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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-1
votes
2answers
152 views

Dual space of $L^\infty$ is $L^1$ with the weak-* topology?

A friend of mine found a book in which the author said that the dual space of $L^\infty$ is $L^1$, of course not with the norm topology but with the weak-* topology. Does anyone know where I can find ...
0
votes
1answer
39 views

Haar functions form a complete orthonormal system

I want to show that the Haar functions in $L^2([0,1])$ forms an orthonormal basis: Let $$f = 1_{[0, 1/2)} - 1_{[1/2,0)} \ \ \mbox{,} \ \ f_{j,k}(t) = 2^{j/2}f(2^jt - k).$$ Let $\mathscr{A} = \{(j.k) ...
1
vote
1answer
20 views

the Lp norm of the integral of a measurable function is bounded similar to Holders Condition

I have a final coming up in my Measure Theory class, and I found a question that I couldn't get a clean answer: show that for all $ f \in L^p[\mathbb{R}]$ there exists $C \in \mathbb{R}$ such that ...
1
vote
1answer
29 views

measurability of weak limit? or uniqueness of weak limit with sigma-algebras

I have a basic question about weak limits that I hope someone can clarify. Let $(\Omega,\mathcal{F},P)$ where $\Omega \subseteq \mathbb{R}^k$ be a probability space and let $\{f_n\}$ be a sequence of ...
-2
votes
2answers
66 views

How to show the convergence in $L^p$ spaces? [closed]

Let $f_{n}\in L^{p}(\Bbb{R})\ (1\le p\le\infty)$. For $0<r<1$, if there exists a positive constant $C$ such that $||f_{n+1}-f_{n}||_{p}<Cr^{n}$, how to show $(f_{n})$ converges in ...
0
votes
1answer
63 views

Are the three statements the same?

Let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz function on $\mathbb{R}^n$. Consider two statements which have the same proof. $$ f\in \mathcal{S}(\mathbb{R}^n)\,\,\Longrightarrow\,\,f\in ...
5
votes
1answer
53 views

Uniqueness in Riesz representation $L^p$-spaces.

We are in $L^p(\mu)$. Where $(\Omega, \mathcal{A},\mu)$ is an arbitrary measure-space. We have a bounded linear functional in this space $l$, by Riesz representation theorem we have that for $f\in ...
2
votes
0answers
35 views

Is the set of linear combinations dense in the set of the dual space of $l_p$?

Good day, Right now I'm working with the book "Functional Analysis" by Bachman and Narici, it is available on Google Books, see ...
2
votes
0answers
20 views

Fréchet differentiability of Nemyckij operator defined on $L^2$

I have been told the following. Suppose $\Omega\subseteq\mathbb{R}^n$ is a bounded borel set, $f$ is Carathéodory function on $\Omega\times\mathbb{R}=\{(x,s):x\in\Omega,s\in\mathbb{R}\}$, $f_s$, ...
3
votes
1answer
70 views

$L^\infty$ is complete - proof from exercises in Royden & Fitzpatrick's Real Analysis.

I've been reading up on some Analysis for my comp exams, and I couldn't find in my texts a proof of $L^\infty$ being Banach. Someone pointed me to the following exercise in Royden & Fitzpatrick. ...
1
vote
2answers
43 views

Form functions that are continuous at one point in L^\infty a Banach space.

Is the subspace $\{f \in L^\infty(\mathbb{R}) ~|~ f \text{ is continuous at } x=0 \}$ a Banach space? The norm is of course the essential supremum. Does the essential supremum even notice a single ...
1
vote
1answer
22 views

Convergence of second derivative in $L^2(\mathbb{R}^+)$

Could you find a sequence $f_n$ of smooth functions with compact support over the half line $\mathbb{R}^+$ such that $f_n$ converges in $L^2(\mathbb{R}^+)$ but such that the second derivatives ...
4
votes
1answer
58 views

Showing that the operator is bounded and find its norm.

I have this operator $T: L^p(0,\infty)\rightarrow L^p(0,\infty)$, $1<p<\infty$ : $(Tf)(x)=1/x\int_0^xf(t)dt$. I am supposed to show that it is bounded and fint its norm. I had an idea that ...
0
votes
1answer
52 views

Show that there is a measure $\mu$ and a subspace $Y$ of $L_p(\mu)$ such that $\bar{d}(X, Y) \leq \lambda$.

The following is Proposition $7.1$: Let $X$ be a Banach space, let $1 \leq p \leq \infty$ and let $\lambda \geq 1$. Assume that for every finite dimensional subspace $E$ of $X$ there is a subsapce ...
1
vote
1answer
32 views

Show that if $E$ is a finite dimensional subspace of $l_p$, there exists an integer $m$ such that $\| P_m(x) \| \geq (1 - \frac{1}{n}) \| x \|$

Show that if $E$ is a finite dimensional subspace of $l_p$, there exists an integer $m$ such that $$\| P_m(x) \| \geq (1 - \dfrac{1}{n}) \| x \|$$ where $x \in E$ and $P_m$ is a projection map from ...
1
vote
1answer
42 views

Fourier transforms having compact support

As we know, the fourier transform is a map $\mathcal{F}:L^1\rightarrow C_0$ (all with domain $\mathbb{R}$). Can one characterize the space of $f\in L^1$ such that $\mathcal{F}$ has compact support, ...
1
vote
0answers
25 views

Compute of two norms of a function of three variables

Let $f$ be a function defined on $\mathbb R^3$ by $$f(x,y,z)=\exp(-2\mathbb i\pi (x+y+z)) |x|^{1-k} |y|^{k-1} \operatorname{sign}(x) \operatorname{sign}(z),$$ where $sign(x)$ means the sign of $x$ and ...
3
votes
1answer
41 views

$L^2$ mapping is necessarily onto or not?

For $f \in L^2(\mathbb{R})$, let $$Tf(x) := \int_0^1 f(x+y)\,dy.$$Do we necessarily have that$$S: L^2(\mathbb{R}) \to L^2(\mathbb{R}),\text{ }Sf = f - Tf$$is onto?
2
votes
0answers
48 views

Disproving that a particular space is Banach

Let $E$ be a measurable set of finite measure and $1\leq p_1<p_2<\infty$. Consider the linear space $L^{p_2}(E)$ normed by $\|.\|_{p_1}$. Is this normed linear space a Banach space? My ...
0
votes
1answer
50 views

What values of $p$ give convergence to $0$ in $l^p$

Given a sequence $x_n \in l^p$ whose first $n^2$ members equal $\frac {1}{n}$, and all other entries $=0$, for what values of $p$ does the sequence converge to the zero sequence in $l^p$? So do I ...
1
vote
1answer
23 views

Embedding $\mathcal{l}^1 \subset \mathcal{l}^p$ continuous

Is the embedding of $\mathcal{l}^1$ in $\mathcal{l}^p$ for any $p\geq 1$ continuous?
1
vote
1answer
50 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
votes
1answer
19 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 $\| ...
1
vote
1answer
43 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 $E \subset \mathbb{R}^n$. ...
1
vote
1answer
43 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 ...
1
vote
1answer
39 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
votes
1answer
108 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
votes
0answers
41 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 ...
1
vote
1answer
23 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 ...
0
votes
1answer
41 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 ...
2
votes
1answer
42 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
vote
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 ...
1
vote
0answers
49 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
vote
1answer
33 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 ...
0
votes
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 ...
2
votes
1answer
80 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
votes
1answer
85 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 ...
5
votes
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
vote
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 ...
1
vote
1answer
16 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
vote
2answers
41 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
votes
1answer
55 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
votes
1answer
36 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
vote
1answer
30 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
votes
1answer
48 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
vote
2answers
65 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
votes
1answer
46 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
votes
1answer
43 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
votes
1answer
31 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
votes
1answer
47 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} ...