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
10 views

The convergence rate of the derivative of a sequence of function

Let $v_\delta$ be a sequence of continuous diff'able function on $(-1,1)$ and $0\leq v_\delta\leq 1$. For each $\delta>0$, assume that $v_\delta(\delta)=v_\delta(-\delta)=1$ and $v_\delta(0)=0$. We ...
0
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1answer
22 views

Doesn't this $L^p$ norm estimate for all $p$ give me an $L^\infty$ bound?

Let $r_n \to \infty$ as $n \to \infty$. We have that $$\lVert v \rVert_{L^{r_n}(\Omega)} \leq C\lVert v \rVert_{L^{r_0}(\Omega)} < \infty$$ for all $n$, where $C$ is independent of $v$ and ...
-1
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1answer
38 views

Density of $L^\infty(\Omega)h$ in $L^p(\Omega)$ where $h \in L^p(\Omega)$ [on hold]

Let $(\Omega,\mu)$ be a finite measure space. Suppose $1\leq p <\infty$. Let $h$ be an element of $L^p(\Omega)$ with $h >0$ a.e.. How show that the subspace $L^\infty(\Omega)h=\{ f h\ :\ f\in ...
0
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1answer
28 views

Haar functions form a complete orthonormal system

I want to show that the Haar functions in $L^2([0,1])$ forms an orthonormal basis: Let $$f = 1_{[0, 1/2)} - 1_{[1/2,0)} \ \ \mbox{,} \ \ f_{j,k}(t) = 2^{j/2}f(2^jt - k).$$ Let $\mathscr{A} = \{(j.k) ...
0
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1answer
6 views

What is the parametrization of the set of points in $\mathbb{R}^2$ with $L^p$-(semi)norm $1$ for any $p$?

I'm looking for a curve $t_p: [0,L] \rightarrow \mathbb{R}^2$ that describes the set $T_p = \{ (x,y) \in \mathbb{R}^2 : |x|^p + |y|^p = 1\}.$
3
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1answer
19 views

Showing that the map that takes $u_0$ to solution $u(t)$ is self-adjoint

Let $u$ and $v$ be the solution of the heat equation $$w'(t) - \Delta w(t) =0$$ with initial data $u_0$ and $v_0$ respectively, and with either homogeneous Dirichlet or Neumann BCs on a bounded domain ...
7
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1answer
156 views

Convergence of Riemann sums of a periodic function

Short version for people who don't like reading: Let $f\colon\mathbb{R}\to\mathbb{R}$ be $1$-periodic, measurable and bounded. Is it true that, for almost all $x$, the average of $f(x)$, ...
0
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0answers
29 views

Does this inequality imply a uniform $L^\infty$ bound?

Suppose I have the estimate for $t > 0$ $$\lVert u(t) \rVert_{L^\infty(\Omega)} \leq Ct^{-1}\lVert u_0 \rVert_{L^1(\Omega)}$$ for the solution $u$ of a parabolic PDE with initial data $u_0$ on a ...
0
votes
2answers
70 views

Criteria to be in weak $L^{p}$ space

Let $X$ be a $\sigma$-finite measure space. Let $f : X \rightarrow \mathbb{C}$ be a measurable function and $1 < p < \infty$. Suppose for $f$ there is a constant $C$ such that ...
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1answer
25 views
3
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1answer
42 views

Prove a vector in $\ell^2(\mathbb{Z})$ is zero

Suupose we take a vector $\vec{c}\in\ell^2(\mathbb{Z})$ where $$c(i)=\sum_{k=1}^\infty\frac{c(-k+i)+c(k+i)}{k+1}$$ That is, every elements of the vector is a series with the other terms in $\vec{c}$. ...
0
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0answers
20 views

Does every $\mathcal{L}_{2}$ signal have bounded $\mathcal{L}_{2}$ derivative?

Let a real signal $f(t) \in \mathcal{L}_{2}$. Does it always imply that $\dot{f}(t) \in \mathcal{L}_{2}$? It is assumed that $\dot{f}(t)$ exists for all $t \in \mathbb{R}^{+}$.
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1answer
34 views

$L^2$ convergence by the sequence of domain

Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Assume $u\in L^\infty(\Omega)$. We know a sequence $u_n\in L^\infty(\Omega)$ such that $$ \sup_{n}\|u_n\|_{L^\infty}<+\infty $$ ...
1
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1answer
33 views

Proof $\mathcal{C}^1(-1,1)$ is not a closed subspace of Sobolev space $H_0^1 \left[-1,1\right]$

Give a sequence of functions $\varphi_n\in \mathcal{C}^\infty(-1,1)$, Cauchy with respect to the Sobolev space $H^1_0$ norm $$|| \varphi||_1=\sqrt{\int_{-1}^1 (\varphi')^2+\int_{-1}^1 \varphi^2}$$ ...
2
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0answers
26 views

Closed subset for $L^2$ strong and weak convergence

I was trying to solve the following exercise. Let $K$ a closed subset of $\mathbb{R}$. $$X:=\{f\in L^2[0,1]:f(x)\in K \:a.e.\:x\in [0,1] \}$$ Then: 1)X is closed under strong convergence in $L^2$. ...
1
vote
1answer
20 views

Inequality for $u \in L^r(\Omega)$: $\int_{\Omega} (a_1+a_2|u|^{r/s})^s dx \leq a_3 \int_{\Omega} (1+|u|^r)dx$

This question is from one of the steps of the Proof of Proposition B.1 in Appendix B of P. H. Rabinowitz's "Minimax Methods in Critical Point Theory with Applications to Differential Equations." Let ...
2
votes
1answer
46 views

If a sequence of functionals converges weakly then it is bounded.

Let $f_k, f \in L^{\infty}(R)$ and $f_k \overset * \to f$ in $L^{\infty}(R)$. Is $f_k$ a bounded sequence in $L^{\infty}(R^n)$? (Definition: if $(v_n)$ is a sequence in $V = X^*$, we say that $v_n ...
0
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1answer
43 views

If $\sum_{n=1}^{\infty}x_n^2<\infty$ and $\sum_{m=1}^{\infty}x_n^2<\infty$, is $\sum_{k=1}^{\infty}(x_n)_k^2(x_m)_k^2<\infty$? [duplicate]

Let $$l^2=\left\{(x_n):\sum_{n=1}^{\infty}x_n^2<\infty\right\}$$ equipped with the norm $$\|(x_n)\|=\left(\sum_{n=1}^{\infty}x_n^2\right)^{1/2}.$$ Prove that $l^2$ is complete with ...
2
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1answer
391 views

Completeness of $\ell^2$ space

I was reading up and it says that $(\ell^2,\|.\|_2)$ is complete. I know that a metric space $X$ in which every Cauchy sequence converges to an element of $X$ is called complete. And I know that a ...
0
votes
3answers
180 views

show that $l^2$ is a Hilbert space

Let $l^2$ be the space of square summable sequences with the inner product $\langle x,y\rangle=\sum_\limits{i=1}^\infty x_iy_i$. (a) show that $l^2$ is H Hilbert space. To show that it's a ...
0
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1answer
44 views

Suppose $X_n \to_{p} X$, if $\limsup_n E|X_n|^r \leq E|X|^r$, how can I show that $X_n \to_r X$?

If I have that $X_n \to_p X$ (convergence in probability), and if $\limsup_n E|X_n|^r \leq E|X|^r$ for all $r \geq 1$, how can I show that $X_n \to_r X$ (this means $L^{r}$ convergence)? My goal is to ...
1
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3answers
34 views

Justify or provide counterexamples [closed]

Justify or provide counterexamples $(i)$ If $f \in L^1[0,\infty)$ and $f\geq0$ then $\lim\limits_{x\rightarrow\infty}{f(x)}=0$ $(ii)$ If $\lim\limits_{x\rightarrow\infty} f(x)=0$ and $f\geq0$ on ...
0
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0answers
21 views

$f_k\rightarrow g$ in $L^p(\mathbb{R}^n)$ and $f_k\rightarrow h$ in $L^r(\mathbb{R}^n)$, then $g=h$ $a.e.$ in $\mathbb{R}^n$

If $f_k\in L^p(\mathbb{R}^n)\cap L^r(\mathbb{R}^n) $ for some $p,r\in [1,\infty), f_k\rightarrow g$ in $L^p(\mathbb{R}^n)$ and $f_k\rightarrow h$ in $L^r(\mathbb{R}^n)$, then $g=h$ $a.e.$ in ...
0
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0answers
22 views

Integral inequality $L^p$ spaces

I'm trying to solve this problem: Let $1<p<\infty$. Then let $f:(0,\infty)\to [0,\infty]$ a measurable non negative function. It's true the following inequality: $$\int_0^\infty ( ...
0
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1answer
32 views

Showing that $(G' \circ u)u' \in L^p(I)$ where $G \in C^1(\mathbb{R})$ and $u \in W^{1,p}(I)$

I am trying to prove the rule of differentiation of a composition for weak derivatives in Sobolev spaces following the proof given in Corolary 8.11 in Functional Analysis, Sobolev Spaces and Partial ...
0
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1answer
24 views

Some insight about this integral limit

Given $u:\mathbb{R}^N \rightarrow \mathbb{R}$ is continuous and has compact support, we define the set $$K_u: = \{x\in \mathbb{R}^N : u(x) = \|u\|_\infty\}.$$ Looking at the following limit ...
1
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0answers
41 views

banach space bigger than $L^p$

we know that $L^p$ is banach space for any $p\geq 1$. My question: Is there any other banach space that is bigger than $L^p$?. In fact, I have an exercice that I don't have any idea: prove that ...
0
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1answer
33 views

Maple plot the $l^p$ balls: $\{(x,y) \mid |x|^p + |y|^p \le 1\}$ for $p = 1,…,10$

I have been asked to construct this plot in maple for an analysis assignment. I have been given no other instructions on how to do this. I am not familiar at all with defining or plotting these kinds ...
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2answers
39 views

Measure Theory and $L^{p}$ spaces

I have the two following very simple questions regarding measure theory that I want to show: If $f \in L^{p}(X, \mathcal{M}, \mu)$ for $1 \leq p < \infty$, then $f < \infty$ $\mu$-almost ...
2
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0answers
22 views

Multiplier operators on anisotropic weighted $L^2$ spaces

Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$, $\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$ where the scalar complex function ...
2
votes
3answers
130 views

Translating an integrable function creates a sequence that converges to $0$ almost everywhere

Let $a_n$ be a sequence and $N\in\mathbb{N}$ be fixed. Assume that no more than $N$ of the $a_n$'s lie in any interval of lenght $1$. Show that for any $f\in L^1(\mathbb{R})$, we have ...
1
vote
1answer
31 views

Let $f \in L^{1}([0,1])$ be a real valued function. Prove the following

Let $f \in L^{1}([0,1])$ be a real valued function. Prove the following $1) x^k f(x) \in L^1([0,1])$ for all $k\in \mathbb{N}$ $2) \lim_{k\rightarrow\infty}\int_{0}^{1}x^k f(x) dx = 0$ $3)$ If ...
2
votes
1answer
36 views

Does a function $f^p$ belong to $L^ {\infty}$ if $f \in L^{\infty}$ for $1 < p < \infty$?

I understand it should be so, considering the definition: $L^{\infty}( \Omega)= \{ f: \Omega \to \mathbb{R}\, \mid f$ is measurable and there is $C \in \mathbb{R^{+}}$ such that $|f(x)| \leq C$ ...
3
votes
1answer
60 views

$L^2(\mathbb R, \mu) $ a finite dimensional space.

Hi I find the following exercise. Honestly I'm not sure about my "answer", is incredible simple.t I don't know if make sense (in what part is necessary to use $L^2$?). I'd appreciate if someone can ...
2
votes
1answer
51 views

Properties of mollification

We have this theorem For any $1\le p<\infty$ and $f\in L^p(\mathbb{R}^k)$, then $\|f*\phi_\delta - f\|_p\to 0$ as $\delta\to0$, where $\phi$ is any nonnegative measurable function on ...
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1answer
22 views

Prove $F$ is in $L^1$

This is an old qualifier exam question at my school Let $f \in L^{1}([0,\infty))$ and for $x\geq 0$, define $F(x) = \int_{(x,\infty)} f(t) e^{x-t} dm(t) $ Show that $F \in L^{1}([0,\infty))$ The ...
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1answer
37 views

can you prove this theorem An introduction to wavelet Analysis? [closed]

Definition. The sequence $\{f_n(x)\}$, $n\in \mathbb{N}$ defined on an interval $I$ converges in mean-square to the function $f(x)$ on $I$ if $\lim_{n\to \infty} \int_I {|f_n(x)-f(x)|}^2\, dx =0$. We ...
1
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1answer
71 views

A Cauchy sequence has a rapidly Cauchy subsequence

I am trying to fill in the details of a proof related to the Riesz-Fischer Theorem. We need to show that every Cauchy sequence $\{f_n\}$ has a rapidly Cauchy subsequence. My text claims that we can ...
1
vote
1answer
41 views

$\left\|f\right\|_{L^1(μ_1)}<∞$ $μ_2$-a.e.,$\left\|f\right\|_{L^1(μ_2)}<∞$ $μ_1$-a.e. $⇒$ $\left\|f\right\|_{L^1(μ_1\otimesμ_2)}<∞$

Let $(\Omega_i,\mathcal A_i,\mu_i)$ be a $\sigma$-finite measure space and $f:\Omega_1\times\Omega_2\to\mathbb R$ be measurable with respect to $\mathcal A_1\otimes\mathcal A_2$. Can we conclude, that ...
40
votes
2answers
3k views

If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$

Let $1\leq p < \infty$. Suppose that $\{f_k\} \subset L^p$ (the domain here does not necessarily have to be finite), $f_k \to f$ almost everywhere, and $\|f_k\|_{L^p} \to \|f\|_{L^p}$. Why ...
2
votes
1answer
44 views

about a product of random variables that converges weakly

Let $(\Omega,\mathcal{F},P)$ be a probability space. Suppose $f_n,g_n, n\in \mathbb{N}$ are sequences of functions on this space such that their product $f_ng_n$ converges weakly in $L^2$ to $h$, say. ...
1
vote
0answers
36 views

Prove that this function is in $L^\infty$ with $\lVert g\rVert_\infty \le C$.

My professor used the following lemma in the proof that $L^1(X,\mu)^* = L^\infty(X,\mu)$ but left the proof as an exercise. Lemma. Assume that $(X,\mathcal A, \mu)$ is a measure space and $g \in ...
2
votes
1answer
28 views

I do not understand a point in the proof of completness of $L^{\infty}$

do not understand a point in the proof of completness of $L^{\infty}$. I have this proof. We consider the sets $$A_{n,m}=\{x\in E:|f_{n}(x)-f_{m}(x)\|\leq\|f_{n}-f_{m}\|_\infty\}$$ for all ...
5
votes
1answer
51 views

uniform boundedness principle for $L^{1}$

i read this theorem from V.I.Bogachev vol 1 Measure Theory. A family $\mathcal{F}\subset L_{1}(\mu)$,where the measure $\mu$ takes values in $[0,+\infty]$, is norm bounded in $L_{1}(\mu)$ precisely ...
2
votes
1answer
66 views

There is no bounded linear surjection between $\ell_p$ spaces

For $1\leq p,q<\infty$, $p\ne q$, how to prove that there is no bounded linear operator $T:\ell_p\to \ell_q$ such that $T$ is surjective? I've tried to use Pitt's theorem, but without success.
0
votes
1answer
32 views

Does the weak divergence exist for each $\mathcal L^2(\Omega;\mathbb R^d)$-function?

Let $\Omega\subseteq\mathbb R^d$ be open. $v:\Omega\to\mathbb R$ is called weak divergence of $u:\Omega\to\mathbb R^d$ $:\Leftrightarrow$ $$\int_\Omega v\varphi\;{\rm d}\lambda=-\int_\Omega\langle ...
2
votes
0answers
97 views

Test functions are dense in $L^p$?

I was wondering about the following: If we say that the test functions are dense in $L^p$, does this imply that there is also always a sequence of them converging pointwise and in $L^p$ norm to such a ...
0
votes
1answer
39 views

Show that $(\int\left(\sum_k|f_k|\right)^p)^{1/p}\le\int(\sum_k|f_k|^p)^{1/p}$

How to show that $(\int\left(\sum_k|f_k|\right)^p)^{1/p}\le\int(\sum_k|f_k|^p)^{1/p}$, or is it not true in general ? $\{f_k\}\subset L^p, G_n=\sum_1^n|f_k|$, I think $|G_n|_p$ is the sum above ...
1
vote
2answers
35 views

Does the function $f(x)=\frac{1}{\sqrt x}$ belong to $L^p( \mathbb N , P(\mathbb N), \mu),p=1,2,\infty?$

Does the function $f(x)=\frac{1}{\sqrt x}$ belong to $L^p( \mathbb N , P(\mathbb N), \mu),p=1,2,\infty?$ $\mathbb N$- set of natural numbers, $P(\mathbb N)$- the partitive set of natural numbers. I ...
0
votes
2answers
38 views

Definition of $L^p$ norm of a vector-valued function

If $u$ is a vector the definition of the discrete norm will be $$\|u\|_{l^p}=(\sum |u_i|^p)^{1/p},$$ If $u$ is a function, $$\|u\|_{L^p}=\left(\int|u|^p\right)^{1/p}$$ But when $u$ is a vector-valued ...