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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23
votes
2answers
6k views

$L^p$ and $L^q$ space inclusion

Let $(X, \mathcal B, m)$ be a measure space. For $1 \leq p < q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample ...
19
votes
2answers
422 views

In $\ell^p$, if an operator commutes with left shift, it is continuous?

Our professor put this one in our exam, taking it out along the way though because it seemed too tricky. Still we wasted nearly an hour on it and can't stop thinking about a solution. What we have: ...
18
votes
0answers
441 views

How many points of intersection between an ellipse and an $L_p$-circle?

Consider an ellipse $E$ in the plane, centered at the origin. (In my case, the minor axis points into the nonnegative quadrant.) Let S be an "$L_p$-circle": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, ...
14
votes
2answers
4k views

Limit of $L^p$ norm

Could someone help me prove that given a measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^{\infty}$ and some $L^{q}$, ...
12
votes
1answer
693 views

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic Maybe I would have to use the Rademachers.
12
votes
1answer
132 views

Ideals of the algebra of all bounded linear operators on $\ell_p \oplus \ell_q$

Let $\mathcal{L}(X)$ be the algebra of all bounded linear operators from $X$ to $X$ for Banach space $X$. I need to show that $\mathcal{L}(\ell_p \oplus \ell_q)$ for $p \neq q$ contains at least two ...
11
votes
1answer
1k views

Strong and weak convergence in $\ell^1$

Let $\ell^1$ be the space of absolutely summable real or complex sequences. Let us say that a sequence $(x_1, x_2, \ldots)$ of vectors in $\ell^1$ converges weakly to $x \in \ell^1$ if for every ...
10
votes
4answers
820 views

Convergence of integrals in $L^p$

Stuck with this problem from Zgymund's book. Suppose that $f_{n} \rightarrow f$ almost everywhere and that $f_{n}, f \in L^{p}$ where $1<p<\infty$. Assume that $\|f_{n}\|_{p} \leq M < ...
10
votes
2answers
978 views

“Scaled $L^p$ norm” and geometric mean

The $L^p$ norm in $\mathbb{R}^n$ is \begin{align} \|x\|_p = \left(\sum_{j=1}^{n} |x_j|^p\right)^{1/p}. \end{align} Playing around with WolframAlpha, I noticed that, if we define the "scaled" $L^p$ ...
10
votes
2answers
145 views

Finite dimensional subspace of $C([0,1])$

Let $S$ be a subspace of $C([0,1])$, i.e. the continuous real functions on $[0,1]$. Assume that there exists $c>0$ such that $\|f\|_\infty\leq c \|f\|_2$ for all $f\in S$. Then $S$ must be ...
10
votes
4answers
124 views

$l^\infty(I)$ and $l^\infty(J)$ isometrically isomorphic with $|I| \not= |J|.$

Is it possible for $l^\infty (I)$ and $l^{\infty} (J)$ to be isometrically isomorphic with the cardinality of $I$ not equal to the cardinality of $J$? I'm able to show that if $1\le p < \infty,$ ...
9
votes
2answers
644 views

When exactly is the dual of $L^1$ isomorphic to $L^\infty$ via the natural map?

The dual space to the Banach space $L^1(\mu)$ for a sigma-finite measure $\mu$ is $L^\infty(\mu)$, given by the correspondence $\phi \in L^\infty(\mu) \mapsto I_\phi$, where $I_\phi(f) = \int f ...
9
votes
3answers
235 views

Properties of $\bigcap_{p > 1} \ell_p$

Consider the following space of sequences $$\left\{a=(a_n)_{n\in\mathbb{N}}:a\in\bigcap_{p>1}\ell_p, a_n\in\mathbb{R}\right\}$$ What are some of its properties? What is its relation to $\ell_1$ and ...
9
votes
1answer
385 views

Distance minimizers in $L^1$ and $L^{\infty}$

If $H$ is a Hilbert space, we have the Hilbert Projection Theorem, which tells us that given a nonempty, closed, convex subset $K \subset H$, and a point $x \in H$, there is a unique point $y \in K$ ...
9
votes
2answers
137 views

Various kinds of derivatives

Let $f\colon \mathbb{R}\to \mathbb{R}$ be a measurable function. Let us introduce the following notions of "derivative" of $f$. Classical derivative. The unique function $f'_c$ defined pointwise by ...
9
votes
0answers
91 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 ...
8
votes
2answers
760 views

$L^1$ and $L^{\infty}$ are not reflexive

I want some proof for the following statement : $L^1$ and $L^{\infty}$ are not reflexive. Can anyone help me, please? or reference me?
8
votes
1answer
483 views

How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function

We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$. ...
8
votes
1answer
282 views

Convergence in $L^1$ space

Suppose that $f_{n}$ is a sequence of measurable functions, in a finite measure space, $f_{n}\to f $ in $m$-measure and that there exists $g$ in $L^1$ such that $\vert f_n\vert \le g$. Prove that ...
7
votes
2answers
163 views

What does the $L^p$ norm tend to as $p\to 0$?

This is something I was thinking about, so I'm going to post it as a question and post my own answer. I hope that anyone who wants will comment, correct me if I'm wrong, and add their own knowledge ...
7
votes
1answer
531 views

Why is $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$?

In Lieb and Loss's Analysis, I saw that they mentioned $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$ (dense wrt the $L^2$ norm, I think). But I didn't find its proof in the ...
7
votes
2answers
164 views

Is the injection $\ell^p \subset \ell^q$ continuous for $p<q$?

It is easy to show that $\ell^p \subset \ell^q$ when $1 \leq p<q \leq + \infty$, but is the injection continuous? If so is $\ell^{\infty}$ the direct limit $\lim\limits_{\rightarrow} \ \ell^p$ as ...
7
votes
1answer
376 views

Isometry between $L_\infty$ and $\ell_\infty$

It is known that there exist some isomorphism between $L_\infty$ and $\ell_\infty$, which is not explicit at all. Could someone tell me whether there exist an isometric isomorphism between ...
7
votes
1answer
1k views

Inequalities in $l_p$ norm

I'm having difficulty with the following problem. Any help would be appreciated. Problem: Consider the sequence spaces $l_p$ with the usual norm. If $1\le p\le q\le \infty$, I want to show the ...
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 ...
7
votes
4answers
358 views

$\ell^p$ is not isometric to $\ell^q$

The problem is this: if $1\le p<q<\infty$ then $\ell^p$ and $\ell^q$ are not isometric (as Banach spaces). This is an exercise but I'd like to see an elegant proof.
7
votes
2answers
158 views

On $L^p$ and $\ell^p$

If a continuous and infinitely differentiable function $f(x): \mathbb{R}\to\mathbb{C}$ is in $L^p$, is it also true that $f(n),\ n\in \mathbb{Z}$ is in $\ell^p$?
7
votes
3answers
218 views

Convergence of functions in $L^p$

Let $\{f_k\} \subset L^2(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain and suppose that $f_k \to f$ in $L^2(\Omega)$. Now if $a \geq 1$ is some constant, is it possible to say ...
7
votes
1answer
202 views

$L^{p}$ functions from Rudin Exercises 3.5

I am attempting a question from Rudin's "Real and Complex Analysis" Chapter 3 question 5. I shall summarise the question as below: Suppose that $f$ is a complex measurable function on $X$, $\mu$ a ...
7
votes
1answer
336 views

Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.

Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$. The following is from the book "Sobolev spaces" ...
6
votes
2answers
248 views

Subspaces of $L^p$

So studying Qualifying Exam problems in Analysis I cam across this one: For $1\lt r \lt p \lt s \lt \infty$ where $\mu$ denotes Lebesgue measure, a) Construct a subspace of $L^p([0,1],\mu)$ such ...
6
votes
1answer
234 views

pythagoras theorem for $L_p$ spaces

Let's consider $L_2(\mathbb{R}^n)$. Let $Y$ be a non empty closed subspace of $L_2(\mathbb{R}^n)$. Let $x\notin Y$. Let $y^*$ be the best approximation of $x$ on $Y$, i.e., $\|x-y^*\|_2=\inf_{y\in ...
6
votes
1answer
555 views

How do you prove that $\ell_p$ is not isomorphic to $\ell_q$?

I guess that for all $1\le p,q<\infty $, such that $p\ne q$ , the spaces $\ell_p$ and $\ell_q$ are not isomorphic, but how do you prove this?
6
votes
1answer
79 views

$L_{p}(\mathbb{T})$ is not uniformly convex if $p \in \{1,\infty\}$.

How can I prove that $L_{p}(\mathbb{T})$ is not uniformly convex if $p \in \{1,\infty\}$. Here $\mathbb{T} = \mathbb{R}/\mathbb{Z}$
6
votes
1answer
49 views

How to obtain the inequality $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p} $ from Jensen's inequality?

Let $f$ be a positive function with $\lVert f \rVert_{L^2}=1$. Let $p= 2n/(n-2)$. How to obtain $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p}$ from Jensen's inequality? Here all ...
6
votes
1answer
51 views

Under what conditions on $f$ does $\|f\|_r = \|f\|_s$ for $0 < r < s < \infty$.

Question: If $f$ is a complex measurable function on $X$, such that $\mu(X) = 1$, and $\|f\|_{\infty} \neq 0$ when can we say that $\|f\|_r = \|f||_s$ given $0 < r < s \le \infty$? What I ...
6
votes
1answer
74 views

Proof of equivalence of Sobolev Space and Lipschitz functions

The attachment is a proof from Evans book "Measure Theory and Fine Properties of Functions" pg 132 Theorem 5. The statement of the theorem is: Let $f:U \rightarrow \mathbb{R}$. Then $f$ is locally ...
6
votes
1answer
139 views

Extend a function by convolution

Let $f \in \mathcal{C}^{\infty}(\mathbb{R})$ be a compactly supported function ($supp(f)\Subset\mathbb{R})$. I am wondering about the existence of a $g \in L^p(\mathbb{R})$, for some $p$, such that ...
5
votes
4answers
143 views

$\int f_k\to 0 $ but $f_k $ does not converge to $0 $ ae, where $ f_k $ is defined in $[0, 1] $

Give an exemple, in [0, 1], of a sequence of functions $ f_k $ such that $||f_k||_ 1=\int |f|_k \to 0 $ but $ f_k $ does not converge to $0 $ a.e.
5
votes
2answers
2k views

Smooth functions with compact support are dense in $L^1$

Here is another homework question that I did and I'd be glad if you could tell me if it's right. We now strengthen the result of Question Two for $R$ where we have the notion of differentiability. ...
5
votes
2answers
206 views

Are integrable, essentially bounded functions in L^p?

Given an arbitrary measure space (of possibly infinite measure), if $f \in L^1 \cap L^\infty$, then by Hölder's inequality, $f^2 \in L^1$, so $f \in L^2$. Intuition suggests that $f \in L^p$ even for ...
5
votes
2answers
61 views

Do non-$\ell^2$ sequences have an $\ell^2$ functional that takes them to infinity?

Suppose $\{a_n\}_{n=1}^{\infty}$ is a sequence of real numbers (suppose also positive for simplicity) so that $$\sum_{n=0}^{\infty} a_n^2 = \infty$$ i.e. the sum diverges. Can you necessarily find a ...
5
votes
1answer
186 views

Compact linear operator from $L^p (\mathbb R)$ to $L^p (\mathbb R)$

Wanted to prove the following question since one week but couldn't get even single idea on it . Here is the question : if $m : \mathbb R \to \mathbb R$ a measurable function $1 \le p <\infty $ and ...
5
votes
1answer
62 views

An inequality of $L^p$ norms of linear combinations of characteristic functions of balls

Let $1<p<\infty$. Let $(a_n)_{n=1}^\infty$ be a sequence of nonnegative real numbers and $\{B_{r_i}(x_i)\}_{i=1}^\infty$ be a sequence of open balls in $\mathbb{R}^n$. Prove that there exists ...
5
votes
2answers
186 views

Closed set in $\ell^1$

Show that the set $$ B = \left\lbrace(x_n) \in \ell^1 : \sum_{n\geq 1} n|x_n|\leq 1\right\rbrace$$ is compact in $\ell^1$. Hint: You can use without proof the diagonalization process to ...
5
votes
1answer
111 views

Showing that $\|f\|_p\to\|f\|_{\infty}$

I know this question has been asked a lot in this site, I've been checking those questions myself, however according to the theory we use in class there are things that I can't use and/or I don't know ...
5
votes
1answer
116 views

Subsequence convergence in $L^p$

I recall a fact that for functions $f_1,f_2,\ldots\in L^1$ such that $\|f_n-f\|_1\rightarrow 0$ as $n\rightarrow\infty$, there exists a subsequence $f_{n_i}$ that converges to $f$ almost everywhere. ...
5
votes
1answer
159 views

Is $(l^1 ,\|.\|)$ a Banach space?

Suppose $x=\{x_n\}\in l^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$, let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $l^1$ . Is $(l^1 ,\|.\|)$ a Banach space?
5
votes
1answer
293 views

Is there anyway to bound the $L^\infty$ norm by other $L^p$ norm?

If $f\in L^\infty(\mathbb R^2)$ (in my particular exercise, $f\in H^2(\mathbb R^2)$, the sobolev space), I want to bound $|f|_{L^\infty}= $ esssup $|f|\leq c|f|_{L^p}$ for some p, what kind of number ...
5
votes
1answer
136 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)\, ...