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
2answers
51 views

Unit sphere in $L^p([0,1])$ is not compact.

I am studying $L^p$ spaces and I would like a proof why the unit sphere in $L^p([0,1])$ is not compact. I know that unit sphere is not compact in infinite dimensional spaces, but I think there is an ...
1
vote
1answer
21 views

Composing a continuous function with an $L^p$ function.

Let $U \subset \mathbb{R}^n$ be bounded and $1 \leq p < \infty$. Assume $u \in L^p(U)$ and $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Is it true that $f(u) \in L^p(U)$? I think it's ...
1
vote
1answer
35 views

Question on $ L^p $ spaces inequalities to prove limit exists

in my class on real analysis we are currently dealing with $ L^p $ spaces and I have been tackled with this problem from Folland's real analysis stating this: If $ f $ is absolutely continuous on ...
1
vote
1answer
38 views

What does it mean the notation $L^2(0,T;U)$?

I am studying second order parabolic equations, but i can't find in my book what the notation $L^2(0,T;U)$ means. Can anybody explain this to me? Thanks
-1
votes
1answer
31 views

In the space $l_1$,$0$ is not in $conv^* \{e_i\}$

Show that in the space $l_1$ , $0$ is not in $conv^* \{e_i\}$ where $e_i$ 's are the standard basis vectors of $l_1$. I was trying to solve this by contradiction but I failed..Need some help.
1
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0answers
25 views

a Direct computation of infinitesimal generator

Could anyone show me a way to directly compute the generator of a strongly continuous semigroup $S(t)$ in $L^p$ $$\lim_{t\to 0}\frac{S(t)f-f}{t}$$ where the limit shall lies in $L^p$ Here $f\in ...
1
vote
1answer
47 views

If $f \in L^{p_1}$ and $f \in L^{p_2}$ with $1 \le p_1 \lt p_2 \lt \infty$, then $f \in L^{p}$ for all $p$ such that $p_1 \leq p \leq p_2$.

Let $(X, \Sigma, \mu)$ be a measure space. If $f \in L^{p_1}$ and $f \in L^{p_2}$ with $1 \le p_1 \lt p_2 \lt \infty$, then $f \in L^{p}$ for all $p$ such that $p_1 \leq p \leq p_2$. My attempt Let ...
2
votes
0answers
39 views

Averages of integral and$ L^p$ space problem

Let $f: \mathbb R \to \mathbb R$ be an integrable function, for each $h>0$ let $$f_h(t)=\dfrac{1}{h}\int_{t-\frac{h}{2}}^{t+\frac{h}{2}}f(x)dx$$ Suppose $f \in L^P$, prove the following (1) $f_h ...
1
vote
1answer
16 views

Weak convergence in $L^2(0,T;L^2)$ and a $\limsup$

Let $u_n \rightharpoonup u$ in $L^2(0,T;L^2(\Omega))$ for a bounded domain $\Omega$. We have $$\lVert u_n(t) \rVert_{L^2(\Omega)} \leq C$$ for all $t$ where $C$ is independent of $t$. Does it ...
0
votes
1answer
29 views

Kind of Cauchy-Schwarz inequality

Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain. Define the Hilbert space $$ H(div;\Omega):=\{u\in (L^2(\Omega))^3:\nabla\cdot u\in L^2(\Omega)\} $$ equipped with the graph norm $$ ...
2
votes
1answer
34 views

Maximal Ideals in $L^1(\mathbb{R})$

Define $I_\xi = \{ f \in L^1(\mathbb{R}) : \hat{f}(\xi)=0 \}$. I have to prove that $I_\xi$ is a maximal ideal in $L^1(\mathbb{R})$. The following are my attempts at solution : Attempt 1 : Consider ...
2
votes
0answers
21 views

Is such a function in $L^2$?

Let $L^2$ denote $L^2([0,T];\mathbb{R}^N)$ and let $A:[0,T]\times L^2 \to (L^2)^\prime =L^2$ be a Carathéodory function satisfying the following estimate: $$\exists \alpha \in ...
4
votes
0answers
83 views

Limit of $L_p$ norm as $ p \rightarrow 0$

I have reviewed Ayman Houreih's proof for the limit of the $L_p$ norm as $ p \rightarrow 0$ at "Scaled $L^p$ norm" and geometric mean. While I have found the outline of the proof very ...
1
vote
1answer
26 views

Eigenfunction representation of the L2 derivative

I think the main idea of the definitions that follow is to define some sort of generalized double derivative on a subset of $L^2[0,1]$ Define $D(K)$ to be the subset of $C^1[0,1]$ made up of ...
2
votes
0answers
20 views

Find K s.t. $\int_{\Omega}|K-f(x)|(K-f(x))=0$

I'm trying to solve the following problem: Let $\Omega$ a bounded set of $\mathbb{R}$, $f\in L^3(\Omega)$. Find $K\in\mathbb{R}$ such that $\int_{\Omega}|K-f(x)|(K-f(x))=0.$ I'dont know how to ...
1
vote
1answer
16 views

Norm of projection map on $L^p(\mathbb{R}^n)$

$1\leq p < \infty$. Space is $L^p(\mathbb{R}^n)$. Let $\delta >0,\ R>0$ be constants. $Q$ is the open cube centered at origin such that $||y||<\frac{\delta}{2}, \forall y \in Q$. Let ...
2
votes
1answer
28 views

Inverse continuity of an operator

Let $X$ be a Banach space (it is in fact an $L^p$ space) and let $T:X \to X$ be a linear continuous operator (which is not injective and not surjective). I am trying to figure out if the following is ...
0
votes
1answer
49 views

Prove by contradiction that $|f(x)| < \infty$ almost everywhere for $f \in L^p$ ($p \ge 1)$

Let $f \in L^p$, where $p \ge 1$, then $|f(x)| < \infty$ almost everywhere. Does anyone know how to prove this by contradition?
0
votes
1answer
26 views

Closed and bounded but not compact in $L^p(\mathbb{R}^n)$

I am reading the paper "The Kolmogorov-Riesz Compactness Theorem", which gives a characterisation for totally bounded subsets of $L^P(\mathbb{R}^n)$ for $1 \le p <\infty$. A subset $\mathcal{F} ...
2
votes
0answers
34 views

Khintchine inequality (question about Holder inequality)

I'm reading a proof of a Khintchine inequality : Let $(r_{1}, \dots , r_{n})$ be iid random variables with $P(r_{i} = \pm1) = \frac{1}{2}$. Let $f = \sum\limits_{j=1}^{n}a_{j}r_{j}$, where ...
3
votes
0answers
48 views

$L^p$ norm converges to $L^\infty$ norm

The question is: Let $ (X,\mathcal M,\mu) $ be an arbitrary measure space. Let $f$ be a function in $L^r$ for some $0<r<\infty$. Show that $||f||_p$ converges to $||f||_\infty$ as $p\to \infty$. ...
2
votes
0answers
20 views

$L^p$ spaces inclusion [duplicate]

Show that for any measurable function $f$ in a measure space, we have: $$ ||f||_p \leq \max\{||f||_r ,||f||_s \} $$ whenever $0<r<p<s$. Now by splitting the integrals into parts where ...
1
vote
0answers
21 views

An inequality involved $L_p $ functions [duplicate]

If $p\ge 2$ & $f,g $ are $L_p $ functions, prove that : $||\frac {f+g}{2}||_p^p +||\frac {f-g}{2}||_p^p \le \frac {1}{2}[||f||^p_p +||g||_p^p ]$ At first glance , I thought it is easy noticing ...
2
votes
1answer
41 views

The distance from a point in $l_\infty$ to $c_0$

$l_\infty$ is the space of bounded sequences and $c_0$ is the space of sequences converge to $0$, is a closed subspace of $l_\infty$. I am trying to prove that for any $x \in l_\infty$, $\;d(x,c_0) = ...
1
vote
1answer
34 views

$ \lVert {\bf f} \rVert_{p} \leqslant (m(E))^{1-1/p} \lVert {\bf f} \rVert_{\infty} $ for any $ f \in L_{\infty}(E) $

If $m(E)$ is finite and ${\bf f}\in L_\infty(E)$ then for any $p\geqslant 1$, $$ \lVert {\bf f} \rVert_{p} \leqslant (m(E))^{1-1/p} \lVert {\bf f} \rVert_{\infty} .$$ I tried to apply Hölder's ...
3
votes
1answer
33 views

Normalize a function and a measure so that the $L^p$ norm is $1$ for two values of $p$

I'm reading Tao's book on the interpolation of $l^p $ spaces and one part writes "if $\|f\| _{ L^{p _ 0} } = \|f \|_ {L ^{ p _ 1 }} =1 $ then we are done. To obtain the general case, one can multiply ...
2
votes
2answers
78 views

Uniform Convergence Implies $L^2$ Convergence and $L^2$ Convergence Implies $L^1$ Convergence

Some of the books that discuss convergence say that uniform convergence implies $L^2$ convergence and $L^2$ convergence implies $L^1$ convergence, both while taken over a bounded interval I. While I ...
0
votes
0answers
25 views

$L^1$ integrable but not $L^p $integrable for all $p>1$ [duplicate]

Does there exist a function on the real line such that it is $L^1$ integrable but not $L^p $integrable for all $p> 1$? I encountered this problem in Rudin's Real and Complex analysis which ask us ...
1
vote
1answer
39 views

Convolution with imaginary Gaussian cannot be a $(p, p)$ operator unless $p=2$

Let $g=e^{-ix^2}, x\in \mathbb{R}$. Let $T$ be an operator defined as $T(f)=f*g$. Show that $T$ cannot satisfy a $(p,p)$ inequality unless $p=2$. Note: We say an operator $T$ satisfies a ...
1
vote
0answers
34 views

exercise: dual of an Lp-space

I have this exercise: Let $(\Omega,\mathcal{A},\mu)$ be an arbitrary measure space and $1 <p <\infty$. Show that if $l \in \mathcal{L}^p(\mu)^*$, then there exists a sequence ...
10
votes
1answer
135 views

Does $L^p$ have a basis for which the Pythagorean identity with exponent $p$ holds?

In the $\ell^p$ spaces with $1\leq p<\infty$, let $\{e_n\}$ be the standard basis. If $x=\sum_{n=1}^\infty a_ne_n$ is in $\ell^p$, then for any $k$ we can write $$||x||^p=\sum_{n=1}^k ...
2
votes
0answers
23 views

function $f\notin L^{\infty}(\Omega)$, but $f\in L^p(\Omega)$ for all $1\le p <\infty$ [duplicate]

I'm searching for a function $f\notin L^{\infty}(\Omega)$, but $f\in L^p(\Omega)$ for all $1\le p <\infty$. And it has to be $|\Omega |<\infty$. I tried $f(x)=\frac{1}{2\sqrt{x}}$ and $\Omega= ...
0
votes
1answer
69 views

$L^p$ convergence of infinitesimal generator

I have trouble express explicitly the following $$ \lim_{h\to 0^+} h^{-1}\|f_h-f-hg\|_{L^p(\mathbb R)}=0. $$ which relates to the earlier post in finding generator. How to find the infinitesimal ...
1
vote
1answer
32 views

If $0<\mu(X)<\infty$ and $0<p<q< \infty$. Is $(\int_X |f|^p) ^\frac{1}{p}\leq \int_X |f|^q d\mu) ^\frac{1}{q}$ true?

If $0<\mu(X)<\infty$ and $0<p<q< \infty$ Is $(\int_X |f|^p d\mu) ^\frac{1}{p}\leq \int_X |f|^q d\mu) ^\frac{1}{q}$ true. I should say that $\frac{1}{p}+\frac{1}{q}=1$ I think it is ...
0
votes
1answer
56 views

How to prove that a Schwartz function belongs to $L^p$?

If I have a function $f$ belongs to the Schwartz space, i.e. $f\in \mathcal{S}$, how can I prove $f\in L^p$ ? I know that $\mathcal{S}\subset L^p$ hence the above should make sense. But I need a ...
1
vote
1answer
33 views

Proving fact about shifts of $L^\infty$ functions

I am asked to prove that $\tau_{h_n}f\to f$ weakly-* in $L^\infty$, where $f\in L^\infty$, $\tau_{h_n}f(x)=f(x-h_n)$ and $h_n\to0$ in $\mathbb{R}^n$. I managed to prove it assuming $f$ is a.e. equal ...
1
vote
2answers
58 views

Property of functions: do there exist counterexamples?

I was assigned the following: Let $\tau_hf(x)=f(x-h)$, with $f:\mathbb{R}^N\to\mathbb R$ in $L^\infty$ and $x,h\in\mathbb R^n$. Prove translations of $f$ by $h_n\to0$ converge weakly-* in ...
1
vote
2answers
61 views

Understanding this integral from a measure theory perspective.

I know that $$\int_1^\infty \frac{1}{x} dx$$ does not converge in regular calculus. But I'm looking at $L^p$ spaces now and this integral is a good counter example for some things, but what is the ...
1
vote
0answers
40 views

Compact matrix integral operator bound via its kernel

Let $\mathcal{H} = L^p_{n}(0,a)$, where $p \in \mathbb{R}^+$, $p \geq 1$ and $n \in \mathbb{N}$, be the Hilbert space of vector-valued functions defined on the finite interval $(0,a) \subset ...
3
votes
2answers
65 views

Hint for Lebesgue theory/functional analysis type of problem

I am trying to solve the following problem, but I am not too familiar with functional analysis. Could you guys tell me where I should start? Thanks! Let $f \in L^1(\mathbb{R})$ and define $$f_n(x) = ...
3
votes
1answer
54 views

Is $(\oplus\ell_2^n)_{\ell_1}$ complemented in $\ell_1\oplus_\infty\ell_p$?

Fix any $1<p\leq 2$. Let us recall that $E:=(\oplus\ell_2^n)_{\ell_1}$ is just the space of sequences $(x_n)_{n=1}^\infty$, $x_n\in\ell_2^n$, such that $(\|x_n\|_{\ell_2^n})_{n=1}^\infty\in ...
3
votes
1answer
44 views

Prove an equality ($L^P$ spaces)

Prove the equality $$\int |f(x)|^p dx=\int_0^\infty pt^{p-1}m(\left\lbrace x:|f(x)|\geq t\right\rbrace)dt$$ for $p\geq 1$. My first idea was to try to prove this via induction. For the case $p=1$, ...
4
votes
1answer
71 views

Let $a_f=\text{ arg} \min_{a} \int \left|f(x)-a\right| dx$ and $a_g= \text{ arg} \min_{a} \int \left|g(x)-a\right| dx$, is $a_f \le a_g$?

Let $ f(x) \le g(x) $ and assume that $g(x),f(x) \in L^1$ let \begin{align} a_f= \text{ arg} \min_{a } \int_A \left|f(x)-a\right| dx\\ a_g=\text{ arg} \min_{a } \int_A \left|g(x)-a\right| dx ...
0
votes
1answer
36 views

The asymptotic behavior of an absolutely continuous function with square integrable derivative

Let $f : \mathbb R \to \mathbb R$ be absolutely continuous and assume that $f'\in L^2([0,1])$ and that $f(0) = 0.$ Show that the following limit exists and compute its value: $ \lim_{x \to 0} ...
1
vote
0answers
22 views

Pairs $(p,q)$ such that $id: l_p\to l_q$ is bounded [duplicate]

find all pairs $p,q\in [1,\infty)$ such that $id: l_p\to l_q$ is bounded. This just means I must find all $(p,q)$ such that $\|x\|_q \le C\|x\|_p$ for some $C$ dependent on $p$, $q$. I don't know ...
1
vote
2answers
38 views

Prove that $\|f\|_p \leq \liminf \|f_n\|_p$ under weak convergence

Let $1<p<\infty$ and $q$ its conjugate. Given a sequence $(f_k)_{k \in \mathbb N}$ and $f$ in $L^p(\mathbb R^d)$, I am trying to show that if for all $g \in L^q(\mathbb R^d)$, $$\lim_{k \to ...
1
vote
3answers
95 views

Why is $C=\{x\in l^2:|x_n|\leq 2^{-n},n=1,2,3,\dots\}$ compact?

Why is $C=\{x\in l^2:|x_n|\leq 2^{-n},n=1,2,3,\dots\}$ compact? I tried to show that $C$ is totally bounded and closed. I showed that is closed but I don't know how to show that is totally bounded. ...
3
votes
0answers
75 views

Universal property of l^p-spaces

The category $\mathsf{Ban_1}$ of Banach spaces together with short linear maps (i.e. those of norm $\leq 1$) seems to have a natural construction which interpolates between coproduct and product: Let ...
1
vote
0answers
35 views

Problem related to $L^p$ and distribution function

I am trying to solve the following problem: Let $\lambda_f(t) := \mu(\{x \in X: |f(x)| > t\})$ Prove that $f \in L^p(\mathbb R^d)$ for $0<p<\infty$ if and only if ...
0
votes
1answer
15 views

Showing $\ell^p$ space is linear

from what I understand, it is easy to show that $\ell^p$ space is linear, without the famous inequalities (Minkowsky, Holder, etc.). The fact that is a metric space is not of interest for the moment. ...