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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3
votes
2answers
66 views

Hint for Lebesgue theory/functional analysis type of problem

I am trying to solve the following problem, but I am not too familiar with functional analysis. Could you guys tell me where I should start? Thanks! Let $f \in L^1(\mathbb{R})$ and define $$f_n(x) = ...
3
votes
1answer
55 views

Is $(\oplus\ell_2^n)_{\ell_1}$ complemented in $\ell_1\oplus_\infty\ell_p$?

Fix any $1<p\leq 2$. Let us recall that $E:=(\oplus\ell_2^n)_{\ell_1}$ is just the space of sequences $(x_n)_{n=1}^\infty$, $x_n\in\ell_2^n$, such that $(\|x_n\|_{\ell_2^n})_{n=1}^\infty\in ...
3
votes
1answer
45 views

Prove an equality ($L^P$ spaces)

Prove the equality $$\int |f(x)|^p dx=\int_0^\infty pt^{p-1}m(\left\lbrace x:|f(x)|\geq t\right\rbrace)dt$$ for $p\geq 1$. My first idea was to try to prove this via induction. For the case $p=1$, ...
4
votes
1answer
71 views

Let $a_f=\text{ arg} \min_{a} \int \left|f(x)-a\right| dx$ and $a_g= \text{ arg} \min_{a} \int \left|g(x)-a\right| dx$, is $a_f \le a_g$?

Let $ f(x) \le g(x) $ and assume that $g(x),f(x) \in L^1$ let \begin{align} a_f= \text{ arg} \min_{a } \int_A \left|f(x)-a\right| dx\\ a_g=\text{ arg} \min_{a } \int_A \left|g(x)-a\right| dx ...
0
votes
1answer
37 views

The asymptotic behavior of an absolutely continuous function with square integrable derivative

Let $f : \mathbb R \to \mathbb R$ be absolutely continuous and assume that $f'\in L^2([0,1])$ and that $f(0) = 0.$ Show that the following limit exists and compute its value: $ \lim_{x \to 0} ...
1
vote
0answers
22 views

Pairs $(p,q)$ such that $id: l_p\to l_q$ is bounded [duplicate]

find all pairs $p,q\in [1,\infty)$ such that $id: l_p\to l_q$ is bounded. This just means I must find all $(p,q)$ such that $\|x\|_q \le C\|x\|_p$ for some $C$ dependent on $p$, $q$. I don't know ...
1
vote
2answers
39 views

Prove that $\|f\|_p \leq \liminf \|f_n\|_p$ under weak convergence

Let $1<p<\infty$ and $q$ its conjugate. Given a sequence $(f_k)_{k \in \mathbb N}$ and $f$ in $L^p(\mathbb R^d)$, I am trying to show that if for all $g \in L^q(\mathbb R^d)$, $$\lim_{k \to ...
1
vote
3answers
96 views

Why is $C=\{x\in l^2:|x_n|\leq 2^{-n},n=1,2,3,\dots\}$ compact?

Why is $C=\{x\in l^2:|x_n|\leq 2^{-n},n=1,2,3,\dots\}$ compact? I tried to show that $C$ is totally bounded and closed. I showed that is closed but I don't know how to show that is totally bounded. ...
3
votes
0answers
76 views

Universal property of l^p-spaces

The category $\mathsf{Ban_1}$ of Banach spaces together with short linear maps (i.e. those of norm $\leq 1$) seems to have a natural construction which interpolates between coproduct and product: Let ...
1
vote
0answers
35 views

Problem related to $L^p$ and distribution function

I am trying to solve the following problem: Let $\lambda_f(t) := \mu(\{x \in X: |f(x)| > t\})$ Prove that $f \in L^p(\mathbb R^d)$ for $0<p<\infty$ if and only if ...
0
votes
1answer
15 views

Showing $\ell^p$ space is linear

from what I understand, it is easy to show that $\ell^p$ space is linear, without the famous inequalities (Minkowsky, Holder, etc.). The fact that is a metric space is not of interest for the moment. ...
1
vote
1answer
21 views

Strong convergence due to Compact Operator [duplicate]

Given a sequence $u_{n}$ such that: $u_{n} \rightharpoonup 0$ in $L^{2}(\mathbb R^{n})$ & $A$ is a compact operator. The problem is to show that : $Au_{n} \rightarrow 0$ in $L^{2}(\mathbb R^{n})$ ...
1
vote
0answers
34 views

Convergence in L^p and convergence in norm

I am trying to show these two statements: Let $(\mathbb R^n,\mathcal M,m)$ where $m$ is the Lebesgue measure and $\mathcal M$ are the Lebesgue measurable sets, and let $(f_n)_{n \in \mathbb N}$ and ...
0
votes
1answer
32 views

A simple lp inequality

Suppose $f : \mathbf R\rightarrow\mathbf R$ is in $L^p$ for some $p>1$ and also in $L^1$. Show that $\exists c>0$ and $\alpha\in(0,1)$ such that $$\int_A|f(x)| \, dx\leq c m(A)^\alpha$$ where ...
2
votes
2answers
30 views

Pointwise convergence implies $L^p$

Simply, why is it that convergence pointwise, $u_j \rightarrow u$, implies convergence in $L^p$ if $|u_j(x)| \le g(x)$ for some $g$ in $L_+^p$?
0
votes
3answers
56 views

Having trouble showing a subspace of $\ell^2$ is closed.

Let $M =\{ (x_n)_{n \in \mathbb{N}} \in \ell^{2} \mid x_{2j} = 0, \text{ for all } j \in \mathbb{N}\}$. $M$ is a subspace of the Hilbert space $\ell^{2}$ and I'm supposed to show it's closed. ...
0
votes
2answers
36 views

$L^p$ space and map $T_h(f)(x)=f(x+h)$ on $L^p(\mathbb R^n)$

I am trying to show these two statements: Let $f \in L^p(\mathbb R^d)$, $1 \leq p < \infty$, then \begin{align}(a) & \quad \left(\int_{\mathbb ...
0
votes
1answer
19 views

Non-existence of convex neighbourhoods in $L^p(0,1)$

Problem Let $0<p<1$, show that the neighbourhoods $\{f \in L^p(0,1):||f||_p<\epsilon\}$ of the zero function are not convex. I am pretty stuck with this problem. If I've understood the ...
1
vote
1answer
37 views

The inclusions between unit balls in $\ell^p$ spaces

I need to show that $d_∞(x, y) ≤ d_2(x, y) ≤ d_1(x, y) ≤ nd_∞(x, y)$ where $d_1=|x_1-0|+|y_1-0|$ and I'm setting $|x_1-0|+|y_1-0|<0$. Illustrating the $B(0,1)$ balls (centered at 0 with radius 1) ...
0
votes
1answer
29 views

$f(x) = \frac{1}{x^a+x^b}$ or $f(x) = \frac{1}{(x^a+x^b)^p}$?

Let $0< a\leq b$, for which values $p$, does the function: $$f(x) = \frac{1}{x^a+x^b}dx$$ belong to $L_p(0,\infty)$? The first step in the given (potentially incorrectly written) solution is: ...
1
vote
1answer
47 views

The coordinate-wise limit of a sequence in the unit ball of $\ell^p$ is also in the unit ball

Let $1\le p<\infty$, and let $U$ be the open unit ball in $\ell^p$. Let $\{x_k\}\subseteq\overline U$ be a sequence such that for every $n$ the limit $x(n):=\lim_{k\to\infty}x_k(n)$ exists. Can I ...
1
vote
1answer
28 views

Definition of the Lebesgue space $L^p{}_n$

I just came across the notation of the Lebesgue space $L^p_n$. However, I have never seen this before with the index $n$. Could anybody give a definition? I suspect it has something to do with some ...
2
votes
2answers
55 views

Primitive of an $L^1$ function is continuous

The primitive of a continuous function on a compact interval is continuous via the Fundamental Theorem of Calculus. Let $I \subset \mathbb{R}$ be open and let $u': \overline{I} \mapsto \mathbb{R}$ be ...
0
votes
1answer
43 views

Schwartz functions have finite $L^p$ norm

It is known that the Schwartz space is dense in $L^p$. And I was told that Schwartz functions are bounded in $L^p$. Could anyone show me "Every Schwartz function is bounded in $L^p$" by explicitly ...
1
vote
1answer
44 views

Composition with a continuous map preserves $L^p$ convergence?

Assume we have $f_n \rightarrow f$ in $L^p(U)$, where $U$ is a bounded set of an Euclidean space. Is it true that for any continuous $F$, $F(f_n) \rightarrow F(f)$ in $L^p(U)$ as well? It is clear ...
0
votes
1answer
28 views

dual pairs in $\mathbb R^n$ with non-standard norms $l_1$ and $l_{\infty}$

Suppose $X=Y=\mathbb R^n$. Usually, to apply a separating hyperplane theorem in $X$ (the primal space), we associate both $X$ and $Y$ with the standard Euclidean norm. My question is that could I ...
0
votes
1answer
40 views

Weak convergence in finite-dimensional normed spaces

Consider a sequence of finite vectors with length N, i.e. $$\underline{u}_n=\begin{pmatrix}u^1_n\\\vdots\\u^N_n\end{pmatrix},$$ simply for $u^j_n\in \mathbb{R}$. We also have the vector norm $\ell_p$. ...
2
votes
1answer
16 views

some calculations need to be verified relates to $L^p$ norm

Hi I am having trouble verifying the equivalence of the following two inequalities. \begin{align*}\tag{1} \|u(t)\|_{L^2(\Omega)}&\le \|u_0\|_{L^2(\Omega)} \end{align*} \begin{align*}\tag{2} ...
0
votes
1answer
33 views

Agmon's Inequality in higher dimensions

We have the Agmon inequality on $[0,a]$ $$\| u \|_{L^{\infty}} \leq \|u \|_{L^2}^{1/2} \|u_x\|_{L^2}^{1/2}$$ Is there a version in two dimensions, say on $[0,a]^2$? I know there is a multidimensional ...
0
votes
1answer
33 views

Relation between conjugate exponents

I have $r'=q/2$, $1<r<2$. From these I want to derive $$1+\frac{1}{(p/r)'}\ge r-1+\frac{1}{p/r}$$ and it is the same as $$3p'\le q.$$ I've tried several times, but I failed to derive second ...
1
vote
1answer
48 views

Is a bounded function always the Hilbert transform of some other function?

Given $f \in L^\infty(\mathbb R)$, does there always exist a $g$ (in some space) such that \begin{equation*} Hg=f, \end{equation*} where $Hg$ is the Hilbert transform of $g$ ? In other words, is the ...
1
vote
1answer
73 views

$f \mapsto f(0)$ is not continuous on $L^2$

Show that the the linear map $f \mapsto f(0)$ is not a continuous on $L^2(\mathbb{R},m)$. In order to show that the map $L^2(\mathbb{R}) \ni f \mapsto f(0) \in \mathbb{R}$ is not continuous, it ...
2
votes
0answers
37 views

$L^p(\mathbb{R})$ separable.

I'm trying to prove $L^p(\mathbb{R})$, $p \in [1,\infty)$ is separable by showing the collection$$ S:= \{\sum_{i=1}^nr\chi_{(a_i,b_i)}\}_{(a_i,b_i,r) \in \mathbb{Q}^3}$$ is dense in $L^p$. So, since ...
1
vote
1answer
25 views

Splitting integrals in $L^p$

I know that for $f \in L^1$, and $\mu(S) > 0$, we have $$\int_X f \, d\mu = \int_{X \setminus S} f \, d\mu + \int_{S} f \, d\mu.$$ Is the same true in $L^p$? Or do we now get an inequality? ...
1
vote
2answers
35 views

Problem related to $L^p$ space

Problem Let $k: \mathbb R^{d\times d} \to \mathbb R^d$ be a measurable function such that there is $c>0$ with $$\sup_{x \in \mathbb R^d}\int |k(x,y)|dy \leq c, \space \sup_{y \in \mathbb R^d}\int ...
0
votes
2answers
41 views

Convergence in $L^p$ and Minkowski's inequality problem

I am trying to solve the following problem: Let $(\mathbb R^d,\mathcal M,m)$ with $m$ the Lebesgue measure. Let $f_n \to f$ in $L^p$, with $1 \leq p < \infty$, $g_n \to g$ a.e. and ...
1
vote
0answers
34 views

distribution space

What are the differences between function spaces and distribution spaces? I was reading enter link description here But I do not know what are these distribution spaces and how they differ from ...
1
vote
1answer
27 views

In the definition of an $L^p$ space, do we assume $p$ is an integer, or at least rational?

Are $L^p$ spaces defined only when $p$ is an integer? Are rational numbers also acceptable powers? The definition in Rudin's Real and Complex Analysis doesn't seem to specify limitations on $p$ except ...
1
vote
1answer
41 views

Deriviative estimates in finite $L^p$ space

I'm stuck as to how to solve the following exercise: If $U$ is open and bounded with smooth boundary, $1<p<\infty$, $\epsilon>0$, and $u\in C^{\infty}(\bar U)$, show $\exists C$ s.t. ...
0
votes
1answer
32 views

some important density result

I was told (quite often) that $C_0^\infty$ is dense in $L^p$ for all $p\in[1,\infty)$. And I did realise there are some related results posted here. But I would like a standard proof. I have found ...
0
votes
0answers
20 views

Hölder inequality on probability space

I just found the following Hölder inequality on a space $\Omega$ with $\lambda(\Omega)=1$ that I don't understand: It says. Let $(u,v)$ be conjugate exponents, then we get $$||f||_2 \le ...
1
vote
1answer
27 views

An inequality involving a $C^\infty$ function and its Fourier coefficients

Here is the question: (a) Let $f\in C^1(\mathbb{T})$ and $\int_\mathbb{T} f(x)dx=0$. Show that for $1\leq p\leq\infty$: $\|f\|_p\leq\|f'\|_p$ (b) Let $f\in C^\infty(\mathbb{T})$, for ...
1
vote
1answer
42 views

How could I find a non-closed linear subspace $X$ of $l^{2}$ , such that $ l^{2} \ne X + X^{\perp}?$

How could I find a non-closed linear subspace $X$ of $l^{2}$ , such that $l^{2} \ne X + X^{\perp} ?$
2
votes
2answers
51 views

Are $\ell^p$ and $L^p [0,1]$ isometric with p distinct from 2?

For values of $p$ distinct from $2$, are $\ell^p$ and $L^p [0,1]$ isometric under some conditions? In that case, what is the isometry $T:\ell^p \to L^p$?
2
votes
3answers
42 views

Let ${a_n}$ be in $\ell_\infty$ . Prove that $f$ defined by $f({b_n})=\sum_\infty a_nb_n$ is a continuous real valued function on $\ell_1$

Let $a_n$ be in $l^\infty$ . Prove that $f$ defined by \begin{equation} f(b_n) = \sum_{n\geq 0} a_nb_n \end{equation} is a continuous real valued function on $l^1$ My thoughts on the problem are ...
2
votes
1answer
29 views

Growth rate of $g(p)=\int_0^1\log(-\log(x/e))^p\,dx$ and similar functions

Suppose $\Phi:\mathbb R\to\mathbb R$ is a positive function such that $\lim_{p\to\infty}\Phi(p)=\infty$. I want to construct a not essentially bounded function $f:(0,1)\to\mathbb R$ such that if ...
1
vote
2answers
83 views

Convergence in $L^1$ norm, but not point-wise a.e.

As part of a course assignment, I'm asked to find a sequence of functions that converges in $L^1(\Omega \subset \mathbb{R})$, yet does not converge point-wise a.e. My thought is that such a sequence ...
2
votes
4answers
65 views

$ l^1$ not reflexive

Define $ \phi:l^1\to (l^\infty)^* $ by $$ \phi(a_0,a_1,a_2...)(b_0,b_1,b_2...)=\sum_{n=0}^\infty a_nb_n.$$ Prove that $\phi$ is not onto. Here, $(a_0,a_1,a_2...) \in l^1 $ and $(b_0,b_1,b_2...)\in ...
1
vote
1answer
30 views

Completeness of the sum of two $L^p $ spaces

Q:Suppose $L^{p_0}+L^{p_1}$ is defined as the vector space of measurable functions $f$ on a measure space $X$,that can be written as a sum $f=f_0+f_1$ with $f_0\in L^{p_0}$ and $f_1\in L^{p_1}$. ...
1
vote
1answer
37 views

The containment between the Schwartz space, its dual, and the Lebesgue space $L^2$

I read from my note that $$\mathcal{S}(\mathbb{R})\subset L^2(\mathbb{R})\subset\mathcal{S}'(\mathbb{R}).$$ Where $\mathcal{S}$ is the space of rapidly decreasing function on $\mathbb{R}$, ...