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
39 views

Convergences of $ f_n \in L^p$

I'm trying to solve the following task: Let $(\Omega,\mathfrak{A},\mu)$ be a measure space, and $f,f_1,f_2,\dots \in L^P(\Omega,\mathfrak{A},\mu)$ with $p\in [1,\infty[$. Ssuppose that, as ...
0
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1answer
18 views

Problem about Fourier series and $L^p$ spaces

Need some help with this problems: Is there $f \in C(\mathbb{T})$ such that $\hat{f}(k) = \dfrac{1}{|k|^{1/2}}$, if $k \neq 0$? Suppose the $f_n \in L^1(\mathbb{T})$, $n = 1,2,...$ and $\| ...
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1answer
36 views

Show that the space of functions in $L^\infty (E)$ which admit a continuous representative is closed in $L^\infty (E)$

Let $E \subset \Bbb{R}^n$ be a set of positive measure. Let $\mathcal{C}$ be the set of measurable functions $f$ such that there exist a continuous $g$ with $f=g$ a.e. in $E \subset \mathbb{R}^n$. ...
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1answer
41 views

$u\in L^2(\Omega)$ does this imply that $u^p\in L^2(\Omega)$?

Suppose that a function $u:\Omega \rightarrow \mathbb{R}^n$ is such that $u \in L^2(\Omega)$. Does this imply that $u^p \in L^2(\Omega)$? if not can you give a counterexample? Here $\Omega$ is an ...
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1answer
35 views

Cauchy sequence in $L^1$ Space

I am learning about the $L^1$ space (the complete Riemann integrable functions) and I am not used to using $\epsilon, \delta$ in these type of problems yet. Here is my attempt. Below I want to that ...
3
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1answer
96 views

Why is this set compact in $L^2(\mathbb{N})$?

Suppose $L^{2}(\mathbb{N})$ is the Hilbert space of sequences $(a_{n})_{n \in \mathbb N}$ which satisfy $\sum |a_{n}|^{2}$ with $(a,b) = \sum a_{n} \bar{b_{n}}.$ Prove the set of sequences ...
2
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0answers
35 views

The sequence of functions $f_n=\frac{n}{1+n\sqrt{x}}$ in $L^1(0,1)$ and $L^2(0,1)$

During a lecture on $L^p$ spaces, the lecturer made a few comments about the sequence of functions $f_n=\frac{n}{1+n\sqrt{x}}$ that I am not sure I fully understand. 1) First he said $f_n\in ...
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1answer
18 views

A bounded family of functions in $L^p[E]$, where E is a measurable set, is uniformly integrable.

A corollary in Royden & Fitzpatrick's Real Analysis (chapter 7 section 2) reads: Let $E$ a measurable set, and $1<p<\infty$. Suppose $F$ is a family of functions in $L^p(E)$ that is bounded ...
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1answer
24 views

Show that if $f$ is a bounded function on $E$ and $f\in L^{p_1}$ then $f\in L^{p_2}$ for any $p_2>p_1$. [duplicate]

I'm working through Royden & Fitzpatrick's Real Analysis, and one of the questions in the introductory chapters of $L^p$ spaces reads: Show that if $f$ is a bounded function on $E$ and $f\in ...
1
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1answer
35 views

Interchange of $\ell^r$ and $L^p$-norm

Let $(f_i)_{i\in\mathbb{N}}$ be a sequence of $L^p$-functions. What is the relation between $\Vert \Vert (f_i)_{i\in\mathbb{N}}\Vert_{\ell^r}\Vert_{L^p}$ and $\Vert \left(\Vert ...
1
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0answers
20 views

Kernel and cokernel of certain applications in $L^p$ spaces.

Let $X$,$Y$ two finite-dimensional closed linear subspaces of $L^p := (L^p)^n$ (defined in a finite measure space). Define $$L^p_X = \{ f \in L^p : \int \langle f,x \rangle = 0 \quad \forall x \in X ...
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0answers
38 views

$L^p$ space and continuous injection

Let $1\leq p < r < q \leq \infty$ and $E\in \mathbb{R}$. Define $$A = L^p(E) + L^q(E) = \{f=g+h:g\in L^p(E), h\in L^q(E) \}$$ and $$\|f\|_A = \inf_{f=g+h} \|g\|_p+\|h\|_q$$ where the infimum ...
1
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1answer
30 views

Why $L^p$ is strictly contained in $L^{p,\infty}$

I'm reading up on weak $L^p$ spaces (a.k.a. Marcinkiewicz spaces, or $L^{p,\infty}$ spaces), and I have a little trouble seeing why the function $|x|^{-n/p}$ lies in $L^{p,\infty}(\mathbb{R}^n)$ but ...
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2answers
12 views

Proving equivalence of operators imply equivalence of measures

Let $A:L^2([0,1],\mu)\to L^2([0,1],\nu)$ an unitary operator. Prove that $$d\mu=\rho(x) d\nu$$ for some $L^1(\mu) \ni \rho(x) >0 (\mu\text{ a.e})$ I thought maybe saying ...
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1answer
47 views

Relation between Besov and Sobolev spaces (Littlewood-Paley-theory)

For the Sobolev-norm there holds $\Vert f\Vert_{W^{s,p}(\mathbb{R}^n)}\sim_{n,p,s}\left\Vert\left(\sum_{k\in\mathbb{N}_0}\left\vert (1+2^k)^sP_kf(x)\right\vert^2\right)^\frac{1}{2}\right\Vert_{L^p}$ ...
-1
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1answer
82 views

Nowhere dense subset of $L^1$

Why is $B_n = \{f \in L^1 : \int |f|^2 < n \}$, $n \in \mathbb{N}$ a nowhere dense subset of $L^1$? Please provide a proof without assuming that $L^2 \subsetneq L^1$. Clarification: $L^p$ here ...
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0answers
34 views

$f_n \to f$ strongly in $L^2(\mathbb{R}$)? [duplicate]

Let $f_n \rightharpoonup f$ in $L^2(\mathbb{R})$ and $\|f_n\|_2 \to \|f\|_2$ as $n \to \infty$. Do we have that $f_n \to f$ strongly in $L^2(\mathbb{R})$?
1
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1answer
19 views

Minimize the uniform ($L^\infty$) distance to the space of functions with zero integral

Let $g\in C^0[0,1]$. Minimize $||f-g||_{\infty}$ for all $f\in L^\infty [0,1]$ such that $\int_0^1 fdx = 0$. Considering $|\int_0^1 g-f dx| \ge |\int_0^1 gdx|$ by the requirement on $f$. In the ...
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1answer
13 views

Are Morrey spaces reflexive?

Since $L^{p,0}=L^p$ and $L^1$ is not reflexive, thus in general Morrey space is not reflexive, but how about for $L^{p,\lambda}$ with $1<p<+\infty$ and $0<\lambda<n$, where $n$ is the ...
1
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2answers
22 views

Show that the set of functions in $L^2[0,1]$ with a zero integral on $[0,1]$ is a closed vector subspace of $L^2[0,1]$.

Let $H = L^2([0, 1])$ and let $K \subset H$ be defined as $K = \{f \in H \, : \, \int_{[0,1]} f \, \mathrm{d}m = 0\}$. Show that $K$ is a closed vector subspace of $H$. Find the element of $K$ that ...
2
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1answer
41 views

Give an example of a function which is in $L^2 (\mathbb{R})$ but not in $L^p(\mathbb{R})$ for any $p \in [1, 2) \cup (2, \infty]$.

This question was on a problem set regarding $L^p$ spaces in an undergraduate-level real analysis course. I actually used an answer on StackExchange to help me provide an example, but I couldn't ...
0
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1answer
27 views

If $f_n\to f$ in the $L^1$ norm, show that there is a subsequence $f_{n_k}$ which converges a.e. to $f$.

This question is from a problem set on $L^p$ spaces in my undergraduate-level real analysis course. I said that $f_n$ converges if and only if it is Cauchy. Therefore, $\exists \, N\in\mathbb{N} \; ...
1
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1answer
27 views

Example of $\{f_n\}$ which converges to $f(x)\equiv0$ in $L^ 1$ , but so that $f_n(x)\not\to0$ in $[0, 1]$.

Give an example of a sequence of functions $\{f_n\}$ which converges to the constant zero function in $L^1$, but so that $f_n(x)$ does not converge to zero at any point of $[0, 1]$. This is from ...
3
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1answer
44 views

$L^p$ convergence of a bounded sequence which converges almost everywhere

I'm having a little trouble with this homework problem: Suppose $\mu(X)<\infty$, $f_n\in L^1$, $f_n\to f$ a.e., and there exists $p>1$ and a constant $C>0$ such that $$\|f_n\|_p\leq C$$ ...
1
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2answers
30 views

Does $\{\sin (nx)\}_1^\infty$ converge in the $L^1$ norm on $[0,2\pi]$?

This is a homework question from a problem set in an undergraduate-level real analysis course (coming from merely an intro to analysis course) about $L^p$ spaces. Show that $\{\sin ...
0
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1answer
42 views

How can I find a function that is in $L^1(\mathbb{R})$ with its derivative also but its limit tends to zero?

I am trying to find for a function that would full-fill these conditions below: $$f \in L^1(\mathbb{R})$$ $$f' \in L^1(\mathbb{R})$$ but its $\lim_{t \to \infty}=0$. I've tried to find a function ...
2
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1answer
37 views

When is the space $L^\infty(\mu)$ finite-dimensional?

There is a theorem that for a given $p\in [1,\infty)$ a space $L^p(\mu)$ is finite dimensional iff the set of values of $\mu$ is finite. Is a similar theorem for the space $L^\infty(\mu)$ for ...
0
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1answer
27 views

Closure of subset of $\ell^\infty$

I am doing some assignment and I am trying to understand the nature of the closure of $A\subset \ell^\infty$ as I am having trouble exactly getting an image of how the elements in it is. Our set A ...
5
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1answer
46 views

Certain set is dense in $l^p$ if and only if $\{x_n : n \in \mathbb{N}\} \notin l^q$, where $1/p + 1/q = 1$

Assume that $\{x_n : n \in \mathbb{N}\} \subset \mathbb{R}$ is such that $x_n \neq 0$ for some $n$. Let $p \in (1, \infty)$ and$$G := \left\{\{y_n : n \in \mathbb{N}\} \in l^p : \lim_{N \to \infty} ...
13
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1answer
109 views

Does there exist $f \in L^1(\mathbb{R})$ where $\lim_{r \to 0} {1\over{r}} \int_{x-r}^{x+r} f(y)\,dy = \infty$?

If $E \subset \mathbb{R}$ has measure $0$, does there exist $f \in L^1(\mathbb{R})$ such that, for every $x \in E$, $$\lim_{r \to 0} {1\over{r}} \int_{x-r}^{x+r} f(y)\,dy = \infty?$$ What if $E$ has ...
2
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1answer
52 views

Partial converse of the fact: $f\in L^p , g\in L^q \Rightarrow fg\in L^1$

Hölder's inequality says If $p^{-1}+q^{-1}=1$ and $ f\in L^p, g\in L^q$, then $fg\in L^1$. Then how about the following converse: Let $g$ be measurable. If for all $f\in L^p$ we have $fg \in ...
4
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1answer
41 views

$f \in L^p([0, 1])$, for every $1 \le r < p$, we have $\|f_n - f\|_r \to 0$ as $n \to \infty$.

Let $f_n \in L^p([0, 1])$ with $p > 1$. Assume that $\|f_n\|_p \le 1$ and, as $n \to \infty$, $f_n \to f$ almost everywhere on $[0, 1]$. How do I see the following? We have $f \in L^p([0, 1])$. ...
1
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1answer
24 views

Representing a bounded linear functional on $L^p$

I have a question on this problem, and note it is for homework. Let $p,q$ be conjugate exponents. Define $H\colon L^p([0,1])\to \mathbb{R}$ by $$ Hf=2\int_0^1\left(\int_x^1 f(y)\,dy\right) ...
5
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2answers
30 views

$L^2([-1, 1])$, we have $\lim_{j \to \infty} f_j(x) = 1$ for a.e. $x \in [-1, 1]$?

Let$$\{f_j : j \in \mathbb{N}\} \subset L^2([-1, 1])$$be such that$$f_j \ge 0,\,\text{ }\|f_j \|_{L^1([-1, 1])} = 2,\,\text{ }\left|\|f_j\|_{L^2([-1, 1])} - \sqrt{2}\right| \le 2^{-j}.$$How do I see ...
6
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1answer
68 views

For every $f \in L^1(\mathbb{R})$, do we have $\sup_{n \in \mathbb{N}}|T_nf(x)| < \infty$ for a.e. $x$?

Let $K: \mathbb{R} \to \mathbb{R}$ be given by $K(x) := \dfrac {\sqrt {|x|}} {1 + x^2}$ and for $n \in \mathbb{N}$, $K_n(x) := nK(nx)$. For $f \in L^1(\mathbb{R})$ define $T_n (f)(x) := \int \limits ...
3
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0answers
43 views

Is there a way to find the operator norm in this case?

If $k:[a,b]\times[a,b]\rightarrow \mathbb{R}$ is in $L^2([a,b]\times[a,b])$, we can show that the linear operator: $T_k: L[a,b]\rightarrow L[a,b]$, given by ...
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2answers
57 views

Why is this inclusion map continuous?

If $1\le p<q<r\le\infty$, then I want to conclude that the inclusion map $L^p\cap L^r\to L^q$ is continuous, But I have seen this reference here, but I think that I am not understanding the ...
0
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3answers
31 views

Normed Linear Space ,$p \neq 2$ is $\left \| f\right \|_{p}= \sqrt{f,f}$ for each $ f \in L^P([0,1])$?

For $p \neq 2$, is there an inner product $< ., .>$ on $L^P([0,1])$ such that $\left \| f\right \|_{p}= \sqrt{f,f}$ for each $ f \in L^P([0,1])$? is it true?for P=2 norm is induced by inner ...
0
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0answers
40 views

prove Holders and Minkowski inequalities in $\ell ^p$

For $1 \leq p < \infty$ and a sequence $a=(a_1,a_2.......) \in \ell ^p$ define $T_a $to be the function on the interval $[1, \infty )$ that takes value $a_k $on [k,k+1) for k=1,2.... show that ...
1
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0answers
29 views

$f_n\to f$ almost everywhere and $\|f_n\|_{L^2}\to \|f\|_{L^2}$ implies $\|f_n-f\|_{L^2}\to 0$ [duplicate]

Let $f_n\in L^2(\mathbf{R})$ be a sequence of square integrable functions, $f_n\to f$ almost everywhere, and $\|f_n\|_{L^2}\to \|f\|_{L^2}$. Prove that $\|f_n-f\|_{L^2}\to 0$. I want to use the ...
2
votes
0answers
18 views

Does the norm in $L^p$ has continuity to p? [duplicate]

Let $\Omega$ be any measurable ($\sigma$-finite, if necessary) space. let $1\leq p\leq\infty,x\in L^p(\Omega)$. Then how to proof $\forall\varepsilon>0,\exists\delta>0$,so that for any $1\leq ...
2
votes
1answer
66 views

There is no bounded linear surjection between $\ell_p$ spaces

For $1\leq p,q<\infty$, $p\ne q$, how to prove that there is no bounded linear operator $T:\ell_p\to \ell_q$ such that $T$ is surjective? I've tried to use Pitt's theorem, but without success.
2
votes
2answers
53 views

If $f \in L^{p_1}(E) $ is bounded then $f\in L^{p_2}$ for any $p_2>p_1$

Show that if $f$ is a bounded function on $E$ that belongs to $L^{p_1}(E)$ then it belongs to $L^{p_2}(E)$ for any $p_2>p_1$ How can I insert argument about the boundedness of $f$? I can prove ...
1
vote
1answer
110 views

when composition of continuous and Lebesgue integrable function Lebesgue integrable

Suppose $g:[a,b]\to\mathbb R$ is Lebesgue-integrable and $f:\mathbb R\to\mathbb R$ is continuous, then $f\circ g$ Lebesgue-integrable if $|f(x)|<a+b|x|$ for constants $a$ and $b$. How to prove if ...
2
votes
2answers
20 views

$\nabla \sqrt{\rho} \in L^2(\mathbb{R}^3) \implies \rho \in L^3(\mathbb{R}^3)$

I found this in the INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL XXIV, 250 (1983) inside the paper of Elliot H. Lieb with the title Density Functionals for ...
0
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1answer
11 views

{$g_n$}$_{n \in\mathbb{N}}$ converges to $g$ in $L^q(\mathbb{R^n})$ then {$A_f(g_n)$}$_{n \in \mathbb{N}}$ in $\mathbb{C}$ converges to $A_f(g)$.

Let $1<p,q<\infty$ with $\dfrac{1}{p}+\dfrac{1}{q}=1$ and $f \in L^p(\mathbb{R^n})$. Define the functional $A_f: L^q(\mathbb{R^n}) \to \mathbb{C}$ by: $$A_f(g)=\int_{\mathbb{R^n}}f(x)g(x)dx$$ ...
4
votes
3answers
94 views

Function in $L^p$ but not in $L^{\infty}$

Show that if $$f(x) = \ln\left({1\over x}\right),\quad 0<x\le 1$$ then $f\in L^p((0, 1])$ for all $1 \le p < \infty$ but $f\not\in L^{\infty}((0, 1])$. Intuitively I know I have to show that ...
2
votes
1answer
46 views

Convergence in $L^p$ spaces

Let $f_{n} \subseteq L^{p}(X, \mu)$, $1 < p < \infty$, which converge almost everywhere to a function $f$ in $L^{p}(X, \mu)$ and suppose that there is a constant $M$ such that ...
0
votes
1answer
28 views

Injectivity of Fourier transform between $L^1(\mathbb{R})$ and $C_0(\mathbb{R})$

The Fourier transform maps from $L^1(\mathbb{R})$ to $C_0(\mathbb{R})$ where $C_0(\mathbb{R})$ is all continuous functions that vanish as $x \rightarrow \infty$. Now given $f,g \in L^1(\mathbb{R})$, ...
3
votes
1answer
20 views

Conditional expectation of an $L^p$ random variable

If $Y$, $X$ and $X_1,X_2,...$ are real random variables such that $|X_n|\leq|X|$ for every $n$ and both $X$ and $Y$ are in $L^p$ $(E[|X|^p] < \infty$ and $E[|Y|^p] < \infty)$ for some $p \geq ...