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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15 views

Function in Lp Space [duplicate]

Assume that $1<p<∞ $, f is absolutely continuous on $[a,b]$, $f′$ ∈ $L^{p}$ and $a=\frac{1}{q}$, where $q$ is the exponent conjugate to $p$. Prove that $f ∈ Lip$ $a$. I was thinking of using ...
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1answer
38 views

Convergence of sequence of function in norm.

Let $1\leq p<\infty$. Suppose that $\{f_k\}$ is a sequence in $L^p(X,\mathcal{M},\mu)$ such that the limit $f(x)=\lim_{k \to \infty}f_k(x)$ exists for $\mu$-a.e. $x\in X$. Asumme that ...
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3answers
78 views

If $f$ is in $L^{p}$, prove that $\lim \int_{x}^{x+1} f(t) dt = 0$

If $f$ is in $L^p$, prove that $\lim_{x \to \infty} \int_{x}^{x+1} f(t) dt = 0$ It is easy to think that integration must be vanish as $x \to \infty $ but I cannot write them with math. Suppose ...
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1answer
36 views

Sequence does not converge in $L^{p_0}$ but converges in $L^p\ \forall 1\le p<p_0$

Let $1<p_0<\infty$ Find a sequence $\{f_k\}$ such that $f_k \in L^p$ for $1 \leq p < \infty,$ $f_k \rightarrow 0$ in $L^p$ for $1 \leq p <p_0$, but $f_k$ does not converge in $L^{p_0}$ ...
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2answers
75 views

$\mathcal{L}_2$ continuous functions with $f(0)=\alpha$ are dense in $\mathcal{L}_2 [-1,1]$

Let $X=\mathcal{L}_2 [-1,1]$ and for any scalar $\alpha$ we define $E_\alpha=\{f\in \mathcal{L}: f \text{ continuous in } [-1,1] \text{ and } f(0)=\alpha \}$. Prove $E_\alpha$ is convex for any ...
3
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3answers
58 views

Proving that $x,y \in \ell^2(\Bbb N) \implies x+y \in \ell^2(\Bbb N)$.

I want to prove that $x,y \in \ell^2(\Bbb N) \implies x+y \in \ell^2(\Bbb N)$. I'm damn sure that there is a quick way to do this, but I'm not seeing it. I am capable of proving Young, Hölder and ...
2
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2answers
58 views

Convergence in $L^p$ spaces.

Prove that for all integrable functions $g_n, g$, we have the implication $\|g_n-g\|_1\to 0\Rightarrow \|g_n\|_1\to \|g\|_1$ as $n\to \infty$. Is the converse true? It seems like $|g_n-g|_1 \to 0$ ...
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1answer
34 views

Is the Cesaro Operator normal?

The Cesàro operator $T:ℓ_p→ℓ_p$ is defined by $$(Tx)_k=(1/k)\sum_{j=1}^k x_j$$ where $x=(x_j)$. Is this operator normal?
4
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1answer
57 views

Dual space of $L^{\infty}$ - Where is the mistake?

Today I thought about this for the first time and I really cannot see what is going on. I think it is a very stupid question but I really cannot see it. Consider the space $L^{\infty}(\mathbb{R})$ ...
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0answers
15 views

$L_+^p(X,\mu)$ is a closed and convexe subset of $L^p(X,\mu)$.

I have a problem with an exercise: Let $(X,A,\mu)$ a measure place, $p\in[1,\infty)$ and $\mu(X)<\infty$.Prove that the set $$L_+^p(X,\mu):=\{f\in L^p(X,\mu):f(x)\geq 0\ \mu-\text{a.e.}\}$$ is a ...
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1answer
70 views

Prove $\sin(kx) \rightharpoonup 0$ as $k \to \infty$ in $L^2(0,1)$ [duplicate]

I want to show that $u_k(x)= \sin(kx) \rightharpoonup 0$ as $k \to \infty$ in $L^2(0,1)$. We know trivially that $0 \in L^2(0,1)$. I need to show that $\langle u^*,\sin(kx) \rangle \to \langle ...
3
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0answers
38 views

Properties of $L^{\infty}$

I'm trying to get a better grasp on the idea of $L^{\infty}$. What are the implications if we are given that $f \in L^{\infty}$? Also, how do we write $\|f\|_{\infty}$ in terms of the inf of a set of ...
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1answer
44 views

Set derived from definition of $\Vert f \Vert_\infty$

Someone told me that the set $B_n := \{x \in X : \vert f(x) \vert > \Vert f \Vert_\infty - \frac{1}{n}\}$ for $n \in \mathbb{N}$ (where $B_n$ has finite positive measure), is derived from the ...
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0answers
16 views

Step of proof Hardy's Inequality [duplicate]

I trying to prove the Hardy's Inequality, by Evans book. I need of a little help in the step: If $u\in L^1(B(0,r))$ satisfying $$(2-n)\int_{B(0,r)}\frac{u^2}{|x|^2}dx=2\int_{B(0,r)}u ...
3
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1answer
35 views

Convolution product, Lp spaces

I wonder how to prove the following statement, Let p,q be real numbers s.t $1\leq p \leq\infty$, $1\leq q \leq\infty$ and $ \frac{1}{p}+ \frac{1}{q}=1$ Let $f \in L^p(\mathbb R^n)$ and $g \in ...
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1answer
34 views

About the definition of Sobolev Spaces

I'm studying Sobolev Space and I have a question about the definition: Def.: The Sobolev Space $W^{k,p}(U)$ consists of all locally summable functions $u:U\to \mathbb{R}$ such that for each ...
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1answer
25 views

Are continuous functions dense in $L^1$?

It is a well known fact that the continuous compactly supported functions are dense in $L^1(\mathbb R)$. An immediate counterexample to this fact for a non locally compact space is $\mathbb R ...
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2answers
50 views

$L^p$ space corollary

I'm confused about the proof for this theorem: let $E$ be a measurable set s.t. $mE<\infty$ and $1 \leq p_1 < p_2 \leq \infty$. Then, $L^{p_2}(E) \subset L^{p_1}(E)$. Also, $\|f\|_{p_1} \leq ...
3
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1answer
45 views

Definition of functions in $L^p$ space

I know that if we suppose that $1 \leq p \leq \infty$, and if $f$ is in $L^p$, then this means that $\|f\|_p=[\int_X (f^p) dx]^{\frac{1}{p}}$. But I feel as though I'm missing some important ...
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2answers
37 views

$L^1$ functions at $\infty$ to $\infty$

I am having trouble with a practice prelim question: If $f \in L^1(\mathbb{R})$ then $\lim_{n \rightarrow \infty} \int_n^{\infty} f(x)dx = 0$ I know that $f$ is bounded, but I am not if I should add ...
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1answer
23 views

$f_n$ converges in duality with $C_b$ and is uniformly bounded then it converges in duality with $L^1$

Let $(X,m)$ be a metric measure space, $(f_n)_n$ a sequence in $L^\infty, f \in L^\infty$ s.t. $$ \int gf_n \ dm \rightarrow \int g f $$ for every $g : X \rightarrow \mathbb R$ continuous and bounded. ...
4
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1answer
95 views

Converge of a sequence in $L^p(\mathbb{R}^3)$

Let $f(x)\in L^p(\mathbb{R}^3)$ for every $p\in [1, \infty]$. Let $B(n)\subset \mathbb{R}^3$ be the ball of radius $n$ centered at the origin. I want to show that the sequence ...
2
votes
2answers
46 views

Compactness and (global) convergence in measure

Let $B$ denote the unit ball of $L^\infty$. Question: is $B$ sequentially compact for the topology of convergence in measure ? I am not necessarily assuming that the measure is finite (but $\sigma$ ...
4
votes
1answer
36 views

Defining a bounded operator on $l^p$

Let $(c_{jk})_{j,k \in \mathbb{N}} \subset \mathbb{C}$ be such that $a:=\sup_{k \in \mathbb{N}} \sum_{j \in \mathbb{N}}|c_{jk}|<\infty$ and $b:=\sup_{j \in \mathbb{N}} \sum_{k \in ...
2
votes
1answer
110 views

Showing an integral is in $L^1$

Let $0<a<1$ and $f\in L^1([0,1])$. Show $g(x)=\int_0 ^x\frac{1}{(x-t)^a}f(t)dt$ exists a.e. in $[0,1]$ and $g\in L^1([0,1])$. Using Fubini, $$\int_0 ^1 \vert g(x) \vert dx=\int_0 ^1 \int_0 ...
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1answer
28 views

Is the Fourier transform a conformal map on $L^{2}$?

I read that a conformal map is one that preserves the angles. I know nothing more about conformal maps. I don't know where to find a generalized definition of a conformal map, but I guess that if ...
3
votes
1answer
47 views

Showing a sum of $\vert f(x+k)\vert $ belongs to $L^{\infty}$ if $f,f'\in L^1$

I am working on this Suppose that $f,f'\in L^1(\mathbb{R})$. Then $\sum_{k= 0} ^{\infty}\vert f(x+k)\vert\in L^{\infty}([a,b])$ for any $a,b\in \mathbb{R}.$ Idea: Let $i$ be any integer. $\int_i ...
2
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1answer
35 views

Existence of a global limit in $L^1([-N,N])$ for each $N\in \mathbb{N}$

Let $(f_n)_n$ be sequence of functions $f_n\in L^1_{loc}(\mathbb{R})$ such that for each $N\in \mathbb{N}$, $(f_n)_n$ is a Cauchy sequence in $L^1([-N,N])$. Then for each $N$, $(f_n)_n$ converges to a ...
2
votes
1answer
12 views

Showing $\lim \int_0 ^1 f_1(x)f_2(x)\cos (2\pi nx)=0$ for some $f_1,f_2\in L^p$

Suppose $f_1\in L^{3/2}([0,1]),f_2\in L^3([0,1])$. Then $\lim_{n\to \infty} \int_0 ^1 f_1(x)f_2(x)\cos (2\pi nx)=0$ Idea 1: Since I see $L^p$ functions in an integral, Holder's inequality comes ...
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1answer
30 views

$l^p$ space not having inner product

I know that $l^2$ space is a Hilbert space. But for other $l^p$ spaces, where $p\geq1$, I have to show that they do not satisfy the parallelogram equality. But, I can't find appropriate sequences ...
0
votes
2answers
22 views

Predual of $l^1(\Gamma)$

Let $\Gamma$ be an uncountable index set. For example $\Gamma=\mathbb R$. Let $l^1(\Gamma)$ be the set of functions with countable support and finite sum: $$ \sum_{a\in\Gamma}|f(a)|<\infty. $$ The ...
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1answer
24 views

L^p spaces are separable and complete but not compact?

Where is the mistake in my reasoning?: Let X be a separable metric space, then for every $p\in [1,\infty)$ and for every borel measure $\mu$ on $X$: $L^p_{\mu}(X)$ is separable. Therefore by a ...
1
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1answer
25 views

modulation-translation operator continuous in $L^{p}$ norm?

We put, $T_{y}f(x):=f(x-y), \ (x, y\in \mathbb R^{n}).$ It is well-known that $\|T_yf-f\|_{L^{p}} \to 0$ as $y\to 0$ for $1\leq p <\infty.$ Next we put, $M_tT_yf(x):= f(x-ty) e^{i t (x\cdot y)}, ...
1
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1answer
45 views

Intuition behind variance in terms of $L^P$ norms?

I've just started working through Varadhan's Probability lecture notes, and I was wondering if there's any intuitive connection between the variance formula and Holder's inequality/ $L^p$ norms in ...
3
votes
1answer
35 views

If $\sum \|f_n -f \|_{L^1} < \infty$ then $f_n \rightarrow f$ almost uniformly

Consider $(X,m)$ a measure space, $f_n, f : X \rightarrow \mathbb R$ s.t. $\sum_{n=1}^{\infty} \|f_n -f \|_{L^1} < \infty.$ How to show that $f_n \rightarrow f$ almost uniformly? I will have ...
0
votes
1answer
26 views

Approximate identity for periodic integrable functions

I'm studying Fourier analysis now and learned the concept of approximate identity. $$h_n\ge 0,\quad \int_{\mathbb{T}}h_n=1,\quad \lim_{n\to\infty}\int_{\mathbb{T}\setminus[-\delta,\delta]}h_n=0\quad ...
2
votes
1answer
42 views

Uniform integrability and weak L1 convergence

I am working on exercise 4.14 in chapter 3 (on convergence) in the book "Probability and Stochastics" by Erhan Cinlar. The exercise can be found on page 109. First, let me give the necessary ...
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1answer
42 views

Can we conclude that $v_{n}\rightarrow v$ in $L^{\infty}\left(\Omega\right)$ if $p>N$

Let $\Omega\subset\mathbb{R}^{N}$ be a smooth bounded domain, $v_{n}\rightharpoonup v$ in $W_{0}^{1,p}\left(\Omega\right)$ , $\left\Vert v_{n}\right\Vert _{W_{0}^{1,p}}=1$ $\forall n$ . So we ...
3
votes
1answer
36 views

Partial Fourier series for $L^p$ functions, $p\ge 1$

Let $f$ be an $L^2$ function on the unit circle $f \in L^2(S^1, d \theta)$. This is equivalent to giving a Fourier series $\sum_{n \in \mathbb{Z}} a_n e^{i n \theta}$ with $\sum_{n \in \mathbb{Z}} | ...
3
votes
1answer
31 views

Continuity of Translation and Dilation on $L^p$ spaces

Let us consider any $f \in L^p(U)$, where $U \subset \mathbb R^n$ is open, and $1 < p < \infty$. We know the translation operator $f(x) \mapsto f(x+a)$ and the dilation operator $f(x) \mapsto ...
1
vote
1answer
23 views

Why can a function in $L^1(\partial \mathbb{D})$ be represented by a Fourier series?

I am looking for a reference to the claim that for any $f\in L^1(\partial \mathbb{D})$, where $\partial \mathbb{D}$ is the unit circle in $\mathbb{C}$, ...
1
vote
1answer
59 views

Does weak convergence in $W^{1,p}$ imply strong convergence in $L^q$?

Does weak convergence in $W^{1,p}(\Omega)$ imply strong convergence in $L^q(\Omega)$ when $\Omega$ is bounded? If $f_j$ converges weakly to $f$ in $W^{1,p}$, what can we say about the $L^q$ ...
4
votes
1answer
31 views

Is $L^1_{loc}(\mathbb{R})$ complete with the norm $|f|=\sup_{x\in \mathbb{R}}\int_x^{x+1}|f(y)|dy$

Let $BL^1_{loc}$ be the space of locally integrable functions $f:\mathbb{R}\to \mathbb{R}$ such that $|f|=\sup_{x\in \mathbb{R}}\int_x^{x+1}|f(y)|dy<\infty$. Is this space complete ? What I tried: ...
-1
votes
2answers
52 views

$L_p$ spaces and convergence in mesure [closed]

How can one prove the following one: Let $f,f_n\in \cal{L^p}$$(X)$, $\forall n\in\mathbb{N}$. If exists $g\in\cal{L^p}$$(X)$ such that $-g\le f_n\le g, \forall n\in{\mathbb{N}}$ and ...
3
votes
3answers
36 views

Inequality leading to Holder's: $t^\theta \leq \theta t + 1 - \theta$ with $0<\theta<1$ and $t\geq 0$

In the process of proving Holder's Inequality for $l^p$ spaces, as per my instructions it begins by first asking us to prove the following inequality as a first step: If $0<\theta<1$ and ...
3
votes
2answers
38 views

$\overline{L^2(\mathbb R)\cap L^1(\mathbb R)}^{L^2(\mathbb R)}=L^2(\mathbb R)$

While reading a proof in a book they used the following result: $$ \overline{L^2(\mathbb R)\cap L^1(\mathbb R)}^{L^2(\mathbb R)}=L^2(\mathbb R) $$ saying that it's well known !! But all I can see is ...
0
votes
1answer
29 views

Weak derivative and approximation

Let $f \in L^2(\mathbb R)$ be absolutely continuous. Is it true that if $f_n$ is a sequence of Lipschitz functions s.t. $f_n \rightarrow f$ in $L^2$ and $ f_n' \rightarrow g $ in $L^2$ then $ ...
2
votes
0answers
29 views

convergence in $L^p$ implies convergence in measure

I am trying to show that if $f_n$ converges to $f$ in $L^p(X,\mu)$ then $f_n\to f$ in $L^p$ in measure, where $1\le p \le \infty$. Here is my attempt for $p>1$ - Let $\varepsilon>0$ and define ...
2
votes
1answer
42 views

Finding $p$ for which $f\in L^p(\mathbb{R}^2)$

Here's an old qual problem in analysis. Let $s=\Vert x \Vert$ and define $$f(x)=\frac{1}{s(1+s^{1/2}\log s)}$$, $x\in \mathbb{R}$. Find $1\le p\le \infty$ for which $f\in L^p(\mathbb{R}^2)$. Attempt ...
2
votes
1answer
84 views

If $g$ is the $L^2$-derivative of a function $f\in L^2$, then integrating $g$ gives $f$

If $f(x), g(x) \in L^2(\mathbb{R})$ and $\lim\limits_{h\to\infty}\int_{\mathbb{R}}|f_h(x)-g(x)|^2dx=0$, where $f_h(x):=\frac{f(x+h)-f(x)}{h}$ for any $h\neq 0$, show that ...