Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [lp-spaces]

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 problem is a $L^p$ space.

1
vote
0answers
17 views

Existence of functionals on $L^0$

Studying a paper about risk measures by F. Delbaen, I came into this statement: Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space: if $\mathbb{P}$ is atomless, then there exists no ...
3
votes
1answer
37 views

Are $c_0$ and $c$ duals of some spaces?

The (continuous) dual of a normed vector space is always a Banach space, but the converse is not true. That is, not all Banach spaces are isomorphic to the dual space of some normed vector space. ...
1
vote
0answers
20 views

Does $\int^{\infty}_{0}pa^{p-1}1_{\{|f|>a\}}(x)da=\int^{|f(x)|}_{0}pa^{p-1}da$?

Let $f\in L^p(\mathbb{R}^d)$ for $p\in [1,\infty)$. Show that $$\|f\|_p=\left(\int^{\infty}_{0}pa^{p-1}m\{|f|>a\} \right)^{1/p}$$ My attempt: We can use Fubini's theorem in the following way: \...
0
votes
0answers
34 views

Ideas on showing $x^{-\frac{3}{2}}\sin{(x)} \notin L^{p}[0,\infty[$

I want to show that $x^{-\frac{3}{2}}\sin{(x)} \notin L^{p}[0,\infty[$ where $p \geq 2$. My idea (which has not been fruitful): $\int_{0}^{\infty}\vert x^{-\frac{3}{2}}\sin{(x)}\vert ^{p}dx=\int_{0}...
1
vote
1answer
26 views

Weakly convergence

If $g \in L^p(\mathbb{R})$ be a given non-trivial function, show that following sequences converge weakly in $L^p$ but not strongly in $L^p$. (a) $g_k(x)=k^{1/p}g(kx)$. (b) $h_k(x)=g(x+k)$. I need ...
2
votes
3answers
57 views

Hint: Showing $f_{n}(x):=\frac{n\sqrt{x} \sin{(x)}}{1+nx^{2}}$ converges in $L^{p}$ to $x^{-\frac{3}{2}}\sin{(x)}$

Let $f_{n}:]0, \infty[\to \mathbb R$ and $f_{n}(x):=\frac{n\sqrt{x} \sin{(x)}}{1+nx^{2}}$ Show that $f_{n}$ converges in $L^{p}$ to $f$ where $f(x):=x^{-\frac{3}{2}}\sin{(x)}$ and $p \in [1,2[$ My ...
1
vote
0answers
13 views

Unconditional basis in the tensor product of Lp Banach spaces

Assume that X and Y are topological spaces with $\sigma$-finite measure, $L_p(X)$ is a Banach space of complex-valued functions so that $\int_X |f(x)|^p dx < \infty$, $1 \leqslant p < \infty$. ...
1
vote
2answers
35 views

Sufficiency in the proof that $L^p(\mu)$ is complete

In the proof that $L^p(\mu)$ is complete for $p\in[1,\infty]$ (as done in Saxe, Theorem 3.21 or in Folland, Theorem 6.6, the latter of which is outlined here) we make use of the following completeness ...
0
votes
1answer
21 views

Express $|x|^{-p},\;x\in\mathbb{R^+},\;0.5<p<1$ as sum of $L^1(\mathbb{R^+}) + L^2(\mathbb{R^+})$ function

Express $|x|^{-p},\;x\in\mathbb{R^+},\;0.5<p<1$ as sum of $L^1(\mathbb{R^+})+L^2(\mathbb{R^+})$ function. I have been able to show that $|x|^{-p}$ is neither $L^1$ nor $L^2$, but how do I ...
2
votes
1answer
44 views

Convergence of Indicator function in weak*-topology

Let $\omega_n$ and $\omega$ be measurable subsets of $[0,1]$. Also let indicator function $\chi_{\omega_n}$ converge to $\chi_{\omega}$ in weak*-topology on $L^\infty(0,1)$, that is $$\int_0^1f(x)(\...
0
votes
1answer
18 views

Condition on Uniform Convergence of Fourier Series

When I was reading a proof of some problem, it said that "Since $f$ and $f'$ are in $L^2([-\pi,\pi])$, the Fourier Series of $f$ converges to $f$ uniformly". My question is that, is this statement ...
1
vote
0answers
17 views

Proving $g_a\in L^1(\mathbb{R}^d) \iff a<-d$

Prove that $g_a \in L^1(\mathbb{R}^d) \iff a<-d.$ Where $$ g_a(x)=\begin{cases} |x|^a & \text{ if } |x|>1 \\ 0 & \text{ otherwise} \\ \end{cases}$$ We will be using the following ...
0
votes
2answers
22 views

Inverse of a positive $L^{1}$ function on a probability space

Let $(\mathcal{X},\mu)$ be a probability space, and let $f$ be a real-valued, $L^{1}$ function which is also strictly positive $\mu$-a.e. so that we may consider its ($\mu$-a.e.) inverse function $\...
1
vote
0answers
33 views

If $1<p<\infty, p^{-1}+q^{-1}=1,f\in L^{1}(\mathbb{R}^d),$ then there are $g\in L^{p}(\mathbb{R}^d)$ and $h\in L^{q}(\mathbb{R}^d)$ such that $f=gh.$

If $1<p<\infty, p^{-1}+q^{-1}=1,$ and $f\in L^{1}(\mathbb{R}^d),$ show that there are $g\in L^{p}(\mathbb{R}^d)$ and $h\in L^{q}(\mathbb{R}^d)$ such that $f=gh.$ $\textbf{My Thoughts:}$ I ...
2
votes
1answer
32 views

If $\|f_n\|_{p}\leq n^{-2}, f_n\in L^{p},\forall n\in\mathbb{N}$, does pointwise convergence of $f_n$ follow for a.e. $x\in\mathbb{R}$?

Let $p\in[1,\infty)$ and $(f_n)_{n\in\mathbb{N}}\subset L^{p}(\mathbb{R})$ such that $\|f_n\|_{p}\leq n^{-2}$ for all $n\in\mathbb{N}.$ Does $(f_n)_{n\in\mathbb{N}}$ necessarily converge pointwise a.e....
0
votes
1answer
13 views

Correct Formulation of a map between two measurable spaces

Let $\pi: (X,\mathcal{M},\nu) \to (Y,\mathcal{N},\eta)$ be a measurable map i.e. $\pi^{-1}(E) \in \mathcal{M}$ for all $E \in \mathcal{N}$. I want to define a map from $L^{\infty}(Y,\eta)$ to $L^{\...
0
votes
1answer
24 views

If $d_f (t) : (0, \infty) \rightarrow [0,\infty]$ prove $d_f (t) \leq \bigg( \frac{||{f}||_{L^p(X)}}{t} \bigg)^p.$

Let $1 \leq p < \infty$ and suppose that $f \in L^{p}(X).$ Define the function $d_f (t) : (0, \infty) \rightarrow [0,\infty]$ by $$d_f(t)= \mu (\{ x \in X : |f(x)|>t\}).$$ For each $t>0,$ ...
2
votes
1answer
29 views

Proving that $L^{p}(\mathbb{R}^d)\not\subset L^{\infty}(\mathbb{R}^d)$ for $1\leq p<\infty.$

Show that $L^{p}(\mathbb{R}^d)\not\subset L^{\infty}(\mathbb{R}^d)$ for all $p\in[1,\infty).$ My attempt at this problem: I define the following function for $x\in\mathbb{R}^d,$ $$f_a(x)=\begin{...
1
vote
2answers
45 views

Prove that if $f\in L^{p}(E), 1\leq p<\infty, m(E)<\infty$, then $f\in L^{q}(E)$ for all $1\leq q\leq p$

Let $f\in L^{p}(\mathbb{R}^d),$ for $1\leq p<\infty,$ and $f$ is supported on a set $E$, of finite measure. Prove that $f\in L^{q}(\mathbb{R}^d)$ for all $1\leq q\leq p.$ Here are my thoughts so ...
1
vote
1answer
23 views

Given a sequence of Lp functions, does the integral commute with the lp norm?

I have been struggling to prove the following: Let $ \{ f_n \}$ be a sequence in $ L^p(E) $ for some $ p \geq 1 $. Then, $$ \left( \sum_{n=1}^\infty | \int_E f_n \mathrm{d}\mu |^p \right)^{ \frac{1}...
5
votes
2answers
153 views

Using Banach-Alaoglu theorem on $L^1$

Let $C$ be a bounded closed set in $L^1(0,1)$. Let $h_n$ be a sequence in $C$. Prove or disprove that for every $f\in L^\infty(0,1)$ there is a subsequence $h_{n_k}$ such that $$\int_0^t f(s) h_{n_k}(...
2
votes
1answer
29 views

Is every complex Banach space with Schur's property hereditarily $l^1$?

An infinite-dimensional Banach space $X$ is hereditarily $l^1$ if every infinite-dimensional subspace of $X$ contains a subspace isomorphic to $l^1$. And $X$ has Schur's property if every weakly ...
0
votes
0answers
15 views

Showing $||f||_{\infty}=\lim_{p\to\infty}||f||_{p}$ for a probability space. [closed]

Suppose we are given a probability space. Prove that $||f||_{\infty}=\lim_{p\to\infty}||f||_{p}$.
0
votes
0answers
18 views

Factoring Variable in $L^2$

Suppose that $X,Y$ are random-variables in $L^2(\Omega,\mathcal{F},\mathbb{P})$, for some complete probability space $(\Omega,\mathcal{F},\mathbb{P})$, and suppose that there exists a Borel function $...
1
vote
0answers
15 views

Global $L^p$ estimates for the heat equation by approximation

Consider the heat equation $\partial_t u = \Delta u + f $ on $\mathbb{R}^N$ with $u(0) = 0$. Let $u \in C_c^\infty(\mathbb{R}^{N+1})$ be a solution of the heat equation to some $f \in C_c^\infty(\...
0
votes
1answer
21 views

Example showing L1 is not a reflexive space [duplicate]

I know that the $L^p$ spaces are reflexive for $1<p<\infty$. I want to explicitly show that $L^1((0,1),\mathbb{R})$ is not reflexive by finding an element of $L^1$** that is not in $L^1$. To be ...
11
votes
3answers
601 views

Is the rectangular function a convolution of $L^1$ functions?

Do there exist functions $f,g$ in $L^1(\mathbf{R})$ such that the convolution $f \star g$ is (almost everywhere) equal to the indicator function of the interval $[0,1]$ ?
4
votes
1answer
72 views

Obtaining the $L^p$ norm of a function via testing against $L^{p'}$ functions.

Let $f:\mathbb{R}^n\to\mathbb{C}$ be a locally integrable function and let $p\in[1,+\infty)$ and $p'\in(1,+\infty]$ such that $\frac{1}{p}+\frac{1}{p'}=1$. Denoting by $C^\infty_c(\mathbb{R}^n)$ the ...
0
votes
1answer
34 views

Is sum of two square summable sequences is again square summable? [duplicate]

I am trying to prove the set $W=\{(a_n) | \sum a_n^2 <∞\}$ is subspace of vector space $V=\mathbb{R}^∞$ (vector space of all sequences of real numbers) For this, clearly as $0^2+ 0^2+...=0 <∞$ ...
0
votes
0answers
21 views

Functions in $L_p[0,1)$ grow at most polynomially

I just stumbled across the John-Nirenberg inequality which bounds the growth of functions in the BMO space. So I got curious whether such a bound on the growth also exists for $L_p$ functions. Let me ...
1
vote
1answer
28 views

Determine if the limit of the integral is zero

Let $\{A_n\}_{n\in\mathbb{N}}$ be a sequence of disjoint measurable subsets of an interval $[a,b]$. Define the sequence of functions, $$ f_n(x) = \frac{1}{\sqrt{m(A_n)}}\chi_{A_n}(x); \;\;\ x \...
0
votes
1answer
48 views

Is the $L^p$ norm bounded by the $L^\infty$ norm on a bounded space?

Consider the $L^p$ norm defined on some bounded open subset $\Omega\subset\mathbb{R}^n$ with $p>1$. Does there exist a constant $c$ such that $$\Vert f\Vert_{L_p(\Omega)} \le c\Vert f\Vert_{L_\...
2
votes
1answer
24 views

If we have a sequence of measurable functions that is Cauchy with respect to the weak L^p norm, is it Cauchy with respect to the L^p norm?

If $(f_n)$ measurable on $(X,\mathcal{M},\mu)$, $f_n$ Cauchy with respect to weak quasi $L^p$-norm: $[f_n]_p=\sup_{\alpha>0}\alpha \lambda_{f_n}(\alpha)^{\frac{1}{p}} $ where $\lambda_{f}(\alpha)...
0
votes
0answers
18 views

Construction of oscillating sequence in $\mathcal{L}^{\infty}(\Omega, \{z_1, …, z_N\})$

Let $N \in \mathbb{N}$, $\lambda_n \in (0,1)$ , $z_n \in \mathbb{R}^N$ for $n \leq N$ such that \begin{equation} \sum_{n=1}^N \lambda_n = 1,\quad \sum_{n=1}^N \lambda_n z_n = z \in \mathbb{R}^N. \...
2
votes
1answer
37 views

Is this subspace isomorphic to the space $\ell^1$?

Let $X$ be a Banach space. Suppose there exists a sequence $\{x_n\}$ in $X$ such that for all finite $A\subseteq\mathbb{N}$, and for all functions $\alpha:A\to\{-1,1\}$, we have that $\|\sum_{n\...
0
votes
1answer
20 views

If $e_n=\frac{1}{\sqrt{2 \pi}}e^{inx}$ is a basis for $L^2([0, 2\pi])$, then why $\frac{1}{\sqrt{2 \pi}}e^{-inx}$ as well?

If $\frac{1}{\sqrt{2 \pi}}e^{inx}$ is a basis for $L^2([0, 2\pi])$, then why $\frac{1}{\sqrt{2 \pi}}e^{-inx}$ as well? This realization allows one to write Fourier coefficients as: $$\frac{1}{\sqrt{...
1
vote
1answer
33 views

Weak convergence and distribution convergence

I'm a bit confuse with weak convergence and distribution convergence : 1) $f_n\to f$ weakly in $L^p(\mathbb R^n)$ if for all $\varphi \in L^q$ (the conjugate exponent of $p$), $$\lim_{n\to \infty }\...
1
vote
1answer
19 views

Can we generalize lemma Scheffé in $L^p$?

Scheffé lemma says that: If $f_n\to f$ pointwise and if $\int|f_n|\to \int|f|$ then $f_n\to f$ in $L^1$. I have several questions around this result. Q1) If $f_n\to f$ pointwise and that $f_n\in L^1$...
1
vote
0answers
28 views

Statement of Riesz-Thorin theorem

I've seen a number of sources (most I have seen I think) state the Riesz-Thorin theorem in the following way: Let $(\Omega_{1},\Sigma_{1},\mu_{1})$ and $(\Omega_{2},\Sigma_{2},\mu_{2})$ be $\sigma$-...
0
votes
2answers
46 views

Is {-1,1}^N compact in $\ell_\infty$?

I am reading A short Course on Banach Space Theory. And it says that if ∑xn converges unconditionally in a Banach Space X, then the set of all vectors of the form ∑εnxn is a compact subset of X. ...
7
votes
0answers
208 views

Question on a step of the proof of Theorem 1.25 of Introduction to Fourier Analysis on Euclidean Spaces

Theorem 1.25: Suppose $ \phi \in L^1(\mathbb{R^n}) $ and $ \int_{\mathbb{R^n}} \phi =1 $ . Also, let $\phi_{\epsilon}(x)=\frac{\phi\left(\frac{x}{\epsilon}\right)}{\epsilon^n}$.Moreover , suppose ...
5
votes
1answer
49 views

On a sequence $f_k$ in $L^{2+\frac{1}{k}}$

Suppose that $f_k\in L^{2+\frac{1}{k}}(\Omega)$ with the property that $\|f_k\|_{L^{2+\frac{1}{k}}(\Omega)} = 1$ for all $k\ge 1$. $\Omega$ is a bounded domain in $\mathbb{R}^n$. Can such a sequence ...
1
vote
0answers
19 views

Optimality for $L^p$ [duplicate]

Let us say that $f$ is a function belonging to $L^p(0,1)$. Then nothing assures us that $f$ belongs to $L^q (0,1)$ for some $q > p$. I was trying to find an "optimal function" for its $L^p$ space, ...
2
votes
0answers
26 views

Berge's Theorem of the Maximum for correspondences mapping into subsets of $L^1$

I am looking for a way to apply Berge's maximum theorem when my constraint set is a subset of $L^1$. The problem looks like this: $$ g(\theta) = \min_{x\in B(\theta)} \mathbb{E}\left[f\left(x\right)\...
3
votes
4answers
58 views

Show that F vanishes at infinity.

Suppose $1 ≤ p < ∞, f ∈ L^p(R)$, and $F(x) = \int_{x}^{x+1} f(t) dm(t)$ Prove that F vanishes at infinity. We know that $\int_R |f|^p < \infty$, then, of course, for any $x, F(x)< \int_x^{...
1
vote
1answer
40 views

Prove that for $f\in L^1(\mathbb R)$, $\int_{\mathbb R}|f|=0\implies f=0$ a.e.

Let $f\in L^1(\mathbb R)$ s.t. $$\int_{\mathbb R}|f|=0\implies f=0\ a.e.$$ My attempt Suppose $f$ continuous and that there is $y$ s.t. $|f(y)|\neq 0$. In particular, there is $\delta >0$ s.t. $f(...
4
votes
1answer
33 views

Exercise in Holder's inequality

The following is a problem from Royden and Fitzpatrick's Real Analysis book. Find the values of the parameter $\lambda$ for which $$ \lim\limits_{\epsilon\rightarrow0^{+}} \frac{1}{\epsilon^\...
5
votes
1answer
38 views

If $\left(\sqrt{\gamma_n}\right)_{n\in\mathbb N}$ is $L^2(\mu)$-Cauchy, does $\left(\gamma_n\right)_{n\in\mathbb N}$ converge in $L^1(\mu)$?

Let $(\Omega,\mathcal A,\mu)$ be a measurable space Suppose $(g_n)_{n\in\mathbb N}$ is a sequence of bounded $\mathcal A$-measurable functions $g_n:\mathbb R\to\mathbb R$ such that $\left(\sqrt{\...
3
votes
1answer
60 views

Bounded Sequence in $L^\infty$ and Interpolation in $L^p$

a) Let $1\leq p_1\leq p\leq p_2\leq \infty$ and for $\alpha \in [0,1]$ $\frac {1}{p}=\frac {\alpha}{p_1}+\frac {1- \alpha}{p_2}$ Prove that if $f\in L^{p_1}\cap L^{p_2}$, then $f\in L^p$ and we have ...
0
votes
1answer
33 views

Verification of alternative proof of $\lim_{p\to \infty}\|u\|_p=\|u\|_\infty$

I have to show that $$\lim_{p\to \infty}\|u\|_p=\|u\|_\infty$$ Suppose $u\in L^\infty (E) $ for measurable $E \subset \mathbb{R}^d$ having finite measure. I come up with this proof: $$\Big| \|u\|_p -...