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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2answers
40 views

List of counter-examples to $\mathcal{L}_p(\mathbb{R})$ inclusions.

Given $1 \leq p < q \leq \infty$, it is well-known that $$\ell_p \subseteq \ell_q$$ and that $\mathcal{L}_p(\mu) \supseteq \mathcal{L}_q(\mu)$ whenever $\mu$ is finite. However $\mathcal{L}_p(\...
1
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1answer
59 views

The convolution is in $L^1$

According to my notes: The convolution is in $L^1$. Indeed $$\left| \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^n} f(y) g(x-y) dy\right) dx\right| \leq \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} |f(y) ...
0
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1answer
60 views

if $\int{}f$ is finite, then $\int{}f$ exists?

My textbook said, If $\int_E f$ exists then, of course, $-\infty\le\int_E f\le+\infty$. If $\int_E f$ exists and is finite, we say that $f$ is Lebesgue integrable, or simple integrable, on $E$ and ...
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1answer
34 views

$L_p(\mu)\subseteq L_q(\mu)$ [closed]

Given a measure space $(\Omega,\mathfrak A,\mu)$ and $1≤q≤p$, how can I show that $$L_p(\mu)\subseteq L_q(\mu)$$ if the measure $\mu$ is finite, that means $\mu(\Omega)<\infty$?
1
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1answer
38 views

Inequality of $L^p$ type

If $a\geq 1,$ $b\geq c\geq 1$ and $p>0$ then is it true that $$\frac{a+b}{\left\{\int_0^{2\pi}|e^{i\theta}+b|^pd\theta\right\}^{1/p}}\leq \frac{a+c}{\left\{\int_0^{2\pi}|e^{i\theta}+c|^pd\theta\...
0
votes
1answer
18 views

When summation of two sequences is finite, is one finite?

$|\cdot|$ is Lebesgue measure. Let $w(\alpha) := |\{x:f(x)>\alpha\}|$ Let $f$ be a nonnegative function. Then, the proof uses that $\displaystyle\sum_{k=-\infty}^{\infty} 2^{kp}w(2^k)\lt\infty \...
1
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1answer
20 views

For $E \subset \mathbb{R^n}$, is $f^p$ finite almost everywhere in $E$ if $f \in L^p(E)$?

Q1) I didn't learn $L^p(E)$ yet, only learned $L(E)$. In order to solve problems, however, I need to know that the following theorem is correct or not. Let $E \subset \mathbb{R^n}$. Then, if $f \in ...
0
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1answer
48 views

Three questions on measurable functions and $L^p$ spaces

I'm learning about measure theory and $L^P$ spaces and need help with the following questions: True or False (justify): $(1)$ Let $f:(-1, 1) \to \mathbb{R}$ measurable on $(-n, n), \; \forall ...
1
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1answer
22 views

Easy example of $f\in L_1^*\backslash L_\infty$?

If I'm not mistaken the dual of $L_1$ is $L_\infty$ whenever the measured space is $\sigma$-finite. So I know where not to look for an easy example of $f\in L_1^*\backslash L_\infty$. Does anyone know ...
0
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0answers
14 views

Definition of $L^p(C,B)$ where $C$ is a subset of the domain of the function and $B$ is a Banach space.

I am looking for the definition of $L^p(C,B)$ as declared in the title. I know that $C^s(C,B)$ is defined with the norm: $$\frac{ \| f(x) - f(y)\|_B}{|x-y|^s}\le A$$ i.e $f\in C^s(C,B)$ iff the ...
1
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1answer
74 views

In Rudin's proof of the completeness of $L^\infty$

This is closely related to a previous question: In a proof of the completeness of $L^\infty$ The following is a proof of completeness of $L^\infty$ by Rudin in his Real and Complex Analysis: Here ...
5
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2answers
93 views

In a proof of the completeness of $L^\infty$

The following is a proof of the completeness of $L^\infty$ in a lecture note by Hunter: Here are my questions: Can one (literally) replace $1/m$ in the whole proof with $\epsilon$ and replace $...
2
votes
2answers
41 views

If for $u \in L^2(\mathbb{R}^n)$ , we define $v(t)=u(x+th) $ $v: [0,1] \to \mathbb{R}$ $\Rightarrow^?$ $v \in L^2((0,1))$

I have a function $u(x) \in L^2(\mathbb{R}^n)$ ($n \geq 2$). Suppose we define another function $v$ as $$v:[0,1] \to \mathbb{R} $$ $$\quad \quad \quad \quad \quad \quad \ t \to u(x+th)$$ where $h \in ...
4
votes
1answer
48 views

What's an example of a function in $L^1(0,1)$ but not $L^p(0,1)$ for $p>1$?

What's an example of a function in $L^1(0,1)$ but not $L^p(0,1)$ for $p>1$? I've seen this answer but this is on an infinite domain. I'm interested only in $(0,1)$. I tried playing around with $\...
-2
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1answer
24 views

Convergence pw if converges in Lp space

Let $p\in[1,\infty]$ be given. If $f$ and $g$ are non-negative analytic functions such that the following holds: \begin{equation} \int_{\mathbb{R}}|\frac{\partial^ig(t)}{\partial^it}|^pdt=\int_{\...
0
votes
1answer
17 views

Riesz theorem and $L^p$ norm in expectation

I am reading a paper that uses the following fact, which claims to be from the Riesz's theorem: For a continuous stochastic process $\{ X_t \}$, let $u_t$ be its density function at each time ...
1
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1answer
27 views

Theorem 2.14 (The dual of $L^p(\Omega)$) in Lieb's Analysis book

The following pictures are Theorem 2.14 (The dual of $L^p(\Omega)$ in Lieb's Analysis book and its proof of the case $1<p<\infty$. My question is how to get the inequility (3) in the red box? ...
3
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0answers
28 views

Some sort of generalized Jensen inequality?

Let $(X, \mathcal{A},\mu)$ a measure space such that $\mu(X) > 0$ and let $f, g : X \rightarrow (0,\infty)$ be such that $f, g, f\log(f), f\log(g) \in L^1(\mu)$. Show that $$ \|f\|_1\log \|f\|_1 ...
1
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1answer
34 views

Construct an isometric isomorphism between $L_p(\mathbb{R})$ and $L_p[0;1]$

I know that $L_p(\mathbb{R})$ and $L_p[0;1], \; p<+\infty$ are isometrically isomorphic, which means that there is an isomorphism that respects norm. The question is how to construct it?
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1answer
30 views

$L^p$ Norm of product of two bounded functions

If $f$ and $g$ are bounded functions in $L^p[a,b]$, does the following inequality hold in $L_p$ spaces? $$\|fg\|_p\leq\|f\|_p\|g\|_p$$
1
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1answer
50 views

How is this inequality called? (And how to improve this process)

I am reading a book and it mentions the following: Let $u \in H^1_0(G)$; then $$\lVert u\rVert ^2_{L^\infty(G)} \le C \lVert u \rVert_{L^2(G)}\lVert u'\rVert_{L^2(G)}$$ Note: Here $G = (a,b) \subset ...
2
votes
2answers
74 views

$L_p$ space and convergence

Let $f_i\rightarrow f$ $m$-a.e on $[0,1]$, $m$ is a measure and $\int|f_i(x)|^4dm$$\le1$ for all $i$.Then $\int|f_i(x)|^2dm\rightarrow \int|f(x)|^2dm$. how to prove it? in my solution i prove that $|...
0
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0answers
37 views

prove that K is Lp- bounded operator

Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space, $1\leq p <\infty$, and suppose that $k:X\times X\rightarrow \mathbb{F}$ is an $\Omega \times \Omega$ measurable function such that for $f$ ...
4
votes
2answers
139 views

Show that $(L^{p},\|\|_{p})$ is a Banach space.

Show that $(L^{p},\|\|_{p})$ is a Banach space. My approach: I prove the statement for $(L^{1},\|\|_{1})$, of the following way, first all, is easy show that $\|\|_{1}$ is a norm. So, $(L^{1},\|\|_{...
1
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2answers
39 views

$ L^p $ and inequality

I am trying to solve the following problem in Measure Theory. I assume that I have to use Hölder's Inequality but I don't see how. Let $ E $ measurable, $m(E)<+\infty$, $1<p<+\infty$ and $ ...
2
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0answers
24 views

Embedding of Schwartz space onto L^p related spaces

So, i'm trying to prove that a Banach quasinormed space E (with its quasinorm based on $L^p$ and Weak-$L^p$ norms) is between $\mathcal{S}$ and $\mathcal{S'}$, on respect to embeddings, that is, $$ \...
3
votes
0answers
42 views

Why we care about Lp spaces? [closed]

I was reading a survey about the LHS technique when it started to talk about Lp- distance in the same distance, where usually $p$ is usually $1$, $2$ or infinite. What are the possible applications of ...
0
votes
1answer
10 views

Does monomials approximate the constant function in the sense of $L^2$?

I'm trying to show that $\{x,x^2,x^3,\dots\}$ approximate the constant function in the sense of $L^2[0,1]$-convergence; i.e. that there exists a sequence of polynomials $p_n$ with $p_n(0) = 0$ such ...
3
votes
1answer
38 views

Operator norm of an identity map over $l_p$ space

Let $1 \leq p < q \leq \infty$ ($p$ and $q$ are not related) conclude that the identity map I : $ l^n_p → l^n_q$ has operator norm exactly 1. I figured I need to show that given $\|Ix\| \leq c\|x\|...
2
votes
1answer
43 views

Convolution - Hölder inequality

I wonder if you guys can help me out with a question(not homework). I have $\phi(x)=\int_\mathbb{R} |f(t)g(x-t)|dt$ where $f \in L^1(\mathbb{R}) $ and $g \in L^p(\mathbb{R})$ and p and p' are ...
2
votes
2answers
41 views

Schauder bases of subspaces of the sequence space $\ell^p(\mathbb{N})$

Consider the canonical Schauder basis $\{e_i:i\in \mathbb{N}\}$ for $\ell^p(\mathbb{N})$, where $e_i(j)=\delta_{ij}$. Let $M$ be a subspace of $\ell^p(\mathbb{N})$. Is it right that $\{e_i:i\in \...
0
votes
1answer
49 views

$\ell_2$ convergence and $\ell_1$ norm convergence implies $\ell_1$ convergence

Let $x_n \in \ell_2$ converge to $x_\infty \in \ell_2$ and $||x_n||_1$ converge to $||x_\infty||_1$ where $||\cdot||_1$ is $\ell_1$ norm. Is it true, that $x_n$ converge to $x_\infty$ in $\ell_1$?
2
votes
1answer
30 views

Unique ground state of Schrödinger Operators

I'm reading a book and there is an argument that the ground state of a Schrödinger operator is unique. The problem is I think the argument is complete non-sense! These are lecture notes by Witten, I ...
1
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3answers
36 views

Closure of set in $C[0,1]$ with $L_1$ norm

Let $C[0,1]$ be endowed with the $L_1$ norm. I am trying to prove/disprove that $S=\{f\in C[0,1]:\;f\left(\frac{1}{2}\right)=0\}$ is closed. I am pretty sure it is, so I considered a convergent ...
2
votes
1answer
43 views

Must the image of unit ball be bounded?

I am going through a series of true or false questions, one of them is: If $\phi:U\to V$ is a linear, $U,V$ are normed vector spaces, must $\text{im}\,\phi\big|_S$ be bounded, where$S=\{x\in U:\;|...
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1answer
24 views

$\text{supp}(f) = \text{supp} (g) , \|f\|_{L^{2}(\mathbb R)} \leq \|g\|_{L^{2}(\mathbb R)} \implies \|f\|_{L^{2}([0,1])} \leq \|g\|_{L^{2}([0,1])}$?

Let $f,g \in L^{2}(\mathbb R)$. Suppose that $\int_{\mathbb R} |f(t)|^{2} dt \leq \int_{\mathbb R} |g(t)|^{2} dt.$ We also assume that support of $f$ and support of $g$ are equal. My Question is:...
2
votes
0answers
26 views

How fast can Sobolev functions grow?

It is a simple fact that $L^p$-functions cannot grow arbitrarily fast. More precisely, one has for every $\ell>0$ $$ |\{f\geq\ell\}|\leq \frac{\|f\|_{L^p}^p}{\ell^p} $$ for every $f\in L^p$. My ...
2
votes
2answers
43 views

Weak and strong convergence in $L^p$

Another practice qual question: Let $X = [-\pi,\pi]$ and consider the Lebesgue measure. Let $p$ be a real number with $1 \leq p < \infty$. Define for each integer $k \geq 1$ that $f_k(x) = \sin(...
0
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0answers
23 views

Finding the adjoint of the left translations semigroup on $L^p (\Bbb R)$

If $t \mapsto T_l (t)$ is the left translation operator by $t$ on $L^p (\Bbb R)$ given by $\Big( T_l (t) (f) \Big) (s) = f (t + s)$, find the adjoint of the left translations semigroup. Note that on $...
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0answers
18 views

Given an operator $Q$ between a Hilbert space $U$ and $L^2(ℝ^d;ℝ^d)$, is it possible to make sense of $U∋u↦(Qu)(x)$ for a fixed $x∈ℝ^d$?

Let $U$ be a Hilbert space $H:=L^2(\mathbb R^d;\mathbb R^d)$ for some $d\in\left\{2,3\right\}$ $Q$ be a Hilbert-Schmidt operator from $U$ to $H$. I want that $\tilde Q(x)$, where $$\tilde Q(x):=...
3
votes
1answer
43 views

The lack of uniform continuity of shift operator on $L^2$

When studying $c_0-$semigroups, I came accross a statement that if we define shift operator $(S(t)f)(x) = f(x+t)$ for $t>0$ on $L^2(\mathbb{R})$, then $S(t)$ forms a $c_0-$semigroup (that's easy) ...
2
votes
1answer
16 views

How to lift the restriction of sigma-finite by transfinite induction in proving linear functionals separate of $L^\infty(\Omega)$?

The picture below is Theorem 2.10 (Linear functionals separate of $L^p$) in page 56 of Lieb's "Analysis" book. Question: How could I understand that the restriction of sigma-finite can be lifted by ...
1
vote
1answer
30 views

Are $L^p$ topologies compatible?

Consider the subspace topologies on $L^p\cap L^q$ be induced by $L^p$ and $L^q$ respectively. Then, are these subspace topologies compatible? Moreover, I'm curious about the special case $L^1(\mathbb{...
2
votes
1answer
45 views

$Tf = xf(x)$ is not compact in $L^2([0,1])$

I want to prove, in a rather elementary way, that $Tf = xf(x)$ is not compact in $L^2([0,1])$. I cannot find the appropriate bounded sequence whose image has no Cauchy sub-sequences. I have tried ...
4
votes
1answer
107 views

Show that for any $1<p<\infty$ the set $\{ f \in L^p(\mathbb{R}) \cap L^1(\mathbb{R})\}$ where $ \int_{\mathbb{R}} f=0$ is dense in $L^p(\mathbb{R})$. [closed]

Show that for any $1<p<\infty$ the set $\{ f \in L^p(\mathbb{R}) \cap L^1(\mathbb{R})\}$ where $ \int_{\mathbb{R}} f=0$ is dense in $L^p(\mathbb{R})$. Is the statement true if $\mathbb{R}$ is ...
3
votes
3answers
35 views

Show that $A$ and $A^C$ are both dense in $(\ell^2,\lVert \cdot \rVert_2)$, where $A=\{x\in\ell_2:\sum_{k=1}^\infty x_k\neq0\}$.

The title says it all. Showing $A$ is dense in $\ell_2$ seems easy; for any $x\notin A$, for each $n\in\mathbb N$ let $x^n$ in $\ell_2$ where $x^n$ is identical to $x$ except that $x^n_1=x_1 + \frac1n$...
0
votes
1answer
18 views

some detail calculation on the proof of equivalence of norms

We say that two norm $\|x\|_1$ and $\|x\|_2$ on a vector space $X$ are said to be equivalent if there exists $K>0$ and $M>0$ such that $$ K\|x\|_1\le \|x\|_2\le M\|x\|_1 $$ Prove that on a ...
4
votes
1answer
43 views

Prove that following are true for $\phi \in C_c^{\infty}(\mathbb{R}), \phi \ne 0$

Fix a function $ \phi \in C_c^{\infty}(\mathbb{R}), \phi \ne 0$ and set $u_n(x)=\phi(x+n)$. Let $1 \le p \le \infty$. Then Check that $u_n$ is bounded in $W^{1,p}(\mathbb{R})$ Prove that there ...
7
votes
2answers
293 views

Prove series converge for almost every $x$

Let $f\in L^p(\mathbb{R})$, $1<p<\infty$, and let $\alpha>1-\frac{1}{p}$. Show that the series $$\sum_{n=1}^{\infty}\int_n^{n+n^{-\alpha}} |f(x+y)|dy$$ converges for a.e. $x\in \mathbb{R}$. ...
3
votes
0answers
33 views

Lebesgue-integrability of derivatives

Let $f:\mathbb R\to\mathbb [0,\infty)$ be a non-negative, twice-differentiable function. Suppose that $\int_{-\infty}^{\infty}f(x)\,\mathrm dx<\infty$, $\int_{-\infty}^{\infty}|f''(x)|\,\mathrm ...