-2
votes
1answer
108 views

Continuity of $|.|$ in $W^{1,p}_0$

please i dont understand this proof We suppose that $u\rightarrow u$ on $W^{1,p}_0$ i dont understand why we must use the weak compactness and the uniform convexity of $W^{1,p}_0$ ? Thank you
1
vote
1answer
47 views

Prove that a relatively compact subset of $L^p$ is bounded.

Let $p\in [1,\infty)$, $A\subset L^p(\mathbb R^m)$ relatively compact and $\lambda^m$ be the Lebesgue measure on $\mathbb R^m$. Prove: a) $A$ is bounded. b) $\lim_{y \to 0}\sup_{f \in A} ...
2
votes
1answer
35 views

$\|f'(x)\|_{L^p} \le C \|f(x)\|_{L^p}^{1/2} \|f''(x)\|_{L^p}^{1/2}$ for smooth $f$ with compact support

I'm trying to prove the following Let $f: \mathbb{R} \to \mathbb{R}$ be a smooth function supported on $[a, b]$ where $-\infty < a < b < \infty$. $2 \le p < \infty$. Then $$ ...
0
votes
1answer
53 views

Inequality important in $L^p$ space

If$\,\,$ $0<p<\infty$, put$\,\,$ $\gamma_{p}=\max(1,2^{p-1})$, and show that $$|\alpha-\beta|^p \leq \gamma_{p}(|\alpha|^p + |\beta|^p)$$ for arbitrary complex numbers $\alpha$ and $\beta$. ...
1
vote
1answer
17 views

Relation between $L^2(\mathbb{R}_+)$ and $L^1_w(\mathbb{R}_+)$

As an exercice, I'm looking to find an inclusion or equality relationship between $L^2(\mathbb{R}_+)$ and $L^1_w(\mathbb{R}_+)$ when $w: x \to x^{-1/2}$. Actually, I think that we have the inclusion ...
1
vote
0answers
46 views

Norm of multiplication operator

I have that $(X,\Omega,\mu)$ is a sigma finite space, and I have that $g$ is a measurable function. Assume that $fg\in L^p$ for all $1\leq p\leq \infty$. I want to show that $g\in L^\infty$. My idea ...
10
votes
4answers
125 views

$l^\infty(I)$ and $l^\infty(J)$ isometrically isomorphic with $|I| \not= |J|.$

Is it possible for $l^\infty (I)$ and $l^{\infty} (J)$ to be isometrically isomorphic with the cardinality of $I$ not equal to the cardinality of $J$? I'm able to show that if $1\le p < \infty,$ ...
1
vote
1answer
144 views

Limit of $\|x\|_p$ as $p\rightarrow\infty$ [duplicate]

I am not sure how to start this hw problem. Here it is: Let $x$ be a given vector in $\ell^p$ with $1\le p\lt\infty$. Show that $x \in \ell^\infty$ and $$\lim_{p \to \infty} ||x||_p = ...
0
votes
1answer
76 views

Bounded subsequence in Sobolev Space

The following is an exercise. Let $I=(0,1)$ and let $(u_n)$ be a bounded sequence in Sobolev space $W^{1,p}$, First question: does "bounded" here means that (for a suitable $M$) $$ \| u_n \|_p ...
2
votes
1answer
39 views

property of a given subset $ A $ of $\mathit{l}^2 $

I was given this exercise Let $\lbrace e_n \rbrace $ be the canonical basis of $\mathit {l}^2 $ (the space of square sommable series with the usual norm) and set $$ A:= \lbrace \sum_{n \ in ...
0
votes
1answer
36 views

Does $u_n(x) \subset L^p(-1,1)$ converge strongly-weak-weak* to something?

Let $\lbrace u_n(x) := [ \sin(nx)]^+ \rbrace.$ Does this sequence converge in some sense to something in $L^{p}(-1,1)$? My attempt: for $1<p< +\infty$ $u_n$ is limited so by Banach-Alaoglu + ...
2
votes
1answer
77 views

Show that $f(x)= \frac{x^{-1/2}}{1+ | \log x |} $ is only $L_{p} ((0,\infty])$ for p=2

Show that $ {\rm f}\left(x\right)= {x^{-1/2} \over 1 + \left\vert\,\log\left(x\right)\,\right\vert}$ is only $L_{p}\left(\vphantom{\large A}\left(0,\infty\right]\right)$ for $p = 2$. Where $x > ...
2
votes
2answers
87 views

Show that the operator is bounded in $L_p$

Consider the operator $C$, acting on functions $f$ on the unit circle $S^1 = \left\{ z \in \mathbb C \mid |z| = 1 \right\}$ by the rule $$ (Cf)(z) = \frac{1}{2\pi i} ...
2
votes
3answers
454 views

Countably generated $\sigma$-algebra implies separability of $L^p$ spaces

Let $\Sigma = \sigma(\mathcal C)$ be the $\sigma$-algebra generated by the countable collection of sets $\mathcal C \subset \mathcal{P}(X)$. How can I prove that if $\mu$ is a $\sigma$-finite measure ...
1
vote
0answers
54 views

Self-adjointness of differentiation operator

Let's say $\mathcal{H}=L^2([0,1])$ and $p$ is the operator $-i\frac{d}{dx}$ defined on $\mathcal{D}(p)=\{f\in L^2([0,1])\ |\ f'\in L^2, f(1)=e^{i\theta}f(0)\}$. I have to prove that $p$ is ...
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 ...
1
vote
1answer
72 views

Star graph embeddings

This is an homework question which I'm struggling with: Let $S = (V, E, w)$ a star graph, meaning, $S$ is a tree that all it's vertices are leafs except one. I need to : show that every weighted ...
0
votes
1answer
123 views

Isometric embedding of an $n$-point equilateral space

I'm stumped on these questions, and would appreciate a solution: I need to find an isometric embedding of the n-point equilateral space in $l_{p}$. And if $n=2^{d}$, an isometric embedding of the ...
1
vote
1answer
81 views

Norm operator and compactness

For the operator $U\colon \ell_{p}\to\ell_{p},\;\left( 1\leqslant p<\infty \right) :$ \begin{equation*} Ux=U\left( x_{1},x_{2},\dots \right) =\left( 0,x_{1},\frac{x_{2}}{2},\frac{% ...
2
votes
1answer
94 views

A question on Orlicz norms

I was reading Empirical Processes from "Weak Convergence and Empirical processes" by Van Der Waart and Jon Wellner. There I was studying Orlicz norms of random variables which are defined as follows: ...
2
votes
1answer
146 views

Extension of a Bounded Operator on $L^p$ to $L^r$

Let $1<p\leq \infty$, and let $p^{-1} + q^{-1} = 1$. Let $T$ be a bounded operator on $L^p$ such that $\int(Tf)g = \int f(Tg)$ for all $f,g \in L^p \cap L^q.$ Show that $T$ uniquely extends to a ...
1
vote
1answer
24 views

If a function is $L^p$ small, is its expectation with respect to a $\sigma$-algebra $L^p$ small?

This came up in my homework, but isn't strictly my homework. I've just gotten very curious, and I keep going in circles trying to prove it. Consider a probability measure space $(X,\Sigma,\mu)$ and ...
2
votes
2answers
167 views

Compact inclusion in $L^p$

Is it true that there is a compact inclusion from $L^p$ to $L^q$ whith $q<p$? What is the counterexample if what I said is wrong? Thank you.
1
vote
1answer
48 views

Prove that $ A\subset \ell_1 $ is compact iff $A$ satisfies the following property

$A$ is compact iff $ A $ is bounded and, given $\epsilon > 0$, there exists $ n_0$ such that $ \sum_ {k=n}^\infty |x_k|\le\epsilon $ for all $n \geq n_0 $ and for all $ x\in A $. To prove ...
3
votes
1answer
136 views

$W^{1,p}$ compact in $L^\infty$?

Is $W^{1,p}(0,1)$ compactly contained in $L^\infty(0,1)$? Can I use this to show that I can select a sequence $(u_{n_k})$ from every bounded sequence $(u_n)$ in $W^{1,p}(0,1)$ such that $\lVert ...
1
vote
1answer
75 views

Inclusion in $L_p$ space

I have been wondering how to prove the following statement, and would greatly appreciate your help: If $f$ is a bounded function on $E$ that belongs to $L_{p_1}(E)$, then it belongs to $L_{p_2}(E)$ ...
0
votes
1answer
192 views

Essential Supremum

For $f\in L^\infty[a,b]$, show that $$\|f\|_\infty = \min \big\{M : m\{x \in [a, b] : |f(x)|>M\} = 0\big\}\;,$$ and if, furthermore, $f$ is continuous on $[a, b]$, that $\|f\|_\infty = ...
3
votes
1answer
223 views

Pointwise a.e. convergence implies strong convergence?

Let $ 1 \leq p_1 < p_2 < \infty$, and suppose that $f_n$ is a sequence of functions in $L^{p_1}[a,b]$ such that $f_n \to f$ pointwise a.e. on $[a,b]$. Suppose in addition that $ ||f_n||_{p_2} ...
3
votes
2answers
183 views

Weak convergence of a sequence of characteristic functions

I am trying to produce a sequence of sets $A_n \subseteq [0,1] $ such that their characteristic functions $\chi_{A_n}$ converge weakly in $L^2[0,1]$ to $\frac{1}{2}\chi_{[0,1]}$. The sequence of ...
1
vote
1answer
122 views

Is $L^{p_2}$ complete under the $L^{p_1}$ norm?

Given $p_1,p_2$ such that $1 \leqslant p_1 < p_2 < \infty$ and a measurable set $E$ of finite measure, I'm trying to determine whether the space $L^{p_2}(E) $, which I know to be contained in ...
1
vote
1answer
155 views

Closed graph theorem to prove that a sequence is in $\ell^q$

Let $\{a_n\}$ be a sequence of complex numbers such that $\sum \limits _{n=1}^{\infty} a_nb_n$ converges for every complex sequence $b_n \in \ell^p$. Show that $\{a_n\} \in \ell^q$ where ...
8
votes
1answer
488 views

How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function

We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$. ...
4
votes
1answer
245 views

$L^2$ norm inequality

I need some help with this homework question. I was asked to provide an example of a $n$-dimensional subspace $W$ of $L^2[0,1]$ such that all functions in that subspace with $L^2$ norm equal to $1$ ...
1
vote
1answer
107 views

Show a function is in $L_\infty$

Let's assume we're working on a measure space $(X,\Sigma,\mu)$, where $\mu$ is a $\sigma$-finite measure. Suppose that $g$ is a measurable function such that $\forall f\in L^2$, $||fg||_2\leq ...
1
vote
1answer
173 views

prove a subset of squence space $l^p$ closed in strong topology

Let $l^p$ be the space of $p$-summable sequences. von Neumann constructed a subset of $l^p$ space $$S=\{X_{mn}: m,n≥1\}$$ where $X_{mn}\in l^p$ are defined by $X_{mn}(m)=1, X_{mn}(n)=m$ and ...
1
vote
1answer
67 views

How to bound $L^p$ norm of a product

I am trying to show that if I can approximate two characteristic functions $\chi_A,\chi_B$ by simple functions involving only a particular set of characteristic functions, then I can approximate ...
1
vote
1answer
753 views

$\ell_p$ is Hilbert space if and only if $p=2$

Can anybody please help me to prove this.. Let p greater than or equal to 1,show that the space of all p-summable sequences is an inner product space if and only if p=2
1
vote
1answer
128 views

The distance between a point and a set

It is a problem in my homework. Let $$ X = \{x \in C[0,1] : x(0) = 0\} $$ with norm $\Vert\cdot\Vert_\infty$. Denote $$ M =\left\{ x \in X : \int\limits_0^1 x(t)=0\right\} $$ If $\Vert ...