For questions about $L^p$ spaces, that is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence class of measurable functions with $p$-th power of the absolute value integrable. Question can be about properties of elements of these spaces, or when the ambient space on a ...

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0answers
25 views

If $p$ is a distribution, what is the meaning of the claim $\nabla p\in L^p(\Omega)^d$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $q\ge 1$ I've seen the following Lemma (without a proof) in a paper and don't understand how I ...
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0answers
25 views

Bounding $L_p$ norms on a convergent $L_1$ sequence

I've encountered a prelim problem on $L_p$ spaces that I'm pretty stuck on. Suppose $1 < p < \infty$ and $f_n \in L_1([0,1]) \cap L_p([0,1])$, with $||f_n||_p$ bounded above by some constant $M$...
0
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1answer
34 views

Convergence in $L^2(\Bbb R)$ implies convergence of the norms [on hold]

If $||f_n-f||_{L^2(\mathbb{R})}\to 0$ is it always true that $||f||_{L^2(\mathbb{R})}=\lim_{n\to\infty}||f_n||_{L^2(\mathbb{R})}$?
2
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0answers
17 views

Poincaré's inequality proof for $u \in W_0^{1,p}(\Omega)$.

I am trying to prove Poincare's inequality for $u \in W_0^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is an open bounded set and $1 \leq p < \infty$. This is Poincare's inequality: $||...
0
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1answer
49 views

Counter Example for Limit of $\|f\|_p$ in infinity convergence, When Measure space is not finite [on hold]

I found a proof for this fact that limit of $\|f\|_p$ when $p \to \infty $ is $\|f\|_{\infty}$ in here when $f:X \to R $ and $X \in L^p$ measure space is finite. But I need a counter example for ...
0
votes
2answers
52 views

Orthogonal of an Hilbert subspace and density

If $V$ is a subspace of an Hilbert space $H$, I know that the orthogonal of $V$, $V$$^o$, is ($V$closed)$^o$, even if $V$ is not closed. Does this mean that $V$ is always dense in $V$$^o$? Thanks!...
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0answers
26 views

What does the notation $L^{\infty}(R^+;L_+^1(-\infty ,b))$ stand for?

If we write $p\in L^{\infty}(R^+)$ that means the set of all bounded function over $R^+$, but i have no idea what is the meaning of this notation \newline $p\in L^{\infty}(R^+;L_+^1(-\infty ,b)), b&...
1
vote
1answer
29 views

Integration of periodic function $f \in L^1([0, 2\pi])$

While studying trigonometric series and $L^p$ spaces I was wondering the following: Let's say we have a $2\pi$-periodic function $f \in L^1([0, 2\pi])$ satisfying $\int_{0}^{2\pi}f(x) \, dx = 0$. Is ...
0
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1answer
42 views

Convergence in $L^p$ and convergence almost everywhere

Why $f_n$ converges to $f$ in $L^p$ space implies that exists subsequence of $f_n$ converging to $f$ almost everywhere?
3
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2answers
59 views

Constructing an $L^2$ space on the unit ring $\mathcal{S^1}$

Revised Question: Starting with $L^2[0,2\pi]$, does the canonical map $$[0,2\pi)\ni\theta\mapsto e^{i\theta}\in\mathcal{S^1}$$(with functions going across in the obvious way) turn $L^2[\mathcal{S^1}]$...
2
votes
1answer
44 views

Characterization of measures such that $\frac{1}{x} \in L^1(H)$

Let $H$ be a finite measure on $(0,1)$. What conditions must $H$ fulfill, such that \begin{equation*} \frac{1}{x} \in L^1(H),\ \ \ \frac{1}{1 - x} \in L^1(H) \end{equation*} I'm trying to characterize ...
4
votes
1answer
67 views

Explanation of spaces of functions in PDE

Let's consider following equation: The problem $$ \begin{cases} -\operatorname{div}\left( p\left(x\right) \nabla{u} \right) + q(x)u = f \quad\text{... on } \Omega \\ u = h(x) \quad\text{... on } \...
2
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0answers
26 views

Show linearity of a functional if it holds for nonnegatives

Consider a functional $G^+:L_p \to \mathbb{R}$. Here $L_p = L_p (X,\textbf{X}, \mu)$ is the collection of all integrable fns (f s.t. $\int \vert f \vert^p d \mu < \infty$ on the measure space $(X,\...
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0answers
24 views

$L^p$ convergence of smooth compactly supported functions

I already checked the similar question here, but want to check slightly different argument. Given $f(x)\in L^p(\mathbb{R}^n)$, can I find $f_n(x) \in C^\infty_0$ s.t. $f_n \rightarrow f$ almost ...
1
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1answer
62 views

Why does Rudin say that $f$ is in $L^\infty$?

The setting is that we have $L^\infty([0,1],m)$, $m$ Lebesgue measure, and we have shown the Gel'fand transform is an isometry onto $\mathcal{C}(\Delta)$, $\Delta$ being the space of maximal ideals / ...
0
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0answers
31 views

Does weak convergence and convergence ae. imply the following?

Let $1\leq p<\infty$ and $f_k,f\in L^p(0,1)$ such that $f_k\rightarrow f$ a.e.. Let $q$ be the conjugate exponent of $p$ and assume that $\int f_k g \rightarrow \int fg$ for all $g\in L^q(0,1)$. ...
2
votes
1answer
68 views

Proof verification: Something similar to Riesz-Fischer Theorem

Question: Suppose $\{f_n\}$ converges to $f$ in $L^p(\mathbb{R})$, $1\leq p<\infty$. Prove that there is a subsequence $\{f_{n_k}\}$ and $g\in L^p(\mathbb{R})$ so that $f_{n_k}\to f$ a.e. and $|f_{...
5
votes
1answer
70 views

Why are there no finitely additive measures on $\ell_\infty$ for which the measure of every ball is positive and finite?

As the question title suggests, why are there no finitely additive measures on $\ell_\infty$ for which the measure of every ball is positive and finite? Here, we do not assume that the measure is ...
2
votes
1answer
29 views

Question on Inequality from Bartle's Elements of Integration: Riesz Fischer Theorem

I am puzzled how did Bartle get $$|g_k|\leq\sum_{j=k}^\infty |g_{j+1}-g_j|$$ (second last line)? I tried using Triangle Inequality and ended up with one extra term: $$\begin{align*} |g_k|&=|g_k-...
4
votes
2answers
82 views

Supremum of absolute value of the Fourier transform equals $1$, and it is attained exactly at $0$

Suppose that $f \in L^1(\mathbb{R}^n)$, $f \ge 0$, $\|f\|_{L^1} = 1$. How do I see that $\sup_{\xi\in\mathbb{R}^n} |\mathcal{F}(f)(\xi)| = 1$, and it is attained exactly at $0$?
0
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1answer
31 views

Writing an operator $T$ defined by $(T f)(t) = \int_{-\pi}^\pi h(t − s)f(s)ds$ as $\sum_{n \in \mathbb Z} \mu_n \langle f, \varphi_n\rangle \varphi_n$

Let $h$ be a continuous function with period $2\pi$. Define $T : L_2[−\pi, \pi] \to L_2[−\pi, \pi]$ by $(T f)(t) = \int \limits _{-\pi}^\pi h(t − s)f(s)ds$. Let $\{\varphi_n(t) =\frac{1}{\sqrt{2\pi}} ...
2
votes
1answer
48 views

Upper and lower bound on $L^1$ norm purely in terms of measure

Suppose $f$ is a measurable almost everywhere finite function on $\mathbb{R}^d$, and let$$E_n = \{x : 2^n \le |f(x)| < 2^{n + 1}\}, \quad n \in \mathbb{Z}.$$What is a non-trivial upper and lower ...
0
votes
1answer
21 views

$L^1$ approximation by a slightly “displaced” copy

Let $f:\Bbb R\to \Bbb R$ be an $L^1$ function and $f_\epsilon(x):=f(x+\epsilon)$, $\mu$ is the Lebesgue measure, prove that $$\lim_{\epsilon\to 0}\int|f_\epsilon-f|\mathrm d\mu=0.$$ I tried to ...
0
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1answer
32 views

If $f_n$ converges to $f$ in $L^1(\mathbb{R})$ and $f_n$ converges to $g$, what relation exists between $f$ and $g$?

Take $(f_n)$ to be a sequence in $L^1(\mathbb{R})$ and suppose it is true that $(f_n)$ converges in $L^1(R)$ to a function $f \in L^1(\mathbb{R})$. Let $g$ be a function such that $(f_n)$ converges to ...
0
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0answers
33 views

Is there any relationship between $\mathcal{l}^p$ spaces and $\mathcal{L}^P$ space?

Due to the similarity of the names, I guessed that there may be some relationship between the two spaces. Is there such a relationship, or is there nothing more to it other than the fact that they ...
0
votes
1answer
40 views

Suppose $f_n\to f$ in $L^1([0,1],\lambda)$. Prove or disprove: $\exists \{f_{n_j}\}$ such that $f_{n_j}(x)\to f(x)$ for almost every $x\in[0,1]$. [duplicate]

This is part of an old preliminary exam in Analysis I am reviewing to prepare for my own prelim. $\lambda$ is the Lebesgue measure. $f_n\to f$ with respect to the $L^1-$norm. I know that there exists ...
6
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0answers
126 views

Inequality for Lévy SDE

Let $X_{s}^{t,x}$ denote the solution at time $s$ of an Ito SDE whose coefficients are Lipschitz continuous with initial condition $X_t=x$. Let $t\leq s\leq T<\infty$. The inequality $$ \mathbb{E}\...
0
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0answers
31 views

Bounding $L^p$ norm of a function defined by averaging

Let $\Delta=\{t_0, t_1, ... t_m\}$ be a partition of $[a, b]$ and let $f{\in}L^{p}[a, b]$ for $1\le p\le\infty$. Let $T\Delta$ be the function on $[a, b]$ defined by $T\Delta(f)(a)=0$ and $$T\Delta(f)...
2
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1answer
28 views

Non-convergent Cauchy sequence in $\ell^1$ with respect to the $\ell^2$ norm

Let $X = \ell^1$, the set of absolutely convergent real valued sequences and let $d_2(\mathbf{x},\mathbf{y}) = \left(\sum_{k=1}^\infty |x_k - y_k|^2\right)^{1/2}.$ This is the $2$-norm on the $1$ ...
1
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0answers
27 views

$f_n\rightharpoonup f$ weakly in $\sigma(L^1,L^{\infty})$ and $\|f_n\|_1\to \|f\|_1$ no implies $\|f_n-f\|_1\nrightarrow 0$ [duplicate]

I'm studying Brezis' book of Functional Analysis. I'm trying to do the exercise $4.19$ and I would like of a little help. Le be $\Omega=(0,1)\in \Bbb{R}.$ Construct $(f_n)$ in $L^1 (\Omega),$ $f_n ...
4
votes
1answer
67 views

Polynomials dense in $L^{2}(\mathbb{R})$?

I was reading something about a week ago and there was a line that said something to the effect of "the space of polynomials is dense in $L^{2}(\mathbb{R})$" and then there was another line that said ...
0
votes
1answer
41 views

If $f \in L^1([a, 1])$ for all $a \in (0, 1)$, is it true that $f \in L^1((0, 1])$?

I'm learning about measure theory and need help with the following question: True or Fasle (justify): If $f \in L^1([a, 1])$ for all $a \in (0, 1)$, then $f \in L^1((0, 1])$. While it is very ...
0
votes
0answers
19 views

Show property of convolution

Proposition: Let $ u, v \in L^2(\mathbb{R}^n) $ then $ \widehat{u \ast v}=\widehat{u} \cdot \widehat{v}$. Proof: We want to show that $\mathcal{F}^{-1} (\widehat{u} \cdot \widehat{v})=u \ast v $. We ...
1
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0answers
45 views

Show $L^2$-convergence

Lemma: Let $\psi \in C_C^{\infty}(\mathbb{R}^n), \psi \geq 0, \int \psi(x)dx=1, \psi_{\epsilon}(x)= \epsilon^{-n} \psi{\left( \frac{x}{\epsilon}\right)}, \epsilon>0$. Let $f \in L^2(\mathbb{R}^n)$ ...
-1
votes
1answer
67 views

Show that the following is a bounded linear operator on $L^2(R_+)$ Calculate the adjoint operator.

The following is a question I have been working on for some time with help from my teacher. Unfortunately we have a solution but are not 100% confident with it. Some guidance on if part(s) of our ...
1
vote
0answers
29 views

Inequalities in weak $L^1$ norm

I have the following lemma: Suppose that for $j=1,2,\dots,$ $g_j(x)$ is a nonnegative function on a measure space for which $\left|\left\{x: g_j (x) >s\right\}\right|<1/s$. Let $\{c_j\}$ be a ...
0
votes
1answer
17 views

Uniform convergence of a sequence of functions given as product and convolution.

Suppose we have, for an open bounded set $\Omega \subset \mathbb{R}^n$: A function $u \in L^p(\mathbb{R}^n) \cap C(\mathbb{R}^n)$. A sequence of mollifiers $(\rho_n) \subset C_c^{\infty}(\mathbb{R}^...
2
votes
0answers
47 views

Let $T(f):=\frac{1}{x}\int_{0}^{x}{f(t)\,\mathrm{d}t}$ (the Hardy operator) find the norm of $T$ on $L^p$ [duplicate]

We have the operator $T: L^p(\mathbb{R}^+) \to L^p(\mathbb{R}^+) $ with $p \in (1,+\infty)$, defined by $T(f):=\frac{1}{x}\int_{0}^{x}{f(t)dt}$. We define $\tilde{f}(x)=e^{x/p}f(e^x)$ for all $f \in ...
3
votes
1answer
40 views

If $\{\nabla u_j\}$ is Cauchy in $L^p(\mathbb{R}^n)$ and $\int_{B(0,1)} u_j dx = 0$, does $\{u_j\}$ converge in $L^p_{\text{loc}}(\mathbb{R}^n)$?

Let $1 < p < \infty$. Let $\{u_j\}_{j=1}^\infty$ be a sequence of functions in $W^{1,p}_{\text{loc}}(\mathbb{R}^n)$ such that $\nabla u_j \in L^p(\mathbb{R}^n)$ for all $j$, $\int_{B(0,1)} u_j ...
4
votes
3answers
63 views

Properties of sequences of bounded functions in $L^2$

Let $f_n: [0, 1] \to \mathbb{R}$ be a bounded sequence of functions in $L^2([0, 1])$. This means that there exists $C_0 > 0$ so that$$\|f_n\|_{L^2} \le C_0 \text{ for all }n.$$Assume that $f_n$ ...
0
votes
0answers
37 views

Subspace of $L^1(\Omega)$ closed

Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space, $\mathcal{A}\subset\mathcal{B}$ two sub-sigma-algebras and $f$ a $\mathcal{B}$-measurable function. I want to show, that the subspace $$\...
4
votes
0answers
27 views

Is it natural how $L^p$ spaces measure local and global sizes the same?

This is a continuation of my question Spaces of functions similar to $L^p$ but with different local and global sizes. I have been bothered by the fact that the $L^p$ norm on $\mathbb R^n$, which is ...
6
votes
1answer
72 views

Spaces of functions similar to $L^p$ but with different local and global sizes

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 ...
1
vote
1answer
41 views

Is it true that $\lim\limits_{r \rightarrow 1}f(r x) = f(x)$ in $L^1$?

Suppose $f \in L^1(\mathbb{R})$ with Lebesgue measure and $r > 0$. Does $f(rx)$ converges to $f$ in $L^1$ as $r \rightarrow 1$ ? Put differently, does $$ \| f(rx) - f(x)\|_1 \rightarrow 0$$ as $r \...
1
vote
2answers
40 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 why I ...
2
votes
1answer
58 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}} ...
2
votes
1answer
95 views

Parseval's identity holds

Theorem: If $u \in L^2(\mathbb{R}^n)$ then the Fourier transform $\widehat{u} \in S'(\mathbb{R}^n)$ is a $L^2(\mathbb{R}^n)$ function and the Parseval's identity holds: $||\widehat{u}||_{L^2(\mathbb{R}...
1
vote
2answers
26 views

Let $f\in L^p(0,1)$ and define $f_h$

Let $f\in L^p(0,1)$ ($1\leq p<\infty$) and define $f_h$ as $$f_h(x)=\begin{cases}f(x+h)&\text{ for } x+h\in [0,1]\\ 0 &\text{ for } x+h\not\in[0,1]\end{cases}$$ Prove that for all $\...
1
vote
2answers
46 views

Inclusion of Schwartz space on $L^p$

I'm looking for a proof of $\mathcal{S}(\mathbb{R}) \subset L^p(\mathbb{R})$ for $1 \leq p \leq \infty$. My informal probe follow like this: For any function $f \in L^p(\mathbb{R})$ exists a ...
0
votes
1answer
36 views

Is the sequence of functions $g_n=ng_1(nx)$ a Cauchy sequence?

Given a function $g_1(x) \in \mathcal L^2(\Bbb R)$ that satisfies: $$\int_{-\infty}^{\infty}dx \space g_1(x)=1$$ one can define a sequence of functions $g_n=ng_1(nx)$. Does $g_n(x)$ define a ...