x}\frac{\partial ...
0
votes
1answer
43 views

Convergence of a sequence in two different $L_p$ spaces

If I have a sequence $\{f_n\}_{n\in \mathbb{N}} \subset L_p\cap L_q$ that is Cauchy under both norms, I am wondering if $f_n \to f$ in $L_p$ implies that $f_n\to f$ in $L_q$. I have been working on ...
2
votes
1answer
78 views

If $u_n \rightharpoonup u$ in $L^2(\Omega)$ and $u_n^2 \rightharpoonup v$ in $L^1(\Omega)$ then is $v=u^2$?

If $u_n \rightharpoonup u$ in $L^2(\Omega)$ and $u_n^2 \rightharpoonup v$ in $L^1(\Omega)$ then is $v=u^2$? We assume that the domain $\Omega$ is bounded. If not is there any way to ensure this?
1
vote
1answer
49 views

Implications of Weak convergence in Sobolev Spaces

A quick question regarding weak convergence in Sobolev Spaces. If $u_{k} \rightharpoonup u$ in $W^{1,p}(\Omega)$ for bounded $\Omega$ then can we show that $\nabla u_{k} \rightharpoonup \nabla u$ in ...
1
vote
0answers
42 views

Convergence of $L^1$ functions

Given that $\Omega$ is bounded and $a_{ij}(u_{k}) \rightarrow a_{ij}(u)$ in $L^{1}(\Omega)$, $a_{i0}(u_{k}) \rightarrow a_{i0}(u)$ in $L^{1}(\Omega)$, $\frac{\partial u_{k}}{\partial x_{j}} ...
1
vote
1answer
33 views

Relation between convergences in $L^{p}$ for probability spaces.

I have read that for a probability space $(\Omega,\Sigma,P)$ it is true that $f \in L^{p}(\Omega,\Sigma,P)$ implies $f \in L^{q}(\Omega,\Sigma,P)$ if $p>q$, and hence $L^{2} \subset L^{1}$. I'm ...
2
votes
1answer
45 views

Integral convergence and weak convergence

Given that $\Omega \subset \mathbb{R}^{n}$ is a connected bounded Lipshitz domain and $u_{k} \rightharpoonup u$ in $W^{1,p}(\Omega)$. We denote $\Gamma$ as the boundary of the domain. We have the ...
2
votes
1answer
85 views

How Lp spaces relate regarding convergence

I know that for a bounded $\Gamma$ it follows that $L^{q}(\Gamma) \subset L^{p}(\Gamma)$ if $q > p$. I have a few questions regarding how $L^{p}$ spaces relate with regard to convergence. Consider ...
-1
votes
1answer
35 views

The convergence in $L^{p_1}$ and $L^{p_2}$

Suppose $f_k$ is a sequence of $M-$measurable function. Let $p_1$ and $p_2\in[1,\infty)$,and $f_k\in L^{p_1}\cap L^{p_2}$.Also suppose $\exists g\in L^{p_1}$ and $h\in L^{p_2}$ s.t. $f_k\rightarrow g$ ...
1
vote
0answers
22 views

Show $\lim_{k \rightarrow \infty} (f_{n_k}, g) = (f,g)$ for a subsequence $f_{n_k}$ of $f_n$ and $g$ in $L^2([0,1])$

I would like to show that for any sequence $f_n(x) \in L^2([0,1])$ with $||f_n||_2=1$ for all $n$, there exists an $f\in L^2([0,1])$ and a subsequence $(f_{n_k})$ such that for every $g\in L^2([0,1])$ ...
5
votes
1answer
125 views

Subsequence convergence in $L^p$

I recall a fact that for functions $f_1,f_2,\ldots\in L^1$ such that $\|f_n-f\|_1\rightarrow 0$ as $n\rightarrow\infty$, there exists a subsequence $f_{n_i}$ that converges to $f$ almost everywhere. ...
3
votes
1answer
79 views

$L^2$ norm of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ The function $$f_r(\theta)=\sum_{n=1}^\infty r^na_ne^{in\theta}$$ is a harmonic ...
0
votes
1answer
71 views

Constructing Sequences in Lp

Consider Banach Space $L^{p}(U)$ where $U$ is bounded open subset in $\mathbb{R}^{n}$. Take a bounded sequence $\{u_{m}\}_{m}^{\infty}$ in $L^{p}(U)$. Consider a subsequence ...
3
votes
1answer
80 views

Bounding for convolution convergence

Suppose $f\in L^p(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_t(x)=\dfrac{1}{t}K\left(\dfrac{x}{t}\right)$$ I'm trying to prove that $\lim_{t\rightarrow ...
0
votes
1answer
116 views

Uniform convergence with Lp functions

I have a convergence question: Say we have a sequence of functions $\{u_{m}\}_{m=1}^{\infty}$ where $\{u_{m}\}_{1}^{\infty} \subset L^{p}(U)$ and where $U$ is bounded. Consider $u^{\epsilon} := ...
0
votes
1answer
42 views

Implication of $L^p$ convergence

Take $U$ as open subset of $\mathbb{R}^{n}$. If $u_{m} \rightarrow u$ in $L^{p}(U)$ then does it follow that $||u_{m}||_{L^{p}(U)} \rightarrow ||u||_{L^{p}(U)}$?
2
votes
1answer
697 views

Does $L^p$-convergence imply pointwise convergence for $C_0^\infty$ functions?

It is stated in my professor's notes that, given a sequence $\{f_j\}$ of $C_0^\infty(\Omega)$ functions (infinitely differentiable with compact support), and a function $g\in C_0^\infty(\Omega)$, all ...
2
votes
1answer
54 views

Lebesgue's convergence for $H(u_n)\nabla u_n$ where $H$ is not everywhere defined

Consider the Heaviside function that is undefined in zero, i.e. $$H(t)=\begin{cases} 1&t>0 \\ 0&t< 0\end{cases}$$ Now consider a sequence of $H^1(\Omega)$-functions $u_n\to u$ in the ...
2
votes
1answer
73 views

$\int_{B} f_n \phi \rightarrow 0$ if the Weak-$L^p$ norm of $f$ tends to zero?

Let $f_n \in L^p(B)$ be a sequence where $B$ is some ball in $\mathbb{R}^n$. Assume that $\|f_n\|_{L^p(B)} \rightarrow 0$ when $n\rightarrow \infty$, then by some $\phi \in C^\infty_0(B)$ applying ...
2
votes
1answer
55 views

Convergence of $\varphi_n(x):=\frac{\varphi(nx)}{n}$ in Schwartz space

I want to find all $\varphi\in\mathcal S(\mathbb R)$ for which the sequence $\varphi_n(x):=\frac{\varphi(nx)}{n}$ converges in $\mathcal S(\mathbb R)$. The first step, I have already managed to do by ...
0
votes
2answers
155 views

Determine if the following sequence converge in the quadratic mean

"For integer $n$ let $f_n(x) = \dfrac{1}{\sqrt{1 + nx^2}}$ say if the sequence $f_n$ converges in quadratic mean." This is what I have concluded so far: $$\lim\limits_{n\rightarrow \infty}f_{n}(0) ...
2
votes
2answers
73 views

Convergence in $L_1$ and Convergence of the Integrals

Am I right with the following argument? (I am a bit confused by all those types of convergence.) Let $f, f_n \in L_1(a,b)$ with $f_n$ converging to $f$ in $L_1$, meaning $$\lVert f_n-f \rVert_1 = ...
1
vote
1answer
79 views

Need help in showing that $F(x)/x^{1/q}$ goes to $0$ as $x$ goes to $0$ and $\infty$.

$1<p<\infty$, $f\in L^{p}(0,\infty)$, $p^{-1}+q^{-1}=1$, define $$F(x)=\int_{0}^{x}f(t)dt,$$ then I need to show that $\frac{F(x)}{x^{\frac{1}{q}}}\rightarrow 0$ as $x\rightarrow 0$ and ...
1
vote
1answer
59 views

Continuity of conditional expectation in $L_p$

I'm looking at a probability space $(\Omega,\mathcal{F},P)$. Let $1\leq p<\infty$, and let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. I'm then asked to show that, for $X\in L_p(P)$, ...
1
vote
1answer
46 views

$L^p$ convergence proof check

I don't have much experience with measure theory, so I want to make sure that I'm not making any bad mistakes. I also want to be sure that the theorem is true so I can use it. Theorem: Let $\{u_i\}$ ...
1
vote
1answer
26 views

Remainder of a series converges uniformly?

Let $B \subset \Bbb R^{\Bbb N}$ and $p \geq 1$. Suppose $$ \sup_{u\in B}\sum_{n=0}^\infty |u_n|^p \leq 1,\qquad \sup_{u\in B}\sum_{n=0}^\infty |u_{n+1}-u_n|^p\leq 1 $$ Is it true that $$ \sup_{u\in ...
2
votes
2answers
159 views

Weak limit of disjoint normalized sequence in $L^p$

I want to prove that the weak limit of a disjoint normalized (pairwise disjoint supports, elements of norm $1$) sequence $(f_n)$ in $L^p$ for $p >1$ is zero ? I started with the measure of ...
1
vote
0answers
501 views

Rademacher function and weak convergence

The function $r_{n}:[0,1]\rightarrow \{-1,1\}$ be defined by $r_{n}(t)=\operatorname{sgn}(\sin(2^{n}\pi t))$ is known as the $n$-th Rademacher function.a) Show that $r_{n}\xrightarrow{w}0$ in ...
2
votes
1answer
136 views

Uniform convergence in $L^p$-spaces

Let $f\in L^p(0,\infty)$, $p>1$. Show that $\int_0^\infty f(x)\frac{\sin xy}{x} dx$ converges uniformly in $y$ in every finite interval. Show also that $|g(t+y)-g(y)|\leqslant M|t|^{\frac{1}{p}}$. ...
1
vote
2answers
39 views

$\ell_{p}$ space closed to addition

I'm trying to show that $\ell_{p}$ is a vector space for any $1\leqslant p<\infty$ . So given two infinite series $\left(x_{n}\right)_{n=1}^{\infty}$ and $\left(y_{n}\right)_{n=1}^{\infty}$ ...