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 classes of measurable functions such that $|f|^p$ is $\mu$-integrable. Questions can be about properties of functions in these spaces, or when the ambient space in a problem is an $L^p$ space.
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Limit of $L^p$ norm
Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, $\displaystyle\lim_{p\to\infty}\|f\|...
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votes
5
answers
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$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 ...
- 1,969
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If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$
Let $1\leq p < \infty$. Suppose that
$\{f_k\} \subset L^p$ (the domain here does not necessarily have to be finite),
$f_k \to f$ almost everywhere, and
$\|f_k\|_{L^p} \to \|f\|_{L^p}$.
Why is ...
- 8,273
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votes
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The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms. [duplicate]
If we are in a sequence space, then the $ l^{p} $-norm of the sequence $ \mathbf{x} = (x_{i})_{i \in \mathbb{N}} $ is $ \displaystyle \left( \sum_{i=1}^{\infty} |x_{i}|^{p} \right)^{1/p} $.
The $ l^{\...
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5
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How do you show monotonicity of the $\ell^p$ norms?
I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it).
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votes
2
answers
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The Duals of $l^\infty$ and $L^{\infty}$
Can we identify the dual space of $l^\infty$ with another "natural space"? If the answer is yes, what can we say about $L^\infty$?
By the dual space I mean the space of all continuous linear ...
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41
votes
2
answers
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"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" $L^p$ ...
- 3,574
35
votes
1
answer
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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 ...
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Show that $\lim _{r \to 0} \|T_rf−f\|_{L_p} =0.$
I am having a hard time with the following real analysis qual problem. Any help would be awesome.
Suppose that $f \in L^p(\mathbb{R})$, where $1\leq p< + \infty$. Let $T_r(f)(t)=f(t−r)$. Show ...
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votes
3
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On the equality case of the Hölder and Minkowski inequalities
I'm following the book Measure and Integral of Richard L. Wheeden and Antoni Zygmund. This is problem 4 of chapter 8.
Consider $E\subseteq \mathbb{R}^n$ a measurable set. In the following all the ...
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1
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A Hamel basis for $\ell^p$?
I am looking for an explicit example for a Hamel basis for $\ell^{p}$?. As we know that for a Banach space a Hamel basis has either finite or uncountably infinite cardinality and for such a basis one ...
- 297
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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 < \...
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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 unless $p=2$.
Maybe I would have to use the Rademacher's functions.
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Why is $L^{\infty}$ not separable?
$l^p (1≤p<{\infty})$ and $L^p (1≤p<∞)$ are separable spaces.
What on earth has changed when the value of $p$ turns from a finite number to ${\infty}$?
Our teacher gave us some hints that ...
- 1,593
24
votes
6
answers
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Dual of $l^\infty$ is not $l^1$
I know that the dual space of $l^\infty$ is not $l^1$, but I didn't understand the reason.
Could you give me a example of an $x \in l^1$ such that if $y \in l^\infty$, then $ f_x(y) = \sum_{k=1}^{\...
- 1,967
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votes
1
answer
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Convergence types in probability theory : Counterexamples
I know that the following implications are true:
$$\text{Almost sure convergence} \Rightarrow \text{ Convergence in probability } \Leftarrow \text{ Convergence in }L^p $$
$$\Downarrow$$
$$\text{...
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2
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$f \in L^1$, but $f \not\in L^p$ for all $p > 1$
"Find an $f \in [0,1]$ such that $f \in L^1$ but $f \not\in L^p$ for any $p > 1$."
I've thought about doing something like $$f(x) = \frac{1}{x}$$ where $|f|^p = \frac{1}{x^p}$ doesn't converge ...
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2
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Translation operator and continuity
I came across a text that proves that translation operator $T_a(f):=f(x-a)$ where $a\in\mathbb{R}^n$ and $f\in L^p(\mathbb{R}^n)$ is continuous. The proof follows:
$$||f(x-a)-f(x)||_p=||f(x-a)-g(x-a)+...
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votes
2
answers
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Proof of Pitt's theorem
I'm reading the book Topics in Banach Space Theory by Albiac F. Kalton N. J. I got stuck at the proof of Pitt's theorem.
In the second paragraphs authors tries to prove ad absurdum that for weakly ...
- 55.4k
1
vote
1
answer
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Question about $\lim _{q \rightarrow \infty}\|f\|_{q}=\|f\|_{\infty}$
Let $(X,B,\mu)$ be a complete measure space,Show that $$\lim _{q \rightarrow \infty}\|f\|_{q}=\|f\|_{\infty}, \quad \forall f \in \bigcup_{p} \bigcap_{p \leqslant q<\infty} L^{q}$$
So,$\lim _{q \...
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votes
2
answers
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Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$?
Is there an explicit isomorphism between $L^\infty[0,1]$ and
$\ell^\infty$?
In some sense, this is a follow-up to my answer to this question where the non-isomorphism between the spaces $L^r$ and $\...
- 76k
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votes
5
answers
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Why is $l^\infty$ not separable?
My functional analysis textbook says
"The metric space $l^\infty$ is not separable."
The metric defined between two sequences $\{a_1,a_2,a_3\dots\}$ and $\{b_1,b_2,b_3,\dots\}$ is $\sup\limits_{i\...
user67803
17
votes
1
answer
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Fourier transform in $L^p$
Let the $f$ be a function in $L^s$ where $s \in [1,\infty) $. For which $r$ Fourier transform $\hat{f}$ belongs to $L^r$?
I'd be grateful for any kind of help including providing a literature or ...
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2
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Inclusion of $l^p$ space for sequences
Inclusion of $L^p$ spaces for functions has been discussed here.
Does this apply to $l^p$ space of sequences similarly?
I tried to show the following: For $1\leq p<q<\infty$, $l^q\subset l^p$ ...
- 1,583
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2
answers
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How do I prove the completeness of $\ell^p$?
Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct:
Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ $\Rightarrow$...
- 201
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votes
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Sharp interpolation inequality for Lebesgue spaces
Suppose $f\in L^{p}(\mathbb{R}^{n}) \cap L^{q}(\mathbb{R}^{n})$. How can I prove that for any $p \lt r \lt q$,
$$
\lVert f \rVert_{r} \leq
(\lVert f \rVert_{p})^{(1/r-1/q)/(1/p-1/q)}
(\lVert f \...
- 129
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Why is every $p$-norm convex?
I know that $p$-norm of $x\in\Bbb{R}^n$ is defined as, for all $p\ge1$,$$\Vert{x}\Vert_p=\left(\sum_{i=1}^{n} \vert{x_i}\vert^p\right)^{1/p}.$$
The textbook refers to "Every norm is convex" for an ...
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$\ell_p$ is Hilbert if and only if $p=2$
Can anybody please help me to prove this:
Let $p$ be greater than or equal to $1$.
Show that for the space $\ell_p=\{(u_n):\sum_{n=1}^\infty |u_n|^p<\infty\}$ of all $p$-summable sequences (with ...
- 1,783
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votes
5
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Show that $l^2$ is a Hilbert space
Let $l^2$ be the space of square summable sequences with the inner product $\langle x,y\rangle=\sum_\limits{i=1}^\infty x_iy_i$.
(a) show that $l^2$ is H Hilbert space.
To show that it's a Hilbert ...
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votes
2
answers
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When does equality hold in the Minkowski's inequality $\|f+g\|_p\leq\|f\|_p+\|g\|_p$?
I would like to see a proof of when equality holds in Minkowski's inequality.
Minkowski's inequality. If $1\le p<\infty$ and $f,g\in L^p$, then $$\|f+g\|_p \le \|f\|_p + \|g\|_p.$$
The proof is ...
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votes
3
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A typical $L^p$ function does not have a well-defined trace on the boundary
This question is from PDE by Evans, 1st edition, Chapter 5, Problem 14. It has been posted here previously, however, I cannot quite put all the information together from the responses there. Hopefully ...
- 2,840
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votes
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Pointwise almost everywhere convergent subsequence of $\{\sin (nx)\}$
Can you prove or disprove that the sequence $\{\sin (nx)\}$ has a pointwise almost everywhere convergent subsequence with respect to the Lebesgue measure on $\mathbb{R}$ ?
Edit:
I am adding my ...
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When $L_p = L_q$?
As we know that $L_p \subseteq L_q$ when $0 < p < q$ for probability measure, I was wondering when $L_p = L_q$ is true and why. Is it to impose some restriction on the domain space? Thanks!
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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 ...
- 45.3k
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1
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convergence in $L^p$ implies convergence in measure
I am trying to show that if $f_n$ converges to $f$ in $L^p(X,\mu)$ then $f_n\to f$ in measure, where $1\le p \le \infty$.
Here is my attempt for $p\ge 1$: Let $\varepsilon>0$ and define $A_{n,\...
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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. ...
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1
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Problems with the proof that $\ell^p$ is complete
By struggling with the proof that $\ell^p$ is complete, I looked up different proofs by different authors, and I ended up focusing on the one given by Kreyszig in his classic book on functional ...
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Space $\mathcal{L}^p(X, \Sigma, \mu)$ is separable iff $(\Sigma, \rho_\Delta)$ is separable
Let's consider the space $\mathcal{L}^p(X, \Sigma, \mu)$ of all functions $f\colon X \to \mathbb{R}$ (or $\mathbb{C}$) for which:
$$
\int\limits_X|f|^p \mu(dx) < \infty.
$$
Here $X$ is a metric ...
- 3,032
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2
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Prove the set of sequences $c_0$ which converge to zero in $l_{\infty}$ is closed.
Prove the set of sequences which converge to zero in $l_{\infty}$ is closed.
Let $x_n(k)\rightarrow x(k)$ as $n\rightarrow\infty$. With $x_n(k)\in c_0$ and $x(k)\in l_{\infty}$.
Let $\varepsilon>...
- 7,277
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1
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Liapunov's Inequality for $L_p$ spaces
Let $1 \leq p,q < \infty$ and $0 \leq \lambda \leq 1$. If $r = \lambda p + (1 - \lambda)q$ and $f \in L_p \cap L_q $, then
$$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q} \tag{*}$$
...
- 1,645
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votes
2
answers
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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 \...
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votes
2
answers
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Measuring $\pi$ with alternate distance metrics (p-norm).
How/why does $\pi$ vary with different metrics in p-norms? Full question is below.
Background
Long ago I did an investigation on Taxicab Geometry using basic geometry. One think I recall is that a ...
- 11.8k
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votes
2
answers
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$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?
user97601
16
votes
2
answers
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Characterization of weak convergence in $\ell_\infty$
Is there some simple characterization of weak convergence of sequences in the space $\ell_\infty$? If yes, is there some similar claim for nets?
I was only able to come up with a characterization of ...
- 51.4k
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2
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What is the predual of $L^1$
Is there a nice characterization of the predual of $L^1$? So, what does the space $X$ look like, such that $X^*=L^1$, where the star denotes the dual of a Banach space. How do you start to find such ...
- 4,067
14
votes
2
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A relative compactness criterion in $\ell^p$
There is a relative compactness criterion for subset of $\ell^p$ that seems to me to be almost unheard-of (I say that because a google search provided no proofs nor references) but that is very useful:...
- 2,266
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votes
2
answers
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Definition of $L^\infty$
It seems like different authors use different definitions for the space $L^\infty$. For example wikipedia starts with bounded functions, and define the seminorm $\lVert f \rVert_\infty$, and take ...
- 1,373
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votes
2
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Why is $L^{1} \cap L^{\infty}$ dense is in $L^{p}$?
It is mentioned that using the interpolation inequality
$$\Vert f \Vert_{p} \leq \Vert f \Vert^{1/p}_{1} \Vert f \Vert_{\infty}^{1-1/p}$$
one can deduce that the space $L^{1} \cap L^{\infty}$ is ...
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votes
1
answer
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$\ell^{\infty}(\mathbb N)$ is not a separable space
I have to prove that $\ell^{\infty}(\mathbb N)$ is not separable.
My attempt
Consider a SUBSET $V$ of $\ell^{\infty}(\mathbb N)$ consisting of bounded sequences that have only $0$, $1$ entries, e.g. ...
- 6,109
6
votes
1
answer
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Proving that the unit ball in $\ell^2(\mathbb{N})$ is non-compact
So on my homework it says that to prove the unit ball in $\ell^2(\mathbb{N})$ is non-compact, it suffices to find countably many elements $x_n$ of $\ell^2(\mathbb{N})$ with $\lVert x_n\rVert \leq \...
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