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
13 views

Proof completeness of $L^p$

I'd like to check if I understood the proof that $L^p$ is complete ($1 \le p <+\infty$). I have to use the following fact: in a metric space, if a Cauchy sequence has a convergent subsequence then ...
1
vote
1answer
18 views

Embedding of Lp spaces

I've managed to prove that for $ 1\leq p < q \leq +\infty $ we have an inclusion (embedding) $ L_q([0,1],\lambda) \rightarrow L_p([0,1], \lambda) ~~ (\lambda $ being Lebesgue measure). The trouble ...
3
votes
1answer
29 views

Holder Inequality

I have a problem with the demonstration of this inequality ('m following Royden) : If $p$ and $q$ are nonnegative numbers such that $\frac{1}{p}+\frac{1}{q}$ and if $f \in L^p$ and $g \in L^q$, then ...
2
votes
1answer
22 views

How to identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? (measurability of function)

How can we identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? For my understanding: $Q_T:=(0, T)\times \Omega$; DEF1: $L^2(Q_T)=\{u: (0, T)\times \Omega \to \mathbb{R}, \mbox{measurable and} ...
1
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2answers
44 views

$L^{p}$ spaces and their properties

I have aquestion :Idont know how to show that if $1<p<q<\infty$ , then $L^{q}$(0,1)$\subset$$L^{p}$(0,1) and $\mid\mid f\mid\mid$$_p$ < $\mid\mid f\mid\mid$$_q$ ,f $\in$$L^{q}$(0,1)? ...
0
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0answers
7 views

Trying to understand $(O(1_{[0,t)}))(s)$ where $O$ is the orthogonal transformaion on $L^{2}[0,\infty)$

Hello I am trying to understand $(O(1_{[0,t)}))(s)$ where $O$ is the orthogonal transformaion on $L^{2}[0,\infty)$. So given $f\in L^{2}$ we have $\left \| O(f) \right \|_{L^{2}[0,\infty)}=\left \| f ...
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0answers
19 views

Polynomial density in $L^p (\mathbb{R},\mu)$

I wanna check a necessary and sufficient condition for a Radon measure witch have the moments of all orders, to say that polynomials are dense in $L^p (\mathbb{R},\mu)$. Or just a paper or an article ...
1
vote
1answer
37 views

Is this map surjective?

Let $B^1(\mathbb{R},\mathbb{R})$ be the set of all locally integrable functions $f:\mathbb{R}\to \mathbb{R}$ such that $$\sup_{t\in \mathbb{R}} \int_t^{t+1}|f(x)|dx<\infty.$$ Consider the map ...
1
vote
1answer
24 views

Are $l_{p} \cap k$ and $l_{p} \cap k_{0}$ complete in $||$ $||_{\infty}$? Are they complete in $l_{p}$ norms?

Let the space $k$ be all convergent sequences of real numbers. Let the space $k_{0}$ be the space of all sequences which converge to zero with $l_{\infty}$ norm. Are $l_{p} \cap k$ and $l_{p} \cap ...
1
vote
2answers
22 views

Functions in $L^p(\mathbb{R}^n)$, are tempered distributions.

How to prove that functions in $L^p(\mathbb{R}^n),1 \leq p \leq \infty$, are tempered distributions.
3
votes
0answers
50 views

Do these limits commute?

Given a sequence of functions $f_{n,m}:\mathbb{R}^{n} \to \mathbb{R}$, suppose that $$\displaystyle lim_{m} f_{n,m}(x)$$ exists almost everywhere (for any fixed n) and also suppose that ...
0
votes
0answers
22 views

Bounding a discrete function

If there is a discrete-time function $u(t)$, where $u(T)=u(0)+\sum_{t=0}^T G(u(t))$, is it possible to prove that $u(t)$ remains bounded for a specific class of $G$ functions? Such as $G\in L^1$ or ...
2
votes
1answer
32 views

Essential range of a function

Let $A_f$ be the set of all averages $\frac{1}{\mu(E)}\intop_{E}\,f\,d\mu$ where $E$ is of positive measure. What is the relationship between $A_f$ and $\mathbb{R}_f$? Is $A_f$ always closed? Are ...
8
votes
2answers
89 views

Why are $L^p$ spaces for $p\not=1,2,\infty$ important?

$L^p$ spaces for arbitrary $1\le p\le\infty$ are a mainstay of basic functional analysis courses, but I've only seen them "in action" when $p$ is 1, 2, or $\infty$. Can anyone give an "elementary" ...
1
vote
3answers
31 views

Set of sequences which converge to zero is a closed subspace of $l^\infty$

Prove that: $$c_0 = \{\{a_n\}_1^\infty:lim_{n\to\infty}a_n = 0\}$$ Is a closed subspace of $l^\infty$.
3
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0answers
59 views

Compact subsets of $L^\infty$

The Riesz Frechet Kolmogorov theorem gives a necessary and sufficient condition for a subset of $L^p(\Omega)$ spaces for $1\leq p<\infty$ and equipped with Lebesgue measure to be relatively compact ...
0
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0answers
24 views

proof in Holders inequality,(equality) [duplicate]

I have this proof in my book: I would like to prove what I underlined in red. but I get stuck. I guess in order to get equality we only need the opposite inequality. However I still don't ...
3
votes
1answer
29 views

Examples of measures that induce certain inclusions in the Lp spaces.

I apologize for the terribly worded title, but I didn't know how else to title this questions (which comes from Rudin's Real & Complex Analysis chapter 3 questions). The question says: For ...
1
vote
1answer
29 views

Compactness of the convergent to zero sequences

I've gotta prove that $$T = \left\{ \left\{ x_i \right\} \in {\ell ^\infty }:\left| x_i \right| < \mu_i,\mathop \lim\limits_{i \to \infty } \mu _i = 0 \right\} \subseteq \ell ^\infty $$ is ...
1
vote
0answers
21 views

A derivative identity, for any multi-index.

I'm trying to prove by induction on the multi-index $\alpha$, that, $$\sum\limits_{j=\frac{|\alpha|}{2}}^{|\alpha|}\sum\limits_{|\beta|=2j-|\alpha|}c_{\beta}x^{\alpha}[m_z(x)]^{j+1}=D^{\alpha}m_z(x)$$ ...
1
vote
1answer
35 views

Hölder norm bounded by $L^p-$norm?

Let $C_0^{\alpha}(\mathbb{R})$, $0<\alpha<1$ denote the space of Hölder-continuous functions on $\mathbb{R}$ with compact support. Is it true that for any $f\in C_b^{\alpha}(\mathbb{R})$ one ...
3
votes
1answer
92 views

An open set in the space of bounded real sequences

Let $X$ denote the set of all bounded real sequences, equipped with the norm $\| (x_n)\|_\infty:= \sup\{|x_1|,|x_2|,|x_3|,\ldots\}$; Let $X_{++}$ denote the set of all bounded positive real sequences ...
2
votes
1answer
38 views

How to prove a tempered distribution is in $L^p(\mathbb{R}^n)$

Given $g \in L^p(\mathbb{R}^n)$, how can I to prove that the tempered distribution $$f=\mathcal{F}^{-1}[(z-4\pi^2|x|^2)^{-1}\mathcal{F}g]$$ is in $L^{p}(\mathbb{R}^n)$ where $z \in \{u \in ...
1
vote
1answer
44 views

Operator norm equality

I came across this problem and am getting stuck on how to prove it. Any help would be appreciated. Suppose $L:C(\textbf{T}) \rightarrow \mathbb{C}$, where $L(f)=\int_0^1 {f(x)g(x)}dx$ for all $f \in ...
1
vote
0answers
31 views

Convergence of Fourier series in $L^\infty$

So if $f\in L^1(\mathbb{T})$ and $S_Nf\rightarrow f$ in $L^\infty(\mathbb{T})$ ($S_Nf$ is the partial sum of the fourier series of $f$), then $f$ is continuous. How do we show that this is true? In ...
3
votes
2answers
30 views

Translations in $L^p(\mathbb{R}^n)$

Let $f,g\in L^p(\mathbb{R}^n)$, $1\leq p< +\infty$. Define (for a.e. $x\in\mathbb{R}^n$) $g_h(x)=g(x-h)$. Show that $$ \lim_{h\to\infty} \lVert f-g_h \rVert_p=(\lVert f\rVert_p^p+\lVert ...
1
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0answers
25 views

L^2 space convolution inequality

How do you use Young's Inequality, and all these convolution formulae to prove the following inequality $$||f*g||^2_{L^2(\mathbb{T})}\leq ||f*f||_{L^2(\mathbb{T})}||g*g||_{L^2(\mathbb{T})}$$ where ...
1
vote
1answer
37 views

Estimating the modulus of continuity of translation in $L^2$ by a Sobolev norm of the function

For any $s\in \mathbb{R}$ define the Hilbert space $H^s(\mathbb{T})$ by means of norm $$\|f\|^2_{H^s}=|\widehat{f}(0)|^2+\sum_{n\in\mathbb{Z}}|n|^{2s}|\widehat{f}(n)|^2.$$ Show that for any $0\leq ...
3
votes
1answer
49 views

tough lp inequalities

Let $1<p<\infty$. If possible, find a positive decreasing sequence $w_1>w_2>\cdots$ such that $\lim w_i=0$, and a (uniform) constant $K>0$, such that ...
1
vote
0answers
29 views

Wiener Algebra, absolute convergence of fourier series

So how do you prove if $f, g\in L^2(\mathbb{T})$, then $f*g\in \mathbb{A}(\mathbb{T})$. $\mathbb{T}$ denote $[0,1)$ and $\mathbb{A}(\mathbb{T})$ denote the Wiener algebra such that if $f\in ...
0
votes
1answer
63 views

The proof of a Sobolev embedding inequality by a compactness argument

I want to prove the following If $N\ge 3$ there exists a constant $c_0=c_0(\Omega)$ such that for all $\alpha\ge 1$ and $z\in H^1(\Omega)$ \begin{align} ...
2
votes
1answer
36 views

Properties of $L^p$ spaces [duplicate]

Let $\Omega$ $\subseteq$ $\Bbb R^{n}$ be bounded and $u$ be a measurable function on $\Omega$ Such that $|u|^{p} \in {L^{1}_{\mathrm{loc}}(\Omega)}$ for some $p \in \Bbb R$. Then how to show the ...
2
votes
1answer
29 views

Supremum of integrals of products of functions in $L^p$ space

Here is the problem I'm dealing with I'm not having success with...well, anything. Any hits on how I could get started and where I would go? edit: information on $L^p$ space
2
votes
2answers
72 views

If $\int u\varphi = 0$ for all $\varphi \in C_c^\infty(M)$ with $\int_M \varphi =0$, is $u=0$ a.e.?

Let $M$ be a compact Riemannian manifold and $u \in L^2(M)$. we know that if for all $\varphi \in C_c^\infty(M)$, $$\int_M \varphi u = 0,$$ then $u=0$ a.e. Suppose $$\int_M \varphi u =0$$ for all ...
2
votes
2answers
35 views

Is $f(x) \in L^p(\mathbb R)$ always bounded for $x\longrightarrow\pm\infty$?

I need to prove the following result on the derivative of an Hilbert transform for $f,f'\in L^p(\mathbb R)$ $$\mathcal H\bigg\{\frac{df(x)}{dx}\bigg\}=\frac{d}{dx}\mathcal Hf(x) $$ In particular ...
3
votes
1answer
15 views

$L^p$ integrability of products of Gaussian variables

Gaussian variables have moments of all orders, so by Hölder's inequality the product of two Gaussian variables $\xi$ and $\eta$ has finite $L^1$-norm: $$ \|\xi \cdot \eta\|_1 \leq \|\xi\|_2 \cdot ...
4
votes
1answer
24 views

BMO functions are $L^p$ Loc for all $1<p<\infty$

In order to motivate my question, I'd like to remember that if $\Omega$ is a bounded domain and $f \in L^q(\Omega)$ for some $q>1$, by Hölder inequality $f \in L^p(\Omega)$ for $p \in (1,q]$ with ...
0
votes
3answers
55 views

$f, f'\in L^{1}(\mathbb R) \implies \lim_{x\to \infty} f(x)=0 ?$

Suppose $f\in L^{1}(\mathbb R) $ and $f'\in L^{1}(\mathbb R).$ My Question is: Can we show, $\lim_{x\to \infty} f(x)=0$ ? Thanks,
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1answer
39 views

Convergence in $l^p$ space?

We know that $\lim_{k \to \infty} a_{n}^{k} = b_n$. All $(a_{n}^{k}) \in l^p$ space. How to show that $(\lim_{k \to \infty} a_{1}^{k},\lim_{k \to \infty} a_{2}^{k},...,\lim_{k \to \infty} ...
4
votes
1answer
40 views

Do $\mathbb{R}^n$ and $\mathbb{C}^n$ valued ordinarily measureable functions form a Banach space under p-norm?

By measureable function I mean an "ordinarily" measureable function, that is measureable in a sense of this definition: a function between measurable spaces is said to be measurable if the preimage of ...
0
votes
1answer
22 views

If a function $F$ belongs to mixed Lebesgue space $L^{p,1}$, does its reflection $G(x,y):=F(y,x)$ also lie in the space?

Let $F:\mathbb R^{2}\to \mathbb C$ be a function. Suppose $F\in L^{p,1}(\mathbb R \times \mathbb R); (1<p< \infty).$ Define $G:\mathbb R^{2}\to \mathbb C$ as follows: $$G(x,y):=F(y,x)$$ My ...
5
votes
1answer
59 views

When is an analytic function in $L^2(\Bbb R)$?

Suppose $f:\Bbb R\to\Bbb C$ is real analytic. In order for $f$ to be in $L^2(\Bbb R)$, clearly all terms in the power series cannot be positive since $f$ would diverge at $\pm\infty$. Likewise, the ...
2
votes
1answer
31 views

composition of $L^{p}$ functions

Suppose $f, g\in L^{p}(\mathbb R), (1\leq p < \infty).$ For simplicity, let us assume that, $g,f:\mathbb R\to \mathbb R$ so that composition of $f$ and $g$, namely, $f\circ g(x)= f(g(x)); (x\in ...
0
votes
1answer
22 views

How is the L$_p$ norm the maximum of its arguments

I am not able to wrap my head around this result. $\lim_{p\to\infty}(x_1^p + x_2^p + x_3^p .. + x_n^p)^{1/p} = \max [x_1, x_2, .. x_n]$ ?
2
votes
0answers
54 views

Prove that $l^{\infty}(\mathbb{Z^+})$ is not separable.

Let $l^{\infty}(\mathbb{Z^+})$ be the set of all bounded complex functions on $\mathbb{Z^+}$. Then prove that $l^{\infty}(\mathbb{Z^+})$ is not separable. My attempt: Suppose $E\subset ...
1
vote
1answer
50 views

Uniform Boundedness Principle for $L^p( \mathbb{R})$

Suppose $\{f_n\}$ is a sequence in $L^p$ such that for each $g\in > L^q$, the sequence $\{\int f_n g\}$ is bounded. Then $\{f_n\}$ is bounded in $L^p$. $(1\leq p<\infty)$ Proof: Argue by ...
0
votes
0answers
43 views

weak convergent sequence in $L^p(\mathbb{R})$ with $(1\leq p < \infty)$ implies norm is bounded

$f_n \rightharpoonup f$ in $L^p(\mathbb{R})$ with $(1\leq p <\infty)$ implies $||f_n||_p$ are bounded. And for $p = \infty$, if $f_n \xrightarrow{w^*} f$, then $||f_n||_\infty$ are ...
1
vote
1answer
20 views

Most general type of $L^p(X,\ V)$ space where compactly-supported continuous functions are dense

Let $(X,\ \tau)$ be a topological (locally compact?) space, $(X,\ \mathcal{F},\ \mu)$ a measure space, $(V,\ \|\cdot\|)$ a Banach space, $1\leq p < \infty$ and $\|\cdot\|_p$ a function defined for ...
4
votes
2answers
61 views

$\sin(nx)$ does not contain Cauchy subsequence in $L^p([0,2\pi]) $ for $1\leq p < \infty$

$\sin(nx)$ does not contain Cauchy subsequence in $L^p([0,2\pi]) $ for $1\leq p < \infty$ My attempt: Set $f_n(x) = \sin(nx)$. Argue by contradiction, suppose there exists a Cauchy ...
9
votes
1answer
63 views

There is no continuous mapping from $L^1([0,1])$ onto $L^\infty([0,1])$

There is no continuous mapping from $L^1([0,1])$ onto $L^\infty([0,1])$. Proof: suppose $T:L^1 \rightarrow L^\infty$ continuous and onto. $L^1$ is separable, let $\{f_n\}$ be a countable dense ...