# Tagged Questions

1answer
37 views

### $L^1(\mathbb{R}^n)$ functions not in $\mathcal{H}^1(\mathbb{R}^n)$

I am wondering how to imagine the Hardy space on $\mathcal{H}^1(\mathbb{R}^n)$ and in particular what sort of functions are in $L^1(\mathbb{R}^n)\backslash\mathcal{H}^1(\mathbb{R}^n)$. Furthermore, is ...
1answer
17 views

### Does $u_n(x) \subset L^p(-1,1)$ converge strongly-weak-weak* to something?

Let $\lbrace u_n(x) := [ \sin(nx)]^+ \rbrace.$ Does this sequence converge in some sense to something in $L^{p}(-1,1)$? My attempt: for $1<p< +\infty$ $u_n$ is limited so by Banach-Alaoglu + ...
2answers
211 views

### $L^1$ and $L^{\infty}$ are not reflexive

I want some proof for the following statement : $L^1$ and $L^{\infty}$ are not reflexive. Can anyone help me, please? or reference me?
1answer
43 views

### Convergence in $L^1$ of a sequence of functions

I have to see if the following sequence of functions is convergent in the space $L^1[(0,\infty)]$ $$f_n(x)= n\frac{\exp\left(-\frac{n}{2x^2}\right)}{x^3}$$ By definition, $f_n(x)$ is convergent in ...
1answer
33 views

### $L^p$ is not uniformly convex for $p=1, \infty$

As the title says: how can we prove that $L^p(\mathbb R^n)$ is not uniformly convex for $p=1$ and $p=\infty$. Does anyone knows a counter-example for the cases $p=1$ and $p = \infty$ for the ...
1answer
47 views

### Constructing Sequences in Lp

Consider Banach Space $L^{p}(U)$ where $U$ is bounded open subset in $\mathbb{R}^{n}$. Take a bounded sequence $\{u_{m}\}_{m}^{\infty}$ in $L^{p}(U)$. Consider a subsequence ...
1answer
103 views

0answers
27 views

### Weak convergence sequence in $\ell^p$ space [duplicate]

Let $e_n = (0,0,\dots,0,1,0,\dots)$ be a sequence, which has $1$ on $n$-th place and zeros elsewhere. Show that $e_n$ weak converges to $0$ in arbitrary space $\ell^p$ for $p \gt 1$, while in $\ell^1$ ...
2answers
84 views

### Are integrable, essentially bounded functions in L^p?

Given an arbitrary measure space (of possibly infinite measure), if $f \in L^1 \cap L^\infty$, then by Hölder's inequality, $f^2 \in L^1$, so $f \in L^2$. Intuition suggests that $f \in L^p$ even for ...
1answer
62 views

### Question on proof of weak compactness of $L^p$

Suppose $L^q(X,\mu)$ is separable (i.e. admits a countable dense subset). I wish to prove that every sequence $\{f_n\}$ in $L^p$ that satisfies $\sup_n \|f_n\|_p < \infty$ has a weakly convergent ...
0answers
54 views

### A stronger form of Young's inequality for convolutions?

Let $G$ be a group and $\nu\in l^p(G)$, $f\in l^q(G)$ where $\frac{1}{p}+\frac{1}{q}=1$. By Young's inequality we know that $$\|\nu * f\|_\infty\leq\|\nu\|_p\|f\|_q.$$ and so $\nu * f\in l^\infty(G)$. ...
2answers
87 views

### Proving that the smooth, compactly supported functions are dense in $L^2$.

I have two problems, one of which depends on the other. (1) I want to prove, cleanly (without too much heavy-weight machinery) that, for some (see (2)) set $\Omega \subseteq \mathbb{R}^n$, the space ...
1answer
90 views

2answers
63 views

0answers
39 views

### A problem with a proof that $L^{p_{2}}\subseteq L^{p_{1}}$ for $1\leq p_{1}\leq p_{2}\leq\infty$

In a functional analysis course I saw a claim that for $1\leq p_{1}\leq p_{2}\leq\infty$we have it that $L^{p_{2}}\subseteq L^{p_{1}}$ I have a few problems with the proof given, and I would ...
1answer
82 views

1answer
53 views

### Weak limits in $L^p$ and $L^q$

When we have a nonnegative, bounded sequence $f_n\in L^p[0,1]$, we can get a nonnegative, bounded sequence $f_n^{p-1}$ in $L^q[0,1]$ (where $q$ is the Hölder conjugate of $p$). Assume furthermore that ...
1answer
102 views

### A inequality involving $L^p$ norm and $L^1$ norm.

Prove $$\lVert f\rVert_p\leq \sup_{\lVert g\rVert_q =1}\lVert fg\rVert_1 ,$$where $$\dfrac {1} {p} +\dfrac {1} {q}=1.$$
1answer
48 views

### Closedness of $L^\infty_+$ in $\sigma(L^\infty,L^1)$

Let $L^\infty_+$ be the set of all $f\in L^\infty$ which are non negative. Our measure can be assumed to be finite. My goal is to prove that $L^\infty_+$ is closed in the weak-star topology ...
2answers
54 views

### Showing that the mean of translations of a function approaches 0 in $L_p$

Given $p \in (1,\infty)$, $f \in L^p(\Bbb R)$ and $T: \Bbb R \to \Bbb R,x \mapsto x+1$. How do I show that for $n \to \infty$ $$\frac{1}{n}\sum_{k=0}^n f \circ T^k \to 0$$ in $L^p$? I see that for ...
1answer
97 views

### Showing that the functions $\sqrt{2x}\exp(2\pi inx^2)$ form a complete orthonormal system for $L^2([0,1])$

How do I show that the functions $$g_n(x) :=\sqrt{2x}\exp(2\pi inx^2)$$ where $n$ is a integer, are a complete orthonormal set in $L^2([0, 1])$? I am relatively new to this and need some help ...
2answers
71 views

### What is name/references of inequality bounding sup-norm by $L_2$ norm (or a similar variant of this)?

I think you have something like the following inequality in most finite dimensional spaces or sufficiently restricted infinite dimensional space: $$\|g\|_{\infty}\lt C\|g\|_2$$ where $C$ would ...