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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0
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1answer
40 views

Prove an identity without using Hölder's inequality

How to prove the following without using Hölder's inequality : $$ \|f\|_{p} = \sup_{\|g\|_q =1} \int |fg| d\mu ; \frac{1}{p} + \frac{1}{q} =1$$
3
votes
1answer
42 views

a function with infinity L^p norm

Let $1\leq p<\infty$, $1/p+1/q=1$. For a function $f$ with $||f||_q=\infty$, can we write $$ ||f||_q=\sup_{g\in L^p(\Omega),||g||_p\neq 0}\frac{\int_\Omega |fg|}{||g||_p}? $$ or $$ ...
7
votes
1answer
91 views

What information is contained in the function $p\mapsto ||f||_p$?

Given a measurable function $f:\mathbb{R}\rightarrow\mathbb{R}$, we obtain a function $\nu_f:(0,\infty)\rightarrow [0,\infty]$ defined by $\nu_f(p):=||f||_p$ This function $\nu_f$ will not ...
3
votes
0answers
44 views

Is $L^p$ separable?

Whether a $L^p(X,\mu)$ space is separable? I understand that the answer depends on $p$ and $X$. It seems to me that it is separable when $1\leq p < \infty, X=\mathbb{R}^n$ or $X=\mathbb{N}$. ...
1
vote
1answer
38 views

If $\| \psi \|_2=1$ can I say something about $\| \psi' \|_2$?

If I have a differentiable $L^2$ function $\psi:\mathbb R\rightarrow \mathbb C$ which is normalised $$ \int |\psi(x)|^2\;\text d x = 1 $$ can I say anything about the order of $$ \int ...
0
votes
0answers
36 views

Dense subsets in $L^1(\mathbb{R})$

Which of the following are dense subsets in metrical space $L^1(\mathbb{R})$? set of smooth functions $C_0^{\infty}(\mathbb{R})$ with compact supports; set of above-mentioned functions' derivatives ...
3
votes
1answer
86 views

Prove that $L^1(\mathbb{N})$ is a Banach space.

I'm trying to prove that $L^1(\mathbb{N}) := \left\{ (x_n)_{n=1}^{\infty} : \sum\limits_{n=1}^{\infty}\left|x_n\right| < \infty \right\} $, the space of all sequences over the field $\mathbb{C}$ ...
2
votes
2answers
58 views

Properties of a sequence of iid rv's

I cannot do part a), and Im fairly sure that b) and c) will follow from it. If possible could I please have a solution to part a) and hints if you feel necessary to parts b) and c).
2
votes
2answers
47 views

Weak convergence of partial sums

I recently came across an interesting problem on weak convergence in $\ell^2 (\Bbb N)$. Suppose that we have canonical basis $\{e_i\}$ in $\ell^2 (\Bbb N)$. We need to prove that the sequence ...
0
votes
2answers
24 views

Let $u_n \to u$ in $L^1(\Omega)$. Does $u_n^p \to u^p$ in $L^1(\Omega)$ if we know $u_n^p \in L^1(\Omega)$?

Suppose $u_n \to u$ in $L^1(\Omega)$ where $\Omega$ is a bounded domain. Suppose that $u_n^p \in L^1(\Omega)$ (actually $L^\infty(\Omega)$ for each $n$). Fix $p \in [1,\infty)$. So $u_n(x) \to u(x)$ ...
3
votes
1answer
33 views

Find a sequence of r.v's satisfying the following conditions

I think part a) can be solved by using $X_n=\frac{1}{n}\chi_{[0,n^2]}$ Not sure about part b).
0
votes
1answer
31 views

If $u_n \rightharpoonup u$ in $L^2(\Omega)$, does $u_n^+ \rightharpoonup u^+$ in $L^2(\Omega)$?

Let $\Omega$ be a bounded domain. If $u_n \rightharpoonup u$ in $L^2(\Omega)$, does $u_n^+ \rightharpoonup u^+$ in $L^2(\Omega)$ where $u_n^+ = \max(0,u_n)$. Note all convergences are weak. My ...
1
vote
2answers
35 views

Are bounded linear functionals on $L^{\infty}$ of “bounded variation?”

Let $(X,\mathscr M,\mu)$ be a measure space and let $L^{\infty}$ be the set of (equivalence classes of) essentially bounded measurable functions on it. Suppose that $\phi\in (L^{\infty})^*$; that is, ...
2
votes
1answer
45 views

Increasing sequence of closed subspaces of $L^2$

Not sure how to start on this question. Writing out a decomposition didnt seem to lead to anything. I think you might have to guess the answer the question first before starting a proof. Please ...
2
votes
1answer
25 views

$L_p$ spaces and tail estimates

I can prove the main identity in this question. Not sure how the "and deduce" bit works. I think $O(\lambda^{-q})$ is some kind of tail estimate.
0
votes
0answers
25 views

How about integral version of Holder's inequality?

In light of the fact that Minkowski's inequality have integral version, I thought there might be one for Holder's as well. I cannot find any through searching (there is an infinite product version in ...
1
vote
1answer
19 views

Unique extension of operators

Let $p,q$ be conjugate exponents with $p\in(1,\infty]$ and consider a measure space $(X,\mathscr M,\mu)$. Suppose that $S:L^p\to L^p$ and $T:L^p\to L^p$ are both bounded, linear operators. Assume that ...
1
vote
1answer
27 views

Equivalent Definitions of the $L_2$ inner product.

If $g \in L_2(\mathbb{R})$, then we can define the $L_2$ norm to have the following relationship: $\|g\|_2^2 = \int_\mathbb{R} g^2$. If $A\subseteq \mathbb{R}$, then we can define the norm of $L_2(A)$ ...
9
votes
0answers
92 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponenent allows for convenient ...
-1
votes
1answer
61 views

Example of open operator but not closed [closed]

Assume that $T:\ell_1\to\ell_2 $ is bounded,linear and one-to-one. Prove that $T(\ell_1)$ is not closed in $\ell_2$
2
votes
1answer
35 views

Convergence in $L^p$ plus bounded gradient implies that the limit belongs to $W^{1,p}$?

I have a question with this problem I have found in the latest edition of the book Functional analysis, Sobolev Spaces author Haim Brezis pag 264 Remark 4 Let $(u_n) \subset W^{1,p} $ such that $u_n ...
0
votes
0answers
22 views

A sufficient condition to be in $\ell^1$ [duplicate]

I would like to show the following, but I'm not sure how exactly to go about it. Let $y = (y_k)_{k \in \mathbb{N}}$ be a sequence in $\mathbb{C}$ such that for all $x = (x_k)_{k \in \mathbb{N}} \in ...
1
vote
1answer
32 views

$L^p$ norm of a measurable function is bounded by its operation on step functions

Let $1\leq p<\infty$, $1/p+1/q=1$. Let $f$ be a measurable function on $[0,1]$ such that for all step functions $g$ on $[0,1]$ $$ \left|\int_0^1 fg d\mu\right|\leq \|g\|_q. $$ Prove that ...
2
votes
1answer
46 views

a condition given by step functions implies the condition holds for L^q space

Let $1\leq p<\infty$, $1/p+1/q=1$. Let $f$ be a measurable function on $[0,1]$ such that for all step functions $g$ on $[0,1]$, $$ |\int_0^1 fg d\mu|\leq ||g||_q. $$ Prove $||f||_p\leq 1$. How ...
0
votes
1answer
50 views

negative Sobolev space contains $L^1$ for a compact domain

I'd like to use something like Aubin-Lions lemma for the following spaces: $$ C^{0, \alpha}(B) \subset L^1(B) \subset W^{-1, q}(B),$$ with $B \subset \mathbb{R}^n$ being a compact, say a closed ball ...
0
votes
1answer
31 views

Isometric isomorphism between $\ell^{1}(\mathbb{R}^{3})$ and $\ell^{\infty}(\mathbb{R}^{3})$

Why is there no isometric isomorphism between $\ell^{1}(\mathbb{R}^{3})$ and $\ell^{\infty}(\mathbb{R}^{3})$? I know that there is such an isomorphism if $\mathbb{R}^{3}$ is replaced with ...
3
votes
1answer
50 views

Another $L^p$ problem, $p\in(0,1)$

Suppose that $0<p<1$. Let $h:(0,1]\to\mathbb C$ be a Lebesgue-measurable function such that $$\int_{x}^1|h(t)|\,\mathrm dt<\infty\quad\forall x\in(0,1].$$ Suppose also that ...
1
vote
1answer
30 views

Functional relation to be in $L^{p}$

Suppose that $f(t)=a(t)g(t)+b(t)$ for $t\geq0$, where $a$ and $b$ are continuous functions. Thus, once can immediately can say that if $b\in{}L^{p}$, $\liminf_{t\to\infty}|a(t)|>0$ and ...
2
votes
2answers
39 views

Rate of divergence of the integral of an $L^q$ function

Let $p,q\in(1,\infty)$ be conjugate exponents (i.e., $1/p+1/q=1$) and let $f:(0,\infty)\to\mathbb R_+$ be a Lebesgue-measurable function such that $\int_0^{\infty} f(x)^q\,\mathrm dx<\infty$. It ...
1
vote
1answer
42 views

Inequality between Fourier coefficients implies inequality for $L^p$ norms on the circle

Given two functions from $L_p [-\pi,\pi]$, where $p\geq 2$, $p$ is an even integer, and $f_n>|g_n|$ for every $n$ (where $f_n$ is the $n$th Fourier coefficient), I need to prove that ...
2
votes
0answers
37 views

Sequence which does not admit weakly convergent subsequences

Let $f_h:[0,1]\to\mathbb{R}$ be defined as $f_h(x)=h$ if $0\le x\le 1$ and $f_h(x)=0$ otherwise. Prove that it does not exist a subsequence $(f_{h_k})_k$ weakly convergent in $L^1(0,1)$. Attempt of ...
0
votes
1answer
32 views

How can you compare two mathematical functions of single variable? [closed]

I have some functions and want to see how close one function is to the other. One way to check is correlation. What are the other ways of checking the "similarness", "closeness" or "nearness"?
1
vote
1answer
34 views

Reference for $f \in L^{p,\infty} \cap L^{q}$ then $f \in L^r$ for $p < r \leq q$

Okay, so I think I've shown that if $f \in L^{p,\infty} \cap L^{q}$ with $p < q$ then $f \in L^r$ for $p < r \leq q$ where $L^{p, \infty}$ denotes the weak $L^p$ space. what I did was I wrote $$ ...
0
votes
0answers
33 views

$L^{p}$ convergence [duplicate]

So... i am a little rusty with measure theory and i need some help with this exercise, if anyone can give me an idea to start Let $(X,\mu)$ a measure space. Let $f_n:X\rightarrow \mathbb{C}$ a ...
0
votes
2answers
23 views

Exercise on abstract integration

Let $f_n$ be a sequence of nonnegative functions defined on $\mathbb{R}^N$ such that $f_n \rightarrow f $ almost everywhere on $\mathbb{R}^N$ and such that $$\int_{\mathbb{R}^N} f_n \rightarrow ...
1
vote
1answer
57 views

If $a_n \to a$ in $L^2$ and $F:\mathbb{R} \to \mathbb{R}$ is continuous, does $\int F(a_{n_j}) \to \int F(a)$?

Suppose $a_n \to a$ in $L^2(\Omega)$. Let $F:\mathbb{R} \to \mathbb{R}$ be continuous with $F(0) = 0$. We have that $F(b) \in L^1(\Omega)$ if $b \in L^2(\Omega)$ and $|F'(x)| \leq C_1 + C_2|x|$. I ...
1
vote
1answer
25 views

Showing a composition is in $L^2$

Let $f$ be a smooth increasing function passing through the origin with $|f(x)| \leq C_1(1+|x|)$. Define $F(r) = \int_0^r f(s)\;ds$ and $F^*(t) = \sup_{r \in \mathbb{R}}tr - F(r).$ Note that ...
-2
votes
1answer
108 views

Continuity of $|.|$ in $W^{1,p}_0$

please i dont understand this proof We suppose that $u\rightarrow u$ on $W^{1,p}_0$ i dont understand why we must use the weak compactness and the uniform convexity of $W^{1,p}_0$ ? Thank you
0
votes
1answer
27 views

Find $\psi\in L^2$ so that the overlap $\langle \psi(x+\delta),\psi(x-\delta)\rangle$ is as small as possible

Is there a way to find / characterise functions $\psi\in L^2$ which make the value of this integral small? $$ \text{Re}\int_{\mathbb R} \overline{ \psi (x+\delta)}\,\psi(x-\delta)\;\text d x $$ ...
4
votes
1answer
94 views

Is $ L^{\infty} $ a direct limit or inverse limit of the directed system $ (L^p , i_{p}^q )_{p,q \in [1 , + \infty [ } $?

Let $X$ be a finite measure space. Then, for any $ 1≤p<q≤+∞ $ : $ L^q(X,B,m)⊂L^p(X,B,m) $. I would like to know if the space $ L^{\infty} ( X , B , m ) $ is the direct limit or the inverse limit of ...
1
vote
1answer
51 views

Explanation of a passage about a smooth approximation to $L^p$ function

I'm reading J.L Vazquez "Porous Medium Equation" book. In it, he says the following: We are given a function $a:\Omega \times (0,T) \to \mathbb{R}$ such that $a \geq 0$. We find a smooth ...
1
vote
1answer
33 views

Dual of $L^1$ when measure is the counting measure [closed]

Let $X$ be an uncountable set, $\mu$ the counting measure on $X$ and $\mathcal{M}$ the $\sigma$- algebra of countable or co-countable sets. How can I prove that the dual of $L^1(\mu|\mathcal{M})$ is ...
4
votes
1answer
46 views

Weak* convergence in $(L^q)^*$ and convergence in $L^p$

Let $\{\phi_n\}$ be a sequence in $L^p(X)$. Assume that $\phi_n\to \phi$ weak* under the natural identification $L^p\cong (L^q)^*$. Of course, it is not true in general that $\phi_n\to \phi$ in $L^p$ ...
1
vote
1answer
47 views

Prove that a relatively compact subset of $L^p$ is bounded.

Let $p\in [1,\infty)$, $A\subset L^p(\mathbb R^m)$ relatively compact and $\lambda^m$ be the Lebesgue measure on $\mathbb R^m$. Prove: a) $A$ is bounded. b) $\lim_{y \to 0}\sup_{f \in A} ...
5
votes
1answer
137 views

If $P$ has marginals $P_1, P_2$, is $L^1(P_1) + L^1(P_2)$ closed in $L^1(P)$?

Suppose that $\mathbb{X}=\mathbb{X}_1\times \mathbb{X}_2$ and suppose that $ P$ is a probability measure on $\mathbb{X}$ with marginals $ P_i$ on $\mathbb{X}_i, i=1,2$, i.e., $$\int f_i(x_i)\, ...
0
votes
0answers
24 views

does the limit of the ratio of $p+1$ norm and $p$ norm equal to $\infty$ norm

Suppose that $f\in L^1(\Omega,\Sigma,\mu)\cap L^\infty(\Omega,\Sigma,\mu)$. Then I have proved that for any $1\leq p\leq \infty$, $f\in L^p(\Omega,\Sigma,\mu)$. Moreover, I have proved ...
1
vote
0answers
38 views

Convergence of product of continuous functions and test functions

I suspect the following result is true but I"m not sure how to go about proving: It is given that $\Omega \subset \mathbb{R}^{n}$ is an open bounded, connected domain.(Not sure if theses conditions ...
2
votes
1answer
41 views

Projection onto finite-dimensional subspace of $L^p$

Let $a_i$ be a basis of $L^p(\Omega)$ and consider $A_n = \text{span}\{a_1, ..., a_n\}$. Take an element $f \in L^p$. We want to define a projection onto the finite-dimensional subspace $A_n$. How do ...
1
vote
1answer
26 views

questions about $L^p$ space with $0<p\leq 1$ parallel to the case $1<p$

Question (1). Riesz-Fischer Theorem: For $1\leq p\leq \infty$, $L^p(\mu)$ is complete. Corollary of proof: Let $1\leq p\leq \infty$. If $(f_n)_{n=1}^\infty$ is a sequence coverging to $f$ with ...
2
votes
2answers
65 views

What is $L^p$-convergence useful for?

Why do people care about $L^p$-convergence $f_n \rightarrow f$? Are there any interesting application of $L^p$-convergence? For example, if $p=\infty$, then the limit $f$ of the sequence $f_n$ of ...