# Questions tagged [lp-spaces]

For questions about $L^p$ spaces. That is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence classes of measurable functions such that $|f|^p$ is $\mu$-integrable. Questions can be about properties of functions in these spaces, or when the ambient space in a problem is an $L^p$ space.

5,638 questions
Filter by
Sorted by
Tagged with
1 vote
33 views

• 35
133 views

### Showing a function is identically zero

Suppose I have a simple function (not compactly supported) $f$ on $\mathbb{R}^n$ that is orthogonal to all polynomials (with respect to the $L^2$ inner product) on $\mathbb{R}^n$. We further know that ...
1 vote
21 views

### For every functional on $l_p$ it exists a sequence that determines it uniquely

I am trying to prove this excercise from my Functional analysis book: Let $p \in (1, + \infty ) \subset \Bbb{R}$, $q = (1- \frac{1}{p})^{-1}$ and denote $l_p$ the space of $p-$bounded sequences of a ...
25 views

### Convergence in $L^\infty$($\Omega$) and almost everywhere

I have a question about the difference between the convergence in $L^\infty$ and convergence almost everywhere. Precisly, let $\mu(\Omega) < \infty$, $f_n \rightarrow f$ in $L^1(\Omega)$, then ...
91 views

### Why does the Haar orthonormal system span the whole $L^2$?

I am reading "Real Analysis with an Introduction to Wavelets and Applications" because I want to understand wavelets better for my work. I got stuck on a detail about the Haar Basis in ...
1 vote
31 views

### Finding the conjugate operator of the following operator

Let $A$ be linear operator from $l^2$ to $l^2$ such that $Ax =y^0 \cdot \sum_{1}^{\infty}{x_k}$ , where $y^0 \in l^2$ — fixed element. Show, that conjugate operator $A^*$ exists and find it. Show, ...
8 views

### When an operator with variable coefficient is bounded?

When the operator $Au(x):=\mathcal{F}^{-1}(m(x,\xi)\widehat{u}(\xi))(x)$ is bounded on $L^p(\mathbb{R}^n)$? I know that, by Mikhlin theorem, if $m(x,\xi)=m(\xi)$ (not dependence on $x$) is a fourier ...
• 3,568
46 views

### If $u\in L^1_{\text{loc}}(\Omega)$ and $\Delta u\in L^1_{\text{loc}}(\Omega)$, then $\nabla u \in L^1_{\text{loc}}(\Omega)$?

(And as well, a transcription for those unable to load images:) Remark 2. There is a local form of Corollary 1, namely if $u\in L^1_{\text{loc}}(\Omega)$ and $\Delta u \in L^1_{\text{loc}}(\Omega)$, ...
• 513
49 views

### $S \in \mathcal{L}(L^1(\Omega))$, find $T^* \in \mathcal{L}(L^\infty(\Omega))$ with $T^*g = Sg \forall g \in L^1(\Omega) \cap L^\infty(\Omega)$

Below I will bring a passage from Heat Kernels by Wolfgang Arendt (Theorem 4.3.3, page 52). I need to understand it and write a more verbose report based on the chapter, however I am stuck at this ...
62 views

### Move Infinite sum inside a limit $t \to \infty$.

This may be simple, but I want to know if my reasoning is ok. I came across a problem whose essential set up is: let $f_k$ be a sequence of functions in $L^1(\mathbb{R})$ (Lebesgue integrable ...
• 457
1 vote
50 views

### Using Hahn Banach for switching between $L^p$ spaces

I want to understand the proof (or under which conditions a proof holds) of the following statement: Let $f$ be a function in $L^p$ and let $q$ be such that $\frac{1}{p}+\frac{1}{q}=1$. Then we can ...
31 views

### Prove operator $(Tf)(x)=sin(x)\cdot f(x)$ is not compact

Given the following operator in $L_2[0,1]$ $$(Tf)(x):=sin(x)\cdot f(x)$$ Prove or Disprove that the opertor is Compact. I thought it is compact and used arzelà–Ascoli theorem, but apparently I am ...
• 593
113 views

### The dual $(L^\infty)^{*}$ is not $L^1$ by constructing example

The problem statement is the same as this post: $L^{\infty *}$ is not isomorphic to $L^1$ . Let $L^\infty = L^\infty(m)$, where $m$ is Lebesgue measure on $I=[0,1]$ . Show that there is a bounded ...
112 views

### Square integrable for universal approximation

Let's consider square-integrable functions $f \in L^2\left(I_n\right)$ with the definition of the $\textit{discriminatory}$: $\textbf{Definition:}$. The activation function $\sigma$ is called ...
74 views

### Showing $\chi_A \ast \chi_B(x)$ is continuous if $m(A),m(B) \in (0,\infty)$. [duplicate]

Let $A,B \subset \Bbb{R}$ be Borel measurable with $m(A),m(B) \in (0,\infty)$, then prove $\chi_A \ast \chi_B$ is continuous. Attempt: Note $$f(x)=\chi_A \ast \chi_B (x)=\int \chi_A(x-y)\chi_B(y)dy.$$ ...
• 4,177
22 views

### Sequence of sequences $\{a^{(n)}\}_n \subseteq \ell^2$ with bounded members $|a^{(n)}_k| \leq 1$ has converging subseq.

I struggling to understand a partial step in the solution to an exercise: Given a seq. of seq. $\{a^{(n)}\}_{n \in \mathbb N} \subseteq \ell^2$ such that $|a^{(n)}_k| \leq 1 \forall n,k \in \mathbb N$...
• 129
33 views

### convergence of functions.

Let a sequence of functions $v_m$ that converges to $v$ in the space $L^{10/3}(\Omega)$, Where $\Omega$ is a bounded domain. Additionally, $\sup\limits_{m} |v_m|<C$, where $C$ is a constant. Can I ...
33 views

• 883
38 views

• 7,673
1 vote
24 views

### For $1 \leq p<q<\infty$ if $X$ contains sets of arb small positive measure, then $L^p \not\subset L^q$.

Its an if and only if but I have one direction down. So suppose $(X,\mathbf{A},\mu)$ is a measure space and $X$ contains sets of arbitrarily small positive measure. Choose $A_1 \in \mathbf{A}$ such ...
• 4,177
1 vote
29 views

• 481
138 views

1 vote
24 views

• 115
1 vote
67 views

### Equivalence of two forms of the Marcinkiewicz interpolation theorem

On this article and Stein & Weiss (statement 1) and the books by Linares & Ponce and Duoandikoetxea (statement 2), I found the following statements of the Marcinkiewicz interpolation theorem: ...
• 1,568
1 vote
141 views

### Does $\sin (nx)$ converge in $L^2$?

I was just introduced the concept that if $(f_n)$ converges in $L^2$ topology to $g(x)\in L^2([0,2\pi])$ then $\lim_{n\to\infty}\int^{2\pi}_0|f_n(x)-g(x)|^2dx=0$. I would appreciate any hint to how to ...
• 191
43 views

### Why is it bad to translate from Lp norm to priors?

It seems like we should start with a probability distribution and infer a log likelihood based on that. Why is it a bad idea to start with regularizers and map this onto priors? For example, if I were ...
239 views

### $L^1- L^\infty$ estimate for the semi group of wave equations

I am looking for a proof of the following lemma for the case where: $y= (y_1,\cdots, y_n)\mapsto P(y) = \|y\|_2= \sqrt{y_1^2+ \cdots + y_n^2}.$ In this case the rank of the mentioned matrix is $n-1$...
• 639
### Trying to understand the proof for the criterion of compactness in $l_p$ space
I have the following theorem about the criterion of compactness in $l_p$ space For the set $K\subset (l_p,||.||_p), p\geq 1$, following conditions are equivalent: i) $K$-totally bounded in \$(l_p,||.||...