# 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.

4,521 questions
Filter by
Sorted by
Tagged with
1answer
81 views

### Hardy's Inequality: Problems $3.14$ and $3.15$ in Rudin's RCA

In Problem $3.14$, we prove (a) Hardy's inequality, (b) the condition for equality, and I shall talk about (c), (d) below. Problem $3.15$ is the discrete case of Hardy's inequality. I have asked three ...
2answers
67 views

### Same on dense subspace implies same on whole space?

I read the whole proof of Theorem 13.18 Bruckner's Real Analysis book and I had no problem understanding the proof except for following two claims inside the proof that are stated without further ...
0answers
29 views

### $L_1$ convergence of holomorphic functions on closed disks to a continuous function

I'm trying to prove that if holomorphic sequence of functions $f_n$ converge to $f$ on closed disks in the $L_1(\Omega)$ sense on an open $\Omega$ and $f$ is continuous, then we can deduce that $f$ is ...
1answer
49 views

### Density of compactly supported smooth functions in Lp

This is an exercise in my Functional Analysis book; I tried to check my solution by referring to this site and elsewhere but nowhere have I found a similar argument to mine (which makes me skeptical ...
2answers
65 views

### Is $L^p$ linear for $0<p<1$?

The following is an exercise from Bruckner's Real Analysis: Show that for all $0 <p< \infty$ the collections $L^p$ of measurable functions defined on a measure space $(X, \mathcal{M},μ)$ such ...
1answer
27 views

0answers
31 views

### Solutions to Mean $L^p$ Errors

$\DeclareMathOperator*{\argmin}{arg\,min}$Lets say we have a random variable $X$ which is absolute integrable. I.e., it is in $L^1$ space and so it is also in any $L^p$ space where $1< p\leq\infty$....
0answers
37 views

1answer
40 views

### Non-canonical examples of divergent sequences that are square summable? [closed]

The canonical example of a divergent sequence that is square summable, $\sum_{n = 1}^\infty a_n$ is finite, is the harmonic sequence: $\sum_{n = 1}^\infty 1/n$. Are there examples of sequences that ...
0answers
22 views

### Convergence in $L^2 (B(0,1))$ of $H^1 (\mathbb{R}^3)$ functions implies the existence of a dominating function?

I am currently reading an article, and there is a passage where I can't understand how to justify what the author says. I'll try to write what I think the relevant informations are, since a precise ...
0answers
33 views

### Can I generalize the classical Bernstein inequality?

I have know the proof of the classical Bernstein inequality as follows. Given some function $f\in S(\mathbb{R}^d)$, where $S(\mathbb{R}^d)$ denotes the Schwartz space of functions. Let $\hat{f}$ ...
0answers
31 views

### Under what conditions is an infinite orthonormal set a basis of $L^2 (\mathbb{R})$?

I have an infinite orthonormal set in $L^2 (\mathbb{R})$ and want to know under which conditions it is as basis of $L^2 (\mathbb{R})$ and how to prove that. In the finite dimensional case, the ...
3answers
39 views

0answers
25 views

0answers
25 views

1answer
30 views

### Necessary and sufficient condition for $l_p$ space inclusion

I am trying to prove the following statement. Given $1<p,q<\infty$, $p<q$ iff $l^p \subset l^q$. The forward direction is easy. I having trouble with the opposite direction and I don't know ...
1answer
51 views

### Implication of Sobolev inequality

For $\varOmega \subset \mathbb{R}^n$, $n\ge2$, a bounded domain and $1 < q < 2 < p < \frac{2n}{(n-2)_+}$ I want to show that \begin{equation*} \| u \|_{L^p(\varOmega)}^2 \le C \left( \| \...
2answers
73 views

### $f\in L^1\implies \lim_{n\rightarrow\infty}f(n^2 x)=0$ a.e. $x\in\mathbb{R}$?

Can someone provide a hint for the following: $f\in L^1(\mathbb{R})\implies \lim_{n\rightarrow\infty}f(n^2 x)=0$ a.e. $x\in\mathbb{R}$ I can't really make heads or tails of it. Something makes me ...