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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3
votes
1answer
59 views

Prove that $\{\sin x, \sin 2x, … , \sin nx\}$ is a linearly independent set

Prove that $\{\sin x, \sin 2x, ... , \sin nx\}$ is linearly independent. The short solution that I do not understand is as follow: For p and q are positive integer, we have $$ ...
1
vote
1answer
15 views

Condition on $f$ in $L^{p, \infty} $ implies $f \in L^q$

If $f \in L^{p, \infty}$ and $\mu ( \{ x : f(x) \neq 0 \} ) < \infty$, then $f \in L^q$ where $q <p$. $\textbf{My Attempt:}$ Let $E = \{x:f(x) \neq 0 \}$ $$ || f ||_q^q = q \int_{0}^{\infty} ...
0
votes
0answers
17 views

Show f is in Lp if and only if sum of measures s.t. |f(x)|>2^n converges [duplicate]

Let $p \in [1,\infty)$ and suppose $\mu$ is a finite measure. Prove that $f \in L^p(\mu)$ if and only if $$\sum^\infty_{n=1} (2^n)^p \mu(\{x:|f(x)| > 2^n\}) < \infty$$ I spent a while trying to ...
2
votes
1answer
29 views

Lebesgue's differentiation theorem for all points

Let $f \in L^2(0,T)$ be such that $f(t)$ is well-defined for every $t$ (not just a.e. $t$). But I have no continuity of $f$. We have by Lebesgue's differentiation theorem that $$\lim_{a \to ...
2
votes
1answer
19 views

If $u_n \to u$ in $L^2(\Omega)$, and $u_n \in L^\infty(\Omega)$, is $u \in L^\infty(\Omega)$?

If $u_n \to u$ in $L^2(\Omega)$, and each $u_n \in L^\infty(\Omega)$, is $u \in L^\infty(\Omega)$ too? There is no uniform bound on $u_n$ in $L^\infty$ though. I don't think it is.
1
vote
1answer
11 views

Linear functionals over a non dense subset in $\ell^2$

I have come across the following statement a couple of times, but cannot figure out quite how to justify it: If $A$ is not a dense subset of $\ell^2$ then its closure is not all of $\ell^2$ so ...
0
votes
1answer
29 views

$\|u\|_{L^{3}(\mathbb R)} \leq C \|Du\|_{L^{2}(\mathbb R)}^{\alpha} \|u\|_{L^{2}(\mathbb R)}^{1-\alpha}$?

Let $u\in H^{1}(\mathbb R).$ Is Gagliardo–Nirenberg interpolation inequality valid for the $p=3, q=r=2, m=1, 0< \alpha < 1$ ; and $j=0$ ? That is, is it true that,$\|u\|_{L^{3}(\mathbb R)} ...
0
votes
1answer
47 views

Let $M=\{f(x) \in C[0,1]\mid f(0)=0\}$. Is $\overline{M}=L^2[0,1]$?

Let $M=\{f(x) \in C[0,1]\mid f(0)=0\}$. Is $\overline {M}=L^2[0,1]$? The closure is taken under the usual norm of $L^2[0,1]$. Moreover,fix n different points$(a_k)\subseteq[0,1]$,k=1...n, ...
0
votes
1answer
37 views

Calculating the norm of a bounded linear functional

Let $C[0,1]$ be the space of all continuous functions on $[0,1]$. Then for any $g\in C[0,1]$ define the norm on $C[0,1]$ as $\|g\|=\max_{t\in[0,1]}|g(t)|$. Then let $f:C[0,1]\to\mathbb{R}$ as $$ ...
2
votes
1answer
33 views

Convolution with Gaussian, without dstributioni theory, part 3

I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following: Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and $$ ...
1
vote
1answer
29 views

Convolution with Gaussian, without distribution theory, part 2

I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following: Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and $$ ...
2
votes
2answers
71 views

$\|f\|_{L^{3}(\mathbb R)}^{4} \leq C \|f\|_{L^{2}(\mathbb R)}^{3} \|\nabla f\|_{L^{2}(\mathbb R)}$ ; for some constant $C$?

Let $f\in H^{1}(\mathbb R).$ (Sobolev space) My Question: Is it true that: $\|f\|_{L^{3}(\mathbb R)}^{4} \leq C \|f\|_{L^{2}(\mathbb R)}^{3} \|\nabla f\|_{L^{2}(\mathbb R)}$ ; for some constant ...
2
votes
1answer
58 views

$\|f\|_{L^{3}(\mathbb R)}^{3} \leq C \|f\|_{L^{2}(\mathbb R)} \|\nabla f\|_{L^{2}(\mathbb R)}^{3}$ ; for some constant $C$?

Let $f\in H^{1}(\mathbb R)$ (Sobolev space). My Question: Is it true that: $\|f\|_{L^{3}(\mathbb R)}^{3} \leq C \|f\|_{L^{2}(\mathbb R)} \|\nabla f\|_{L^{2}(\mathbb R)}^{3}$ ; for some constant ...
5
votes
2answers
128 views

Convolution with Gaussian, without distribution theory, part 1

I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following: Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and $$ ...
5
votes
1answer
124 views

Showing this limit as $x\to 0$

I'm very close to the result and must be missing something basic: $f$ is absolutely continuous on $[\epsilon,1]$ for each $\epsilon \in (0,1)$, $1<p<2$, and, for $0\leq x\leq y\leq 1$, we have ...
1
vote
1answer
40 views

suppose $f:[1,\infty] \rightarrow R, f(1)=0, f'$ is continuous and bounded and $f' \in L^2([1, \infty])$, show $g=f(x)/x \in L^2([1, \infty])$ .

It's a problem in Bass's real analysis book. suppose $f:[1,\infty] \rightarrow R, f(1)=0, f'exist$ and is continuous and bounded and $f' \in L^2([1, \infty])$, let $g(x)=f(x)/x,$ show $g \in L^2([1, ...
3
votes
2answers
43 views

Suppose $\mu(X)=1$ and $f, g $are nonnegative function such that $fg \geq 1$ a.e. prove $(\int fd\mu)(\int gd\mu) \geq 1$.

I have a problem in $L^p$ space. Suppose $\mu(X)=1$ and $f, g $are nonnegative function such that $fg \geq 1$ a.e. prove $(\int fd\mu)(\int gd\mu) \geq 1$. I have no any idea to prove that. Holder ...
1
vote
0answers
20 views

Point wise convergent [duplicate]

Suppose that $f\in L^1(\mathbb{R})$ and that $f(x)=0$ if $x\notin[0,1]$. For $n=1,2,\ldots,$ let $f_n(x)=f\left(x+\frac{1}{n}\right)$. Prove that $\|f_n-f\|_1\to 0$ as $n\to\infty$.
1
vote
1answer
38 views

Why does this set have finite measure?

Fix a measurable $g$ and let $f$ be any simple measurable function that vanishes outside a set of finite measure and satisfies $||f||_p=1$ where $p>1$. If $|\int fg|<M$ for all such $f$, why ...
4
votes
1answer
84 views

Weak convergence in $L^q$ but not in $L^p$, where $1\leq p<q<\infty$?

I've been trying to find a sequence $\{f_n\}$ of functions on $[0,1]$ that converges weakly in $L^q$ but does not converge weakly in $L^p$, where $1\leq p<q<\infty$. I'm stuck and any hints ...
0
votes
1answer
65 views

If $f_n\to f$ a.e. and are bounded in $L^p$ norm, then $\int f_n g\to \int fg$ for any $g\in L^q$

Suppose $p>1$ and $q$ is its conjugate exponent. Suppose $f_n \rightarrow f~a.e.$ and $\sup_n\|f_n\|_p < \infty $. prove that if $g \in L^q,$ then $\lim_{n \rightarrow \infty} \int f_ng=\int ...
2
votes
1answer
56 views

Behavior at $0$ of a function that is absolutely continuous on $[\epsilon, 1]$

The function $f$ on $[0,1]$ is absolutely continuous on $[\epsilon,1]$ for $0<\epsilon<1.$ I further have that $$\int_0^1x|f'(x)|^pdx<\infty.$$ I'm trying to show that $$ \lim_{x\to 0}f(x)\ ...
4
votes
2answers
100 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
41 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
31 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
48 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
32 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
29 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 ...
6
votes
1answer
157 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
43 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 ...
2
votes
0answers
48 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
61 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 = ...
1
vote
1answer
43 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
58 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
38 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 ...
0
votes
0answers
41 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
28 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
52 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
19 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 ...
1
vote
1answer
34 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
33 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
112 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
35 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
32 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
73 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
31 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
26 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
31 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
32 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 ...