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.

$L^p$ spaces are defined for $p\in(0,\infty]$ as follows. Let $(X,\mathcal F,\mu)$ be a measure space. For $p$ with $0 < p<\infty$ we write $L^p(X,\mu)$ ($L^p(X)$ or $L^p$ when there is no ambiguity), for the vector space of equivalence classes (for equality almost everywhere) of measurable functions $f$ such that $\int_X|f|^pd\mu$ is finite. When $1\leq p<\infty$ we endow $L^p$ with the norm $\lVert f\rVert_p:=\left(\int_X|f(x)|^pd\mu(x)\right)^{1/p}$, while for $0 < p < 1$ we write $\lVert f\rVert_p:=\int_X|f(x)|^pd\mu(x)$, which induces a metric on $L^p$.

For $p=+\infty$, $L^\infty$ is the space of equivalence classes of functions $f$ such that we can find a constant $C$ with $|f(x)|\leq C$ almost everywhere. Then $\lVert f\rVert_{\infty}$ is the infimum of constants $C$ satisfying the latter property, and is called the essential supremum of $|f|$.

These spaces are sometimes called Lebesgue spaces. If we work with counting measure on, for example, the set $\mathbb N$, we get sequence spaces $\ell^p$ and $\ell^\infty$ as special cases.

4956 questions
30
votes
2 answers

Hölder's inequality with three functions

Let $p,q,r \in (1,\infty)$ with $1/p+1/q+1/r=1$. Prove that for every functions $f \in L^p(\mathbb{R})$, $g \in L^q(\mathbb{R})$,and $h \in L^r(\mathbb{R})$ $$\int_{\mathbb{R}} |fgh|\leq \|f\|_p\centerdot \|g\|_q \centerdot\|h\|_r.$$ I was…
30
votes
2 answers

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 circle (as defined by all points equal distance from…
Ian Miller
  • 11,630
  • 1
  • 22
  • 40
29
votes
2 answers

Proof of separability of $L^p$ spaces

The following is a proof in Brezis book. It shows the separability of $L^{p}$ spaces: I have a few questions regarding the proof: It says 'it is easy to construct a function $f_{2} \in \varepsilon...$" and it also says " it suffices to split $R$…
28
votes
3 answers

Is the rectangular function a convolution of $L^1$ functions?

Do there exist functions $f,g$ in $L^1(\mathbf{R})$ such that the convolution $f \star g$ is (almost everywhere) equal to the indicator function of the interval $[0,1]$ ?
28
votes
1 answer

$\ell_p$ 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 norm $||u||_p=\sqrt[p]{\sum_{n=1}^\infty |u_n|^p}\…
ccc
  • 1,723
  • 4
  • 16
  • 29
27
votes
1 answer

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 book. So I wonder why it is? Does this…
Tim
  • 43,663
  • 43
  • 199
  • 459
26
votes
1 answer

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 example of convex functions. I failed to prove…
Danny_Kim
  • 3,099
  • 2
  • 16
  • 39
25
votes
2 answers

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 understand how that could be true intuitively, I'm…
AncientSpark
  • 351
  • 1
  • 3
  • 5
23
votes
5 answers

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 space I need to show that the space is…
22
votes
1 answer

Convergence types in probability theory : Counterexamples

I know that the following implications are true: $$\text{Almost sure convergence} \Rightarrow \text{ Convergence in probability } \Leftarrow \text{ Convergence in }L^p $$ $$\Downarrow$$ $$\text{Convergence in distribution}$$ I am looking for some…
21
votes
1 answer

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.
arawarep
  • 269
  • 2
  • 4
21
votes
6 answers

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}^{\infty} x_ky_k$ is not a linear bounded functional on…
Benzio
  • 1,897
  • 2
  • 14
  • 30
21
votes
3 answers

What does it mean to be an $L^1$ function?

I am struggling to understand what the space $L^1$ is, and what it means for a function to be $L^1$. A friend told me that a function $f$ is $L^1$ if $\int_\mathbb{R} |f|$ is finite. It is $L^2$ if $(\int_\mathbb{R} |f|^2)^{1/2}$ Firstly is this…
fourier1234
  • 1,392
  • 1
  • 9
  • 24
21
votes
1 answer

What is rotation when we have a different distance metric?

Background I am looking at the Taxicab metric in 2D and other related norms of the form: $$d(\vec{x}, \vec{y}) = \|\vec{x} - \vec{y}\|_p = \left( \sum_{i = 0}^n |x_i - y_i|^p \right)^{1/p}$$ I haven't formally studied this topic so am investigating…
Ian Miller
  • 11,630
  • 1
  • 22
  • 40
21
votes
1 answer

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,\varepsilon}=\lbrace x: \vert f_n(x)-f(x) \vert \ge…
cap
  • 2,243
  • 3
  • 18
  • 40