Lebesgue space

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### Context
#### Functional analysis
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[[!include functional analysis - contents]]
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#### Integration theory
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[[!include integration theory - contents]]
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# Contents
* table of contents
{: toc}

## Idea

The term 'Lebesgue space' can stand for two distinct notions: one is the general notion of [[measure space]] (compare the [Springer Encyclopaedia of Mathematics](http://eom.springer.de/L/l057910.htm)) and another is the notion of $L^p$ space (or $L_p$ space).  Here we discuss the latter.

Lebesgue spaces $L^p$ in this sense are [[normed vector spaces]] of [[functions]] on a [[measure space]], equipped with the suitable version of the [[p-norm]].

Beware that sometimes the notation '$L_p$' is used as a synonym for $L^p$; sometimes it is used to mean $L^{1/p}$.

## Notation

For historical reasons (starting with the original paper by Riesz), the exponent $p$ is traditionally taken to be the [[reciprocal]] of the “correct” exponent.

If we take $M^p=L^{1/p}$, the spaces
$M^p$ form a $\mathbf{C}$-[[graded algebra]], where $\mathbf{C}$ denotes [[complex numbers]].

This is a conceptual explanation for the appearance of formulas like $1/p+1/q=1/r$ in [[Hölder's inequality]].

In [[differential geometry]], the notion of [[density]] does use the “correct” grading.

In the [[Tomita–Takesaki theory]], the parameter $t$ for [[modular automorphism group]] is almost the “correct” grading, except that it is multiplied by the [[imaginary unit]] $i$.

## Definitions
 {#Definition}

If $1 \leq p \lt \infty$ is a [[real number]] and $(\Omega,\mu)$ is a [[measure space]], one considers the **$L^p$ space** $L_p(\Omega)$, which is the [[vector space]] of [[equivalence classes]] of those [[measurable function|measurable]] (complex- or real-valued) functions $f\colon \Omega \to \mathbb{K}$ whose (absolute values of) $p$th powers are integrable, in that the [[integration|integral]]

$$
  {\|f\|_p}
    \;\coloneqq\;
  \left(
    \textstyle{\int}_\Omega {|f|^p} \,d\mu
  \right)^{1/p}
    \;\lt\;
  \infty
$$

exists. Two such are taken to be equivalent, $f \sim g$, if ${\|f-g\|_p} = 0$. For $p = 2$ this is the space $L^2$ of [[square integrable functions]].

On these spaces $L^p(X)$ of equivalence classes of $p$-power integrable functions, the function ${\|f\|_p}$ satisfies the [[triangle inequality]] (due to [[Minkowski's inequality]], see [below](#MinkowskiInequality)) and hence defines a [[norm]], the _[[p-norm]]_, making them [[normed vector spaces]].

The $L^p$ spaces are examples of [[Banach spaces]]; they are continuous analogues of $l^p$ spaces of $p$-summable series. (Indeed, $l^p(S)$, for $S$ a [[set]], is simply $L^p(S)$ if $S$ is equipped with [[counting measure]].)

For fixed $f$, the norm ${\|f\|_p}$ is continuous in $p$.  Accordingly, for $p = \infty$, one may take the limit of ${\|f\|}_p$ as $p \to \infty$.  However, this turns out to be the same as the [[essential supremum]] norm $\|f\|_\infty$.  Therefore, $L^\infty(\Omega)$ makes sense as long as $\Omega$ is a [[measurable space]] equipped with a family of [[null sets]] (or [[full sets]]); the measure $\mu$ is otherwise irrelevant.

For $0 \leq p \lt 1$, one can modify the definition to make $L^p$ into an [[F-space]] (but not a Banach space).  See the definitions at [[p-norm]].

## Minkowski's inequality
 {#MinkowskiInequality}

We offer here a proof that ${\|f\|_p}$ indeed defines a norm in the case $1 \lt p \lt \infty$, in that it satisfies the [[triangle inequality]]. This is usually known as *[[Minkowski's inequality]]*.

(The cases $p = 1$ and $p = \infty$ follow by continuity and are easy to check from first principles.)

The most usual textbook proofs involve a clever application of [[Hölder's inequality]]; the following proof is more straightforwardly geometric. All functions $f$ may be assumed to be real- or complex-valued.

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###### Theorem

Suppose $1 \leq p \leq \infty$, and suppose $\Omega$ is a [[measure space]] with [[measure]] $\mu$. Then the function ${|(-)|_p} \,\colon\, L^p(\Omega, \mu) \to \mathbb{R}$ defined by
$$
  {\|f\|_p}
    \;\coloneqq\;
  \big(
    \textstyle{\int}_\Omega {|f|^p} \,d\mu
  \big)^{1/p}
$$
defines a [[norm]].

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One must verify three things:

1. Separation axiom: ${\|f\|_p} = 0$ implies $f = 0$.

1. Scaling axiom: ${\|t f\|}_p = {|t|} \, {\|f\|_p}$.

1. Triangle inequality: ${\|f + g\|_p} \leq {\|f\|_p} + {\|g\|_p}$.

The first two properties are obvious, so it remains to prove the last, which is also called __[[Minkowski's inequality]]__.

Our proof of Minkowski's inequality is broken down into a series of simple lemmas. The plan is to boil it down to two things: the scaling axiom, and convexity of the function $x \mapsto {|x|^p}$ (as a function from real or complex numbers to nonnegative real numbers).

First, some generalities. Let $V$ be a (real or complex) vector space equipped with a function ${\|(-)\|}\colon V \to [0, \infty]$ that satisfies the scaling axiom: ${\|t v\|} = {|t|} \, {\|v\|}$ for all scalars $t$, and the separation axiom: ${\|v\|} = 0$ implies $v = 0$. As usual, we define the [[unit ball]] in $V$ to be $\{v \in V \;|\; {\|v\|} \leq 1\}.$

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###### Lemma
Given that the scaling and separation axioms hold, the following conditions are equivalent:

1. The triangle inequality is satisfied.
1. The unit ball is convex.
1. If ${\|u\|} = {\|v\|} = 1$, then ${\|t u + (1-t)v\|} \leq 1$ for all $t \in [0, 1]$.
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###### Proof
Condition 1. implies condition 2. easily: if $u$ and $v$ are in the unit ball and $0 \leq t \leq 1$, we have
$$\array{
{\|t u + (1-t)v\|} & \leq & {\|t u\|} + {\|(1-t)v\|} \\
 & = & t {\|u\|} + (1-t) {\|v\|} \\
 & \leq & t + (1-t) = 1.}$$
Now 2. implies 3. trivially, so it remains to prove that 3. implies 1.  Suppose ${\|v\|}, {\|v'\|} \in (0, \infty)$. Let $u = \frac{v}{{\|v\|}}$ and $u' = \frac{v'}{{\|v'\|}}$ be the associated unit vectors.
Then
$$\array{
\frac{v + v'}{{\|v\|}+{\|v'\|}} & = & (\frac{{\|v\|}}{{\|v\|}+{\|v'\|}})\frac{v}{{\|v\|}} + (\frac{{\|v'\|}}{{\|v\|}+{\|v'\|}})\frac{v'}{{\|v'\|}} \\
 & = & t u + (1-t)u'}$$
where $t = \frac{{\|v\|}}{{\|v\|} + {\|v'\|}}$. If condition 3. holds, then
$${\|t u + (1-t)u'\|} \leq 1$$
but by the scaling axiom, this is the same as saying
$$\frac{{\|v + v'\|}}{{\|v\|} + {\|v'\|}} \leq 1,$$
which is the triangle inequality.
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Consider now $L^p$ with its $p$-norm ${\|f\|} = {|f|_p}$.  By Lemma \ref{conditions}, this inequality is equivalent to

* **Condition 4:** If ${|u|_{p}^{p}} = 1$ and ${|v|_{p}^{p}} = 1$, then ${|t u + (1-t)v|_{p}^{p}} \leq 1$ whenever $0 \leq t \leq 1$.

This allows us to remove the cumbersome exponent $1/p$ in the definition of the $p$-norm.

The next two lemmas may be proven by elementary calculus; we omit the proofs. (But you can also see the [full details](http://ncatlab.org/toddtrimble/published/p-norms).)

+-- {: .num_lemma #2deriv}
###### Lemma
Let $\alpha, \beta$ be two complex numbers, and define
$$\gamma(t) = {|\alpha + \beta t|^p}$$
for _real_ $t$. Then $\gamma''(t)$ is nonnegative.
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###### Lemma
Define $\phi\colon \mathbb{C} \to \mathbb{R}$ by $\phi(x) = |x|^p$.  Then $\phi$ is convex, i.e., for all $x, y$,
$${|t x + (1-t)y|^p} \leq t{|x|^p} + (1-t){|y|^p}$$
for all $t \in [0, 1]$.
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###### Proof of Minkowski's inequality
Let $u$ and $v$ be unit vectors in $L^p$. By condition 4, it suffices to show that ${|t u + (1-t)v|_p} \leq 1$ for all $t \in [0, 1]$. But
$$
  \textstyle{\int}_\Omega {|t u + (1-t)v|^p} \,d\mu
    \;\leq\;
  \textstyle{\int}_\Omega t{|u|}^p + (1-t){|v|}^p \,d\mu
  \,,
$$
by Lemma \ref{convex}. Using $\int {|u|^p} = 1 = \int {|v|^p}$, we are done.
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## Related concepts

* [[square integrable function]]

* [[canonical Hilbert space of half-densities]]

* [[tempered distribution]]

## References

Named after [[Henri Lebesgue]].

* W. Rudin, _Functional analysis_, McGraw Hill 1991.

* L. C. Evans, _Partial differential equations_, Amer. Math. Soc. 1998.

* Wikipedia (English): [Lp space](http://en.wikipedia.org/wiki/Lp_space)

category: analysis

[[!redirects Lebesgue space]]
[[!redirects Lebesgue spaces]]

[[!redirects L-p space]]
[[!redirects L-p spaces]]
[[!redirects L-p-space]]
[[!redirects L-p-spaces]]
[[!redirects L^p space]]
[[!redirects L^p spaces]]
[[!redirects L^p-space]]
[[!redirects L^p-spaces]]
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[[!redirects L_p spaces]]

[[!redirects L-1 space]]
[[!redirects L-1 spaces]]
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[[!redirects L^1-space]]
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[[!redirects L_1 space]]
[[!redirects L_1 spaces]]

[[!redirects L-2 space]]
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[[!redirects L_2 space]]
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Lectures on category theory

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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
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=--
=--

* [[Saunders Mac Lane]],

  _Lectures on category theory_

  Bowdoin Summer School 1969

  Notes taken by [[Ellis Cooper]] ([[MacLaneBowdoinLectures69.pdf:file]])

on [[category theory]].

These lectures are close to what eventually became the classical textbook _[[Categories for the Working Mathematician]]_.  One difference is a long discussion of [[William Lawvere]]'s dissertation results on [[algebraic theories]].

category: reference


+-- {: .rightHandSide}
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###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Representation theory
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[[!include representation theory - contents]]
=--
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This page collects material related to

* [[Stefan Schwede]],

  _Lectures on Equivariant Stable Homotopy Theory_

  ([pdf](http://www.math.uni-bonn.de/people/schwede/equivariant.pdf))

on [[equivariant stable homotopy theory]] modeled on [[orthogonal spectra]].

See also

* [[Stefan Schwede]], _[[Global homotopy theory]]_

#Contents#
* table of contents
{:toc}

> under construction

## 1. Orthogonal spectra

* [[orthogonal spectrum]]

## 2. Equivariant orthogonal spectra

* [[equivariant spectrum]]

* [[RO(G)-grading]]

* [[equivariant sphere spectrum]]

* [[equivariant suspension spectrum]]

## 3. Equivariant homotopy groups

* [[equivariant homotopy groups]]

## 4. Wirthmüller isomorphism and transfers

* [[Wirthmüller isomorphism]]

* [[Mackey functor]]

## 5. Constructions with equivariant spectra

## 6. The tom Dieck splitting

* [[tom Dieck splitting]]

## 7. Fixed points and geometric fixed points

* [[fixed point spectrum]]

## 8. Power constructions

## 9. Norm construction

## 10. Norm map

category: reference


lectures on higher bundle theory

###Higher Bundle Theory###

* ####Introduction on Lie groupoids and Lie algebroids####

  * [[Lie groupoid]]

  * [Lie algebroid](http://ncatlab.org/nlab/show/Lie+groupoid#lie_algebroid)

  * [Morphisms of Lie groupoids](http://ncatlab.org/nlab/show/Lie+groupoid#morphisms_of_lie_groupoids)

  * [Morphisms of Lie algebroids](http://ncatlab.org/nlab/show/Lie+groupoid#morphisms_of_lie_algebroids)

  * [Lie groupoid principal bundles](http://ncatlab.org/nlab/show/groupoid+principal+bundle#lie_groupoid_principal_bundles)

  * [Bibundles of Lie groupoids](http://ncatlab.org/nlab/show/bibundle#SmoothGroupoidPrincipalBibundles)

  * [Differentiating Lie groupoids](https://ncatlab.org/nlab/show/Lie+differentiation#of_a_lie_groupoid)

  * [Integrating Lie algebroids](https://ncatlab.org/nlab/show/Lie+integration#integrating_lie_algebroids_to_stacky_lie_groupoids)

* ####Introduction on stack theory####

  * [[sheaf]]

  * [[2-sheaf]]

  * [sheafification](http://arxiv.org/abs/1004.4558)

* ####Introduction on higher groupoids####

  * [Kan complex](http://arxiv.org/abs/0801.2057)

  * [Stacky Groupoids](http://arxiv.org/abs/1510.09208) (Sect. 3.1)


Lectures on n-Categories and Cohomology

This entry is about the article

* [[John Baez]] and [[Mike Shulman|Michael Shulman]],
  _Lectures on $n$-categories and cohomology_
  ([arXiv](http://arxiv.org/abs/math/0608420))

  in

  [[John Baez]] and [[Peter May]],
  [[John Baez:Towards Higher Categories|Towards Higher Categories]]

#related $n$Lab entries#

* [[higher category theory]]

  * [[n-category]]

* [[cohomology]]

  * [[nonabelian cohomology]]

  * [[group cohomology]]

* [[k-surjective functor]]

  * [[stuff, structure, property]]

  * [[Postnikov system]]

* [[negative thinking]]

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    * [[(-1)-groupoid]]

  * [[(-2)-category]]

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category: reference


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###Context###
#### Physics
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[[!include physicscontents]]
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#### Symplectic geometry
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[[!include symplectic geometry - contents]]
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This entry collects hyperlinks related to the lecture notes

* Sean Bates, [[Alan Weinstein]]:

  \linebreak

  **Lectures on the geometry of quantization**

  \linebreak

  Berkeley Mathematics Lecture Notes series,

  American Mathematical Society (1997)

  [ams:bmln-8](https://bookstore.ams.org/bmln-8)

  [pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)
  [[BatesWeinstein-Lectures.pdf:file]]

on [[classical mechanics]] formalized in [[symplectic geometry]] and its [[quantization]] by the process of [[geometric quantization]].

A related textbook for the symplectic perspective on classical mechanics is

* [[Vladimir Arnold]], _[[Mathematical methods of classical mechanics]]_

category: reference