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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2
votes
0answers
24 views

Do $L^p$ spaces have the approximation property?

A Banach space $X$ has the approximation property if every compact operator $T:X \to X$ is the norm-limit of a sequence of finite-rank operators. My question is if there is a simple proof that the ...
1
vote
1answer
25 views

How do I prove the interior of subspace $\ell^1$?

Let $E:=\ell^1$ is Banach space with standard norm for $\ell^1$, $P:=\{\bar{x}\in\ell^1: \bar{x}=(x_i)=(x_1,x_2,\ldots),x_i \geq 0, \forall i \in \mathbb{N}\}$ and defined that interior of $P$ is ...
1
vote
1answer
27 views

If $f$ and $g$ have the same $L^{2}$ norm, does it imply $hf$ and $hg$ have the smae $L^{2}$ norm?

Let $f,g \in L^{2}(\mathbb R)$ with $\|f\|_{L^{2}}=\|g\|_{L^{2}}$( that is, $f$ and $g$ have the same $L^{2}-$ norm). We choose $h\in \mathcal{S}(\mathbb R)$(= Schwartz space) so that $hf, hg \in ...
4
votes
1answer
32 views

Convergence of $\operatorname E|X_n|^p$ when $0<p<1$

Let $0<p<1$ and $X_1,\ldots,X_n$ be random variables with finite absolute moments of order $p$. Suppose that the random variables $X_1,\ldots,X_n$ converge in mean of order $p$ to a random ...
2
votes
1answer
21 views

Functions such that $\sup_{t\in\mathbb{R}}\int_{\mathbb{R}}e^{a(x)t}|f(x)|dx<\infty$ and …

Can we find a bounded function $a:\mathbb{R}\to\mathbb{R}$ and a function $f\in L^1(\mathbb{R})$ with $f\neq 0$ such that $$\sup_{t\in\mathbb{R}}\int_{\mathbb{R}}e^{a(x)t}|f(x)|dx<\infty$$ and ...
2
votes
1answer
38 views

Proof that v belongs to l_p space under certain conditions

I am struggling with the following problem: Let $M >1$ and $\lambda \in (0,1)$, $\mathbf{z} \in \mathcal{l}_p$. If $|v_t|^p < (M\sum_{s=t}^\infty \lambda^{s-t+1} |z_s| )^p $, then $\mathbf{v} ...
1
vote
2answers
25 views

Continuity of integration in $L^1(X)$.

Let $\varepsilon>0$ and let $f\in L^1(X)$. Show that there exists $\delta>0$ such that for any $E\subseteq X$ with $m(E)<\delta$ we have $$ \int_E \lvert f\rvert<\varepsilon. $$ MY ...
5
votes
1answer
46 views

Convergence types in probability theory : Counterexamples

I know that the following implications are true: $$\text{Almost sure convergence} \Rightarrow \text{ Convergence in probability } \Leftarrow \text{ Convergence in }L^p $$ $$\Downarrow$$ ...
1
vote
0answers
33 views

Approximation of a smooth function by smooth functions with compact support

Let $U \subset \mathbb{R}^N$ be a bounded open set, and let $h \in C^\infty(U)$. Since $h \in L^p(U)$ for every $1 \leq p \leq \infty$, there exists a sequence $(h_j) \subset C_c^\infty(U)$ such that ...
1
vote
0answers
21 views

Sequence in $L^2$ [on hold]

Suppose $f_k$ is a bounded sequence in $W^{1,2}(\mathbb{R}^n)$. We know that $\lim_{k \rightarrow \infty} \sup_{x \in \mathbb{R}^n} \int_{B_1(x)}|f_k(y)|^2dy=0$. Does it follow that $\|f_k\|_q ...
2
votes
0answers
41 views

Constructing a function using the Fourier transform

Pick an integer $n\ge 5$ and let $f\in C_{C}^{\infty}(\mathbb{R}^{N})$. We want to use the Fourier transform to formally construct a function $u\in L^{\infty}(R^{n})$ that solves ...
1
vote
1answer
36 views

Showing an Operator is Well-Defined and Bounded

Let $\{e_n\}_{n \in \mathbb{N}}$ be an orthonormal system within $\ell^2$. Fix a sequence $\lambda = (\lambda_1, \ldots , \lambda_n , \ldots) \in \ell^{\infty}$ and define $ \displaystyle Tf = ...
0
votes
2answers
69 views

How can I study the convergence of the following sequence?

Let we have the following sequence in $L^2$ The sequence is $$X=(x_1,x_2,x_3,......,x_n,......)$$ Such that $$x_1=(1,0,0,0,0,0,,.....)$$ ...
1
vote
1answer
38 views

What is the general method to study the convergence of a sequence

I want to ask questions : First : In general how can I study the convergence of any sequence in $L^p$ ( $L^1$ , $L^2$ , $L^3$ , .......) of course the sequence belongs to the space which I want to ...
2
votes
2answers
46 views

Norm, adjoint operator and compactness os some operators

Let $A_i:\ell^2\rightarrow \ell^2$ be two operators given as follows: $A_1x=(0,x_1,0,\frac{x_2}{2},0,\frac{x_3}{3},...)$ and $A_2x=(x_1,x_1,x_2,x_2,x_3,x_3,...)$ Compute the norm and the adjoint ...
1
vote
2answers
25 views

$\lim_{n \to \infty}\int_{0}^{1}|f_{n}| = 0$ and $\limsup_{n \to \infty} f_{n}(x) = 1$ for all $x \in [0, 1]$?

Is there a sequence $(f_{n})$ of $L^{1}$ functions such that $\lim_{n \to \infty}\int_{0}^{1}|f_{n}| = 0$ and $\limsup_{n \to \infty} f_{n}(x) = 1$ for all $x \in [0, 1]$? I noticed that for every ...
1
vote
0answers
22 views

C^1 is not dense in Holder space

Let $\overline{\Omega}$ be a bounded, closed and convex set of $\mathbb{R}^n$. Prove that $C^1(\overline{\Omega})$, the space of continuously differentiable functions on $\overline{\Omega}$, is not ...
1
vote
1answer
27 views

How can I find a sequence from $l^p\setminus l^1$?

I am trying to find out how to find this sequence Find a sequence $x$ which is in $l^p$ with $p>1$ but $x \not\in l^1$ Thanks
4
votes
0answers
49 views

Why $\|f-g\|=0$ if and only if $f=g$?

I'm learning Fourier Transformation lately, and in the Course Reader page 23, it defines $\|f\|=\left(\int_0^1 \left|f(t)\right|^2 dt\right)^{1/2}$. And then $\|f-g\|=0$ if and only if $f=g$. My ...
1
vote
0answers
16 views

What is the norm of an element in $L^\infty(\mathbb{D})$ using the weak-star topology?

Under the convention that $L^1(\mathbb{D})$ has normalized Lebesgue measure for the unit disc in $\mathbb{C}$, its dual space can be regarded as $L^\infty(\mathbb{D})$. Hence we can equip ...
-2
votes
1answer
56 views

Study the convergence of the following sequence

How can I study the convergence for the following sequences in $L^2$ space ? $x_n=\big(\frac{1}{n},\frac{1}{n},\frac{1}{n},\frac{1}{n},..,\frac{1}{n},0,0,0,...\big)$ $n^2$
1
vote
1answer
20 views

A Cauchy sequence has a rapidly Cauchy subsequence

I am trying to fill in the details of a proof related to the Riesz-Fischer Theorem. We need to show that every Cauchy sequence $\{f_n\}$ has a rapidly Cauchy subsequence. My text claims that we can ...
1
vote
2answers
25 views

A step of change of variables I don't understand when show derivative of a function in $L^n$?

How (16) can goes to (17) and how to show $loglog(1+\frac{1}{|x|})\in L^n$ for $n>1$? My attempt: Let $|x|=r$, then $(x^2)^{\frac{1}{2}}=r$. Differentiate on both sides, I get ...
2
votes
1answer
62 views

Compactness Sobolev embedding for radial functions on $\mathbb{R}^N$

I want to show: Let $N\geq 2$ and $2< q <2^\ast$. Then the embedding \begin{align} H^1_{\text{rad}}(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N) \end{align} is compact. I was able to show ...
2
votes
0answers
33 views

$||f_n-f||_1 \to 0 $ iff $ \int| f_n|\ to \int|f| $ if $f_n \to f $ a.e. [duplicate]

Assume $f_n , f \in L^1 $ and almost every where we have $ f_n \to f$ then I want to show that $\int|f_n-f| \to 0$ iff $\int|f_n| \to \int|f|$ One side is abvious by trinagle inequality , for the ...
-1
votes
1answer
30 views

Showing a convergence in $ L^p$

If $ f_n \in L^p(X,m) $ and $f_n \to 0 $ a.e with respect to $m$ and $||f_n||_p \to ||f||_p $ as $n \to \infty$ then can we say that for all $r \in [1,\infty] $ we have $$ ||f_n-f||_r \to 0 $$ Can ...
3
votes
2answers
66 views

Are $L^\infty$ bounded functions compact in $L^2$?

Is the set $\{ m \in L^2(0,1) : |m|_{L^\infty}\leq A \}$, (i.e. the set of $L^2$ functions with bounded $L^\infty$ norm) a compact subset of $L^2$? (Compact in the topology induced by the ...
1
vote
1answer
21 views

Lower semicontinuity of ${\dot{H}}^1$ norm

I have a in $H^1(\mathbb{R^N})$ uniformly bounded sequence $u_n \in H^1$. I also know $u_n\to u$ in $L^p$ for every $2\leq p < 2^\ast$, where $\ast$ means the Sobolev exponent. Can I conclude that ...
1
vote
1answer
22 views

Proving a certain subset is closed in $L^1$

In an exam, I was asked to prove that, if $A=\{f\in L^1([0,1]):\int_0^1|f(x)|^2\mathrm{d}x\leq1\}$, then $A$ is closed in $L^1$. I tried this approach. $A$ is closed iff for all $f_n\to f$ in $L^1$ ...
2
votes
1answer
27 views

Dual space of weighted $L^p(\omega)$

Let $\omega \in A_p$, where $A_p$ is the family of Muckenhoupt weights. I'm wondering what is the topological dual space of $L^p(\omega)$. Is it isometrically isomorphic to $L^q(\omega)$? (1/p + 1/q = ...
1
vote
1answer
26 views

Has a $L^1$ bounded sequence a weak converging subsequence in $L^2$?

Let $f_n \in L^2(0,1)$ with the property that $\sup_n || f_n ||_{L^1}<A< \infty$, i.e. $f_n$ is a sequence in $L^2$ that is uniformly bounded in the $L^1$-Norm. Does $f_n$ then have a weak ...
3
votes
1answer
95 views

Are $L^\infty$ bounded functions closed in $L^2$?

Is the set $\{ m \in L^2(0,1) : |m|_{L^\infty}\leq A \}$, (i.e. the set of $L^2$ functions with bounded $L^\infty$ norm) a closed subset of $L^2$? (Closed in the topology induced by the $L^2$-norm)
1
vote
1answer
28 views

Does $\int 1_{|m|>A} m^2$ converge to zero for an $L^2$ function?

We assume that $m \in L^2(0,1)$, hence $\int_0^1 m(x)^2 dx< \infty$ but that $m \not \in L^\infty(0,1)$. Hence $\{ x: |m(x)|>A \}$ has always positive measure. Now the question: Does $\int ...
1
vote
0answers
27 views

$f\in L^1(\mathbb{R}^N)$ and Lipschitz continuous, then $f\in L^\infty(\mathbb{R}^N)$.

$f\in L^1(\mathbb{R}^N)$ and Lipschitz continuous, then $f\in L^\infty(\mathbb{R}^N)$. Denote the Lipschitz constant of $f$ as $C$, suppose $f$ is not bounded a.e., then $\mu(\{f> k+C)\}) ...
2
votes
1answer
36 views

Approximating the gradient of a function by $L^2$ functions (cut-offs)

Let $u:(0,\infty) \to \mathbb{R}$ be such that $u' \in L^2(0,\infty)$, but $u \notin L^2(0,\infty)$. However $u \in L^2(0,T)$ for all finite $T$. Is it possible to find a sequence of functions $u_n ...
0
votes
2answers
40 views

Question about this problem If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$

In this problem If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$ I don't understand this part: $$ \int_U \partial_i u^\epsilon\,v\,dx\to 0\tag w$$ for all $v\in L^2(U)$, but now we only ...
3
votes
0answers
26 views

lp and c0 little question

i want to explain that the inclusion $\bigcup_{p < \infty} l^p \subset c_0$ is true. that comes very quickly from the definition of $l^p$. The problem is, that Im sure that this inclusion is ...
1
vote
1answer
32 views

Analogue of Borel-Cantelli Lemma in $L^1(\mathbb{R})$

Let $\mu$ denote Lebesgue measure on the real line. In measure theory, the Borel-Cantelli Lemma states that if $\lbrace E_k \rbrace_{k\ge 1}$ is a collection of measurable sets such that $\sum_{k\ge ...
2
votes
1answer
64 views

If $f\in L^1(\mathbb{R})$, then $\sum_{n\ge 1}f(x+n)$ Converges for a.e. $x$.

I am given that $f\in L^1(\mathbb{R})$ (i.e., $\int_{-\infty}^\infty\vert f \vert<\infty$). I would like to show that $$\sum_{n\ge0}f(x+n)\tag{$*$}$$ converges for almost every (a.e.) $x$, and I am ...
2
votes
0answers
26 views

Discuss the duality relation between $L^p(X)$ and $L^q(X), 1/p + 1/q = 1, $ for $1 ≤ p < ∞.$

Let $(X,M, μ)$ be the measure space where $X = \{a, b\}, M= P(X),$ and $μ(\{a\}) = 1, μ(\{b\}) = ∞.$ Discuss the duality relation between $L^p(X)$ and $L^q(X), 1/p + 1/q = 1, $ for $1 ≤ p < ∞.$ ...
2
votes
1answer
37 views

Showing some Function $fg$ is Integrable when $g(x)\le e^{-\vert x\vert }$ and given some Conditions on $f$.

Suppose $\int_a^b \vert f \vert <\infty$ for all real $a,b$ and that $$\int_{-r}^r \vert f \vert \le (r+1)^a$$ for all real $r$ some real $a$, and that $$g(x)\le e^{-\vert x \vert}$$ I want to show ...
0
votes
2answers
28 views

$μ$ is $σ$-finite iff $L^p(X)$ contains a strictly positive function.

Let $(X,M, μ)$ be a measure space and $0 < p < ∞$. Prove that, $μ$ is $σ$-finite iff $L^p(X)$ contains a strictly positive function. My Work: If I suppose $L^p(X)$ contains a strictly ...
2
votes
3answers
66 views

Adjoint operator of $L^\infty$

Lets denote with $(\Omega,\Sigma,\mu)$ a $\sigma$-finite measurble space with a linear, continuous operator $$T : L^\infty \to L^\infty.$$ Does this always imply the existence of a linear, continuous ...
2
votes
1answer
30 views

Is $L^2(0,\infty;L^2(\Omega)) = L^2((0,\infty)\times \Omega)$?

If $\Omega$ is a bounded $C^1$ domain, is $L^2(0,\infty;L^2(\Omega)) = L^2((0,\infty)\times \Omega)$? Are they the same? I know this is true when instead of $(0,\infty)$ we have a bounded interval.
3
votes
1answer
54 views

If $(a_n)$ is such that $\sum_{n=1}^\infty a_nb_n$ converges for every $b\in\ell_2$, then $a\in\ell_2$

Please help me with this question. I've been thinking about it for almost two days. Let $a_n$ a real series that have the following property: for every series $b_n$ in $l_2$: $\sum_{n=1}^\infty ...
0
votes
2answers
26 views

Continuity of the multiplication map $f\mapsto x^2 f(x)$ between normed spaces

Let $F:C[0,2]\to C[0,2]$ be the map defined by $(F(f))(x)=x^2f(x)$. Show that $F$ is continuous as a function from $(C[0,2],\|\cdot\|_{\sup})$ to $(C[0,2],\|\cdot\|_{2})$. I read this solution: ...
2
votes
1answer
51 views

How to find the norm of $ \Lambda x = \sum_{n=1}^{\infty} \frac{x_n}{n\sqrt{6}}$ in $\ell^2$

Suppose that $(x_n)$ is a sequence in $\ell^2$, i.e. $\displaystyle \sum_{i=1}^{\infty} x_n^2 < \infty$. Define: $$\Lambda x = \sum_{n=1}^{\infty} \frac{x_n}{n\sqrt{6}}$$ Find $\| \Lambda \|_2$ ...
3
votes
1answer
34 views

Dense subset in sequence space

I'm trying to prove that $F=\{x=\{x_n\}_{n\in \mathbb{N}}\in l^2(\mathbb{N}):\sum_{n=1}^{\infty} x_n=0\}$ is dense in the sequence space $l^2(\mathbb{N})$. I think it should be an easy exercise, but ...
0
votes
3answers
57 views

Compute $\lim_{|h|\rightarrow \infty} \int_{\mathbb{R}^n} |f(y+h)+f(y)|^p dy$

Suppose $f\in L^p(\mathbb{R}^n), 0<p<\infty$, and compute $\displaystyle \text{lim}_{|h|\rightarrow \infty} \int_{\mathbb{R}^n} |f(y+h)+f(y)|^p dy$ I have no idea from where to start since ...
3
votes
1answer
45 views

Weak convergence - $f_n$ “goes up the spout”

Fix $1 < p < \infty$. Given $f \in L^p(\mathbb{R})$ define $f_n(x) = n^{1/p}f(nx)$ for $n = 1, 2, \dots$. Prove that $f_n$ converges weakly to $0$ in $L^p$. I'm really confised about this ...