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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1answer
49 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
43 views

Parseval's identity holds

Theorem: If $u \in L^2(\mathbb{R}^n)$ then the fourier transform $\widehat{u} \in S'(\mathbb{R}^n)$ is a $L^2(\mathbb{R}^n)$ function and the Parseval's identity holds: ...
9
votes
1answer
2k views

Proof of separability of $L^p$ spaces

The following is a proof in Brezis book. It shows the separability of $L^{p}$ spaces: I have a few questions regarding the proof: It says 'it is easy to construct a function $f_{2} \in ...
1
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2answers
24 views

Let $f\in L^p(0,1)$ and define $f_h$

Let $f\in L^p(0,1)$ ($1\leq p<\infty$) and define $f_h$ as $$f_h(x)=\begin{cases}f(x+h)&\text{ for } x+h\in [0,1]\\ 0 &\text{ for } x+h\not\in[0,1]\end{cases}$$ Prove that for all ...
1
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2answers
42 views

Inclusion of Schwartz space on $L^p$

I'm looking for a proof of $\mathcal{S}(\mathbb{R}) \subset L^p(\mathbb{R})$ for $1 \leq p \leq \infty$. My informal probe follow like this: For any function $f \in L^p(\mathbb{R})$ exists a ...
0
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1answer
34 views

Is the sequence of functions $g_n=ng_1(nx)$ a Cauchy sequence?

Given a function $g_1(x) \in \mathcal L^2(\Bbb R)$ that satisfies: $$\int_{-\infty}^{\infty}dx \space g_1(x)=1$$ one can define a sequence of functions $g_n=ng_1(nx)$. Does $g_n(x)$ define a ...
0
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2answers
32 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 ...
0
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0answers
23 views

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: $(1)$ True or False (justify): If $f : \mathbb R \to \mathbb{R}$ is measurable on $(-n, n), \, ...
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1answer
22 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} ...
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
29 views

Is $\mathscr{C}(K)$ a subspace of every $L^p(K)$, if $K$ is compact?

Considering a function $f\in\mathscr({C}(K),\lvert\lvert \cdot\rvert\rvert_\infty)$, i.e.$$f:K\rightarrow\mathbb{C}$$ everywhere continuous where $K$ is a compact subset of $\mathbb{R}^n$, does $f$ ...
49
votes
2answers
16k views

$L^p$ and $L^q$ space inclusion

Let $(X, \mathcal B, m)$ be a measure space. For $1 \leq p < q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample ...
0
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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
29 views

$L_p(\mu)\subseteq L_q(\mu)$ [on hold]

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
33 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 ...
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 ...
4
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2answers
134 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, ...
5
votes
2answers
544 views

How do I prove the completeness of $\ell^p$?

Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct: Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ ...
0
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1answer
28 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 ...
20
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2answers
7k views

Why is $L^{\infty}$ not separable?

$l^p (1≤p<{\infty})$ and $L^p (1≤p<∞)$ are separable spaces. What on earth has changed when the value of $p$ turns from a finite number to ${\infty}$? Our teacher gave us some hints that ...
1
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1answer
21 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 ...
5
votes
2answers
88 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 ...
1
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1answer
70 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 ...
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 ...
0
votes
1answer
27 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? $$\|fg\|_p\leq\|f\|_p\|g\|_p$$
2
votes
2answers
39 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 ...
5
votes
1answer
93 views

Integral operator is bounded on $L^p$ if it maps $L^p$ to itself

Here is a homework excercise. Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space,$1\leq p <\infty.$ and suppose that $k:X\times X\rightarrow \mathbb{C}$ is an $\Omega \times \Omega$ ...
5
votes
2answers
63 views

Weak convergence in different $L^p$ spaces

Consider $p \ge \alpha \ge 1.$ If a sequence converges weakly in $L^p,$ say $u_n \rightharpoonup u$, is it true that: $$u_n^{\alpha} \rightharpoonup u^{\alpha} \text{ in $L^{p/ \alpha}$}$$ This ...
4
votes
1answer
40 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 ...
0
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1answer
12 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
26 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? ...
0
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2answers
65 views

Real Analysis , Folland Problem 6.1.5

Problem 6.1.5 - Suppose $0 < p < q < \infty$. Then $L^p \not\subset L^q$ if and only if $X$ contains sets of arbitrary small positive measure, and $L^q\not\subset L^p$ if and only if $X$ ...
3
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0answers
26 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 ...
1
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1answer
30 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?
2
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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 ...
1
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1answer
48 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 ...
0
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0answers
33 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 $f$ ...
1
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2answers
33 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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1answer
42 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
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0answers
22 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
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0answers
41 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
9 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 ...
2
votes
2answers
40 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) = ...
3
votes
1answer
28 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 ...
0
votes
0answers
22 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 ...
2
votes
2answers
39 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
48 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
28 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 ...
2
votes
1answer
42 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 ...
1
vote
3answers
33 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 ...