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.
5,707
questions
225
votes
7
answers
141k
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 ...
- 2,259
204
votes
4
answers
86k
views
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\|...
- 3,354
107
votes
2
answers
19k
views
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
$\{f_k, f\} \subset L^p$ (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 ...
- 8,633
92
votes
2
answers
48k
views
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 ...
- 2,475
66
votes
6
answers
24k
views
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).
- 8,633
60
votes
2
answers
32k
views
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 ...
- 1,663
51
votes
3
answers
24k
views
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 ...
- 10.5k
48
votes
2
answers
20k
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 \...
- 867
47
votes
5
answers
34k
views
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
47
votes
2
answers
5k
views
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 $\...
- 78.7k
43
votes
3
answers
8k
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" $L^p$ ...
- 3,774
42
votes
1
answer
678
views
$\frac{1}{p}+\frac{1}{q}=1$ vs $\sum_{n=0}^\infty \frac{1}{p^n}=q$
It just occurred to me that conjugate exponents, i.e. $p,q\in(1,+\infty)$ such that $$\frac{1}{p}+\frac{1}{q} =1$$
also satisfy the relations:
$\sum_{n=0}^\infty \frac{1}{p^n}=q;$
$\sum_{n=0}^\infty \...
- 5,783
39
votes
1
answer
41k
views
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^{\...
- 2,411
37
votes
3
answers
3k
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 exponent allows for convenient ...
- 7,983
37
votes
1
answer
19k
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 ...
- 90.9k