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 ...
5
votes
1answer
137 views
Is there anyway to bound the $L^\infty$ norm by other $L^p$ norm?
If $f\in L^\infty(\mathbb R^2)$ (in my particular exercise, $f\in H^2(\mathbb R^2)$, the sobolev space), I want to bound $|f|_{L^\infty}= $ esssup $|f|\leq c|f|_{L^p}$ for some p, what kind of number ...
1
vote
2answers
184 views
Proof of Clarkson's Inequality
Trying to find a proof for Clarkson's inequality, which states that if $2 \leq p < \infty$, then for any $f, g \in L^p$, we have that $$\left|\left|f+g\right|\right|_p^p + ...
2
votes
2answers
418 views
convergence of $L^p $ norm [duplicate]
Possible Duplicate:
Limit of $L^p$ norm
If I define $|f|_{L^\infty}= \lim_{n\to \infty} |f|_{L^n}$. How can I prove that this limit is esssup $|f|$?
3
votes
1answer
74 views
How to know a function is in $L^p$.
I am having trouble with this question. It is not an homework question, I am currently trying to practise different problems for an exam.
Let $f$ be a nonnegative measurable function on $\mathbb ...
3
votes
1answer
133 views
Two Real Analysis Questions
If I have $ A = \{a \in \ell_2 : |a(n)| \leqslant c(n)\}$ for $c(n)\geqslant 0$ where $ n \in N $, and I want to show that is $A$ compact in $\ell_2$ iff $\sum{c(n)^2}<\infty$. How do I go about ...
0
votes
1answer
55 views
If f is an $L^p$ function and $\int f(x)g(x)dx=0$ for every $L^p$ function g does that imply that f=0 a.e
If $f$ is an $L^p$ function and $\int f(x)g(x)dx=0$ for every $L^p$
function $g$ does that imply that $f=0$ a.e
2
votes
1answer
138 views
Bounded sequences that form compact sets or not
a) Give an example of a bounded closed subset of
$$ A = \{(x_n) \in \ell^1: \sum_{n\geq1} x_n = 1\}$$
which is not compact. The metric we consider on A is induced by the normal norm on ...
1
vote
1answer
72 views
Is $L^{p_2}$ complete under the $L^{p_1}$ norm?
Given $p_1,p_2$ such that $1 \leqslant p_1 < p_2 < \infty$ and a measurable set $E$ of finite measure, I'm trying to determine whether the space $L^{p_2}(E) $, which I know to be contained in ...
3
votes
1answer
76 views
$f_k \rightarrow f$ in $L^p$ and that $g_k \rightarrow g$ weakly in $L^q$. Show that $f_k g_k \rightarrow fg$ weakly in $L^1$
I want to solve the following exercise, and I thankfully welcome some hints. Note that this is not homework.
Problem: Let $1 < p,q < \infty$ be conjugate exponents. Assume $f_k \rightarrow ...
1
vote
1answer
99 views
Closed graph theorem to prove that a sequence is in $\ell^q$
Let $\{a_n\}$ be a sequence of complex numbers such that $\sum \limits _{n=1}^{\infty} a_nb_n$ converges for every complex sequence $b_n \in \ell^p$.
Show that $\{a_n\} \in \ell^q$ where ...
2
votes
1answer
109 views
Inequality of Lebesgue integral with $L^p$-norm
Let $X_t(\omega)$ be a continuous function $t\rightarrow L^p(\omega)$ (i.e., if we fixed the variable $t$ we obtain a function which belongs to $L^p$), with $t\in[0,T]$ and $\omega\in\mathbb{R}$.
I ...
1
vote
1answer
137 views
Convergence in $\ell^p$ norm provided it weakly converges.
I need some help with the following problem :
$1<p < \infty$ , let $x_n$ be a sequence in $\ell^p$ and also $x\in \ell^p$ . I am interested in showing $$\lim_{n\to \infty} \|x_n-x\|_p\to0$$ ...
3
votes
0answers
74 views
Weak $L^1$ as real interpolation space between $L^p$-spaces?
Let $\Omega$ be a measure space. We denote $L^{p,q}$ the usual Lorentz space. We use a real interpolation method $(\cdot,\cdot)_{\theta,q}$.
Suppose $1\leqslant p,q\leqslant \infty$. I know that if ...
1
vote
0answers
80 views
A continuous embedding.
If $2 \le q \le \infty$ and $2s >n$, then is there a continuous embedding $ H^s (\Bbb R^n ) \hookrightarrow L^q(\Bbb R^n) $ ? Here $H^s$ means a general Sobolev space for $s = 0,1,\cdots$.
4
votes
0answers
75 views
A question about functions in $L^p(E)$
I've been working on a problem. I found a couple of related questions on the site, but I was having a little bit of trouble clarifying everything. The goal is to show that given $f\notin L^p(E)$, ...
2
votes
1answer
71 views
How can I give a bound on the $L^2$ norm of this function?
I came across this question in an old qualifying exam, but I am stumped on how to approach it:
For $f\in L^p((1,\infty), m)$ ($m$ is the Lebesgue measure), $2<p<4$, let
$$(Vf)(x) = ...
4
votes
1answer
454 views
Limit of $L^p$ norm
Could someone help me prove that given a measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^{\infty}$ and some $L^{q}$, ...
1
vote
1answer
69 views
Estimating the integral in norm.
I want to estimate the integral $$\int k(x,y)f(y)dy$$assuming the fact that $k(x,y), f(y)$ are in $L^p, L^q$ respectively.
But I want to bound the the whole integral in $L^r$, $r\in [1,\infty]$.
I ...
1
vote
1answer
43 views
Unit vector basis in $\ell_1$
Can someone illuminate me with a hint about why it is the case that no subsequence of the unit vector basis $(e_n)$ of $\ell_1$ is weak Cauchy?
1
vote
1answer
54 views
Convergence of distributions in $L^p$
If I understand correctly, distributions $F_n \in C^\infty_c(\mathbb{R})^*$ are defined based on how they act on test functions $\phi \in C^\infty_c(\mathbb{R})$.
What does it mean then to say $F_n ...
1
vote
1answer
76 views
$\lim_{n\rightarrow \infty}\int_0^1f_nhdm=\int_0^1fhdm$, prove $f\in L^p(m)$ , where $1\le p<\infty$.
On $[0,1]$, suppose $\|f_n\|_p\le 1$,
$\lim_{n\rightarrow \infty}\int_0^1f_nh\, dm=\int_0^1fh\, dm$, for any $h\in L^\infty(\mu)$,
I need to prove $f\in L^p(\mu)$ , where $1\le p<\infty$.
I ...
3
votes
1answer
125 views
Proof of an inequality of $L^p$ norms
For a general measure space, we define : $\|f\|_p= \left(\int\vert f\vert^p du\right)^{1/p}$.
Let $0 < a < b < c < \infty$ and prove the following:
$$
\|f\|_b \leqslant \max\{\|f\|_a, ...
1
vote
2answers
93 views
Examples that are not Lebesgue integrable for any $p$
I've been trying to think up different examples of functions such that $EZ^p = \infty$ (with $Z>0$) for all $p$, but each time it becomes rather messy. Can anyone suggest some interesting but ...
4
votes
1answer
69 views
Limit problem for $L^p$ function
I am having problems with proving the following:
Let $f$ be a $L^p$ function on $[0,1]$, $f:[0,1] \to \overline{\mathbb{R}}$. Prove that
$$\lim_{t \to \infty} t^p \mu(x: |f(x)| \geq t) = 0.$$
...
7
votes
1answer
189 views
How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function
We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$.
...
3
votes
2answers
139 views
Question from Folland, criteron for a function to belong to $L^p$
This question is from Folland 6.38,
Show that $f \in L^p $ iff $\sum_{k=-\infty}^ {\infty} 2^{pk} \mu \{{x: |f(x)|>2^{k}}\} \lt \infty$
If $f \in L^p $, I applied the Chebyshev's inequality
But ...
1
vote
2answers
76 views
Is the inclusion map of $\ell^1(N)$ in $\ell^2(N)$ bounded and dense?
I am looking for an idea to prove if the inclusion map from $\ell^1(N)$ to $\ell^2(N)$ bounded and does it have a dense image.
And why is the set $A:=\{x: ||x||_1\le 1\} \subset \ell^2(N)$ closed and ...
2
votes
1answer
79 views
Closed subspace of $L^1[0,1]$
The statement I need to prove is following.
Let $S$ be a closed subspace of Lebesgue space $L^1[0,1].$ Assume that for every $f\in S$ there exists a number $p(f)>1$ such that $f\in L^{p(f)}[0,1].$ ...
1
vote
1answer
45 views
Limit and Integral sign in $L^2$.
If $\lim_{k \to \infty} \| u_k - u \|_{L^2(\Bbb R^n)} = 0$ then how can I show that $$ \lim_{k \to \infty} \int_{\Bbb R^n} u_k v = \int_{\Bbb R^n} uv$$ for any $v \in L^2 (\Bbb R^n)$?
2
votes
1answer
78 views
Liapunov's Inequality for $L_p$ spaces
Let $1 \leq p,q < \infty$ and $0 \leq \lambda \leq 1$. If $r = \lambda p + (1 - \lambda)q$ and $f \in L_p \cap L_q $, then
$$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q} \tag{*}$$
...
3
votes
1answer
173 views
When multiplication operator is an isometry/ bounded below?
Let $X$ be a locally compact topological space. Let $\mu,\nu\in M(X)$ - regular finite $\sigma$-additive Borel measures and $\nu$ is absolutely continuous with respect to $\mu$.
Let ...
4
votes
1answer
184 views
$L^2$ norm inequality
I need some help with this homework question. I was asked to provide an example of a $n$-dimensional subspace $W$ of $L^2[0,1]$ such that all functions in that subspace with $L^2$ norm equal to $1$ ...
3
votes
1answer
107 views
How is this book applying Fubini/Tonelli without assuming $\sigma$-finiteness?
I am reading about $L^p$ spaces on this google book and in proposition 1.1.4 (page 4) it writes
$$
p\int_0^\infty \alpha^{p-1} \int_X \chi_{\{x:|f(x)|>\alpha\}}d\mu(x)d\alpha = \int_X ...
2
votes
1answer
74 views
Does $\Vert f \Vert_p = \sup_{\Vert g \Vert_q=1}\int fg d\mu$ fail if $f \notin L^p$?
I know that for $p \in [1,\infty]$ if $X$ is $\sigma$-finite (for the $p=\infty$ case) we have
$$
\Vert f \Vert_p = \sup_{\substack{g \in L^q\\\Vert g \Vert = 1}} \int_X fg d\mu.
$$
I always see it ...
0
votes
1answer
81 views
Convergence in $L^2$ space.
Let $u_k , u \in L^2 (\Bbb R^n)$ for $k \in \Bbb N$. Assume that $f : \Bbb R^n \to \Bbb R$ is continuous and $|f (u_k) | \leqslant M$ , $|f(u) | \leqslant M$ for some $M >0$. If $u_k$ converges to ...
3
votes
1answer
65 views
Negative integral on intervals implies negative function?
Let $f \in L^1([0,1])$ be such that for all $t \geq s$, $\displaystyle \int_s^t f(u)du \leq 0$. Is it true that $f\leq 0$ almost everywhere?
2
votes
1answer
107 views
weak vs. norm compactness in $\ell_1$
So I'm trying to show that weakly compact sets in $\ell_1$ are norm-compact. I've already proven that weak sequential convergence implies norm convergence. I think the idea I want to go with is to ...
1
vote
1answer
89 views
Show a function is in $L_\infty$
Let's assume we're working on a measure space $(X,\Sigma,\mu)$, where $\mu$ is a $\sigma$-finite measure. Suppose that $g$ is a measurable function such that $\forall f\in L^2$, $||fg||_2\leq ...
3
votes
2answers
132 views
In $\ell^1$ but not in $\ell^2$?
Can a sequence $f:\mathbb{Z}\to\mathbb{C}$ be in $\ell^1$ but not $\ell^2$? (any one counter example will suffice)
7
votes
2answers
153 views
On $L^p$ and $\ell^p$
If a continuous and infinitely differentiable function $f(x): \mathbb{R}\to\mathbb{C}$ is in $L^p$, is it also true that $f(n),\ n\in \mathbb{Z}$ is in $\ell^p$?
1
vote
1answer
97 views
prove a subset of squence space lp closed in strong topology
Let $l^p$ be the space of $p$-summable sequences. von Neumann constructed a subset of $l^p$ space
$$S=\{X_{mn}: m,n≥1\}$$
where $X_{mn}\in l^p$ are defined by $X_{mn}(m)=1, X_{mn}(n)=m$ and ...
1
vote
1answer
69 views
Isometric embeddings of $\ell_q^m$ into $\ell_p$ and $L_p$ for $p,q\in[1,+\infty]$
I'm looking for articles describing (or proving nonexistence) of isometric embeddings of $m$-dimensional space $\ell_q^m$ into $L_p$ and $\ell_p$ for $q,p\in[1,+\infty]$.
Since $\ell_q^m$ is finite ...
0
votes
1answer
53 views
Dual space of the function $f$ in Fourier Transform
Let $f\in L^1{(\mathbb{R})}$. Why the Fourier Transform $\hat{f}\in L^{\infty}{(\mathbb{R})}$.
Is it because $(L^1{(\mathbb{R})})^*=L^{\infty}{(\mathbb{R})}$?
3
votes
2answers
139 views
Lp Spaces and limits of translated functions
If $g\in L^p(\mathbb{R}^n)$ and $1\leq p<\infty$ then show $$\lim_{|t|\to \infty}\lVert g_{(t)}+g\rVert_p=2^{1/p}\lVert g\rVert_p,$$
where $g_{(t)}(x):=g(t+x)$.
Any hints? Try to give me only ...
7
votes
2answers
96 views
Is the injection $\ell^p \subset \ell^q$ continuous for $p<q$?
It is easy to show that $\ell^p \subset \ell^q$ when $1 \leq p<q \leq + \infty$, but is the injection continuous?
If so is $\ell^{\infty}$ the direct limit $\lim\limits_{\rightarrow} \ \ell^p$ as ...
1
vote
2answers
81 views
$f_n$ $\in$ $L_2(\mu)$, the limit $ f \in L_2(\mu)$
If $f_n \in L_2(\mu)$, $f_n\rightarrow f$ almost everywhere, this is not enough to conclude $f\in L_1(\mu)$.
But is it enough to conclude whether $f\in L_2(\mu)$ or $$\lim_{n \to ...
1
vote
0answers
55 views
Under what condition two Lp spaces contain the same functions [duplicate]
Possible Duplicate:
When $L_p = L_q$?
Can anyone tell me under what condition these two spaces $L^p(\mu)$ and $L^s(\mu)$ contain the same functions?
1
vote
1answer
52 views
How to bound $L^p$ norm of a product
I am trying to show that if I can approximate two characteristic functions $\chi_A,\chi_B$ by simple functions involving only a particular set of characteristic functions, then I can approximate ...
0
votes
1answer
91 views
what is the closure of $\mathbb{Q}^\mathbb{N}$ in $l^\infty$?
I was wondering that since $l^\infty$ is not separable, which means that there is not a countable dense set in it. However the set $\mathbb{Q}^\mathbb{N}$ is countable (am I right in this?). So what ...
2
votes
2answers
57 views
Convergence of a sequence in $L^1(\mathbb{R}^3)$
All function spaces are over $\mathbb{R}^3$.
Let $u_n \in C^\infty_0$, $u_n\rightarrow u$ in $L^1$. Let $v\in L^1_\text{loc}$ be such that $uv \in L^1$. Does $u_n v \rightarrow uv$ in $L^1$?
What ...
