<p>$L^{\infty}$: the (Banach, usually) space of bounded measurable functions modulo the equality almost everywhere. Like all function spaces not involving holomorphic functions, it's really a space of equivalence classes.</p>

<p>weak-$^*$ topology: the topology on the dual $X^*$ of any topological vector space $X$ induced by the $(X, X^*)$-pairing. </p>

<p>"embedded in $L^{\infty}$": a (say) topological vector space $Y$ is said to be embedded in $L^{\infty}$ if there is an injective continuous linear map from $Y$ to $L^{\infty}$. </p>

<p>(This is the usual meaning. In your paragraph, "embedded" just means viewing those indicator functions as representatives of their classes. The weak-$^*$ topology on $L^{\infty}$ is given by the identification $L^{\infty} = (C_0)^*$.) </p>

CC BY-SA 3.0
2015-04-21T21:03:27.863Z
2015-04-22T02:25:20.990Z
2015-04-22T02:25:20.990Z
Michael
<p>While reading economics papers, especially those related to economy with land, I often encounter the terms $L^\infty$ and the "weak* topology". They seem like very basic terms, but I couldn't find a basic explanation of them. Here is a typical paragraph which I am trying to decipher (it is from page 4 at the introduction of <a href="http://www.sciencedirect.com/science/article/pii/0022053188900087">A foundation of location theory</a>):</p>

<hr>

<p>Let:</p>

<p>$$[0,\frac{1}{n}]\cup[\frac{2}{n},\frac{3}{n}]\cup[\frac{4}{n},\frac{5}{n}]\cup...\cup[\frac{n-2}{n},\frac{n-1}{n}]$$</p>

<p>$(n=1,2,...)$ represent a sequence of commodities of increasing utility. If the indicator functions of these sets are embedded in $L^\infty$ with the weak* topology, then the limit is $\frac{1}{2}1_{[0,1]}$, half the indicator function on the interval $[0,1]$. This, indeed, is a natural limit that is not in the commodity space.</p>

<hr>

<p>So my questions are:</p>

<ul>
<li>What is $L^\infty$?</li>
<li>What is the "weak* topology"?</li>
<li>What is the meaning of "embedded in $L^\infty$"?</li>
</ul>

<p>NOTE: I have some basic (undergraduate) knowledge of topology. I have heard about "weak topology" but never heard of "weak* topology". I will be very thankful for a simple, intuitive explanation that will help me continue reading!</p>

CC BY-SA 3.0
2015-04-12T11:50:28.853Z
For the first one, search LP space. Roughly speaking, $L^\infty$ is a space of measurable functions, in which supreme norm is finite. Weak${}^*$ topology is the weakest topology makes the evaluation mapping of a dual space and the dual of that dual space continuous. The third means there is a homeomorphism between the space of interest and its image, a subset, not the whole space, of $L^\infty$.
Metta World Peace
CC BY-SA 3.0
2015-04-22T02:21:20.543Z
"...weakest topology makes the evaluation mapping of a dual space and the dual of that dual space continuous..." is not weak-*. It's much finer than weak-star. I'm not sure what it's called, maybe Mackey-Arens topology. In that topology, for example, Banach-Alaoglu is not true in general.
Michael
CC BY-SA 3.0
2015-04-22T09:20:50.183Z
@Michael Thank you. You're right. I should have phrased in the way that it's restricted to a subspace of the dual of dual space, which contains all those functionals that can be represented as an evaluation mapping from $X$ to the underlying field.
Metta World Peace
<mathematical-economics>
L-infinity and weak-star topology
Erel Segal-Halevi
# L-infinity and weak-star topology

(Asked by: username_0 on 2015-04-12)

While reading economics papers, especially those related to economy with land, I often encounter the terms $L^\infty$ and the "weak\* topology". They seem like very basic terms, but I couldn't find a basic explanation of them. Here is a typical paragraph which I am trying to decipher (it is from page 4 at the introduction of [A foundation of location theory](http://www.sciencedirect.com/science/article/pii/0022053188900087)):

---

Let:

$$[0,\frac{1}{n}]\cup[\frac{2}{n},\frac{3}{n}]\cup[\frac{4}{n},\frac{5}{n}]\cup...\cup[\frac{n-2}{n},\frac{n-1}{n}]$$

$(n=1,2,...)$ represent a sequence of commodities of increasing utility. If the indicator functions of these sets are embedded in $L^\infty$ with the weak\* topology, then the limit is $\frac{1}{2}1\_{[0,1]}$, half the indicator function on the interval $[0,1]$. This, indeed, is a natural limit that is not in the commodity space.

---

So my questions are:

* What is $L^\infty$?
* What is the "weak\* topology"?
* What is the meaning of "embedded in $L^\infty$"?

NOTE: I have some basic (undergraduate) knowledge of topology. I have heard about "weak topology" but never heard of "weak\* topology". I will be very thankful for a simple, intuitive explanation that will help me continue reading!

**Tags:** mathematical-economics

> **Comments:**
> **username_1**: For the first one, search LP space. Roughly speaking, $L^\infty$ is a space of measurable functions, in which supreme norm is finite. Weak${}^*$ topology is the weakest topology makes the evaluation mapping of a dual space and the dual of that dual space continuous. The third means there is a homeomorphism between the space of interest and its image, a subset, not the whole space, of $L^\infty$.  
> **username_2**: "...weakest topology makes the evaluation mapping of a dual space and the dual of that dual space continuous..." is not weak-*. It's much finer than weak-star. I'm not sure what it's called, maybe Mackey-Arens topology. In that topology, for example, Banach-Alaoglu is not true in general.  
> **username_1**: @username_2 Thank you. You're right. I should have phrased in the way that it's restricted to a subspace of the dual of dual space, which contains all those functionals that can be represented as an evaluation mapping from $X$ to the underlying field.  

---

## 1 answers

### Answer by username_2 on 2015-04-21. Score: 3 | Accepted answer

$L^{\infty}$: the (Banach, usually) space of bounded measurable functions modulo the equality almost everywhere. Like all function spaces not involving holomorphic functions, it's really a space of equivalence classes.

weak-$^\*$ topology: the topology on the dual $X^\*$ of any topological vector space $X$ induced by the $(X, X^\*)$-pairing.

"embedded in $L^{\infty}$": a (say) topological vector space $Y$ is said to be embedded in $L^{\infty}$ if there is an injective continuous linear map from $Y$ to $L^{\infty}$.

(This is the usual meaning. In your paragraph, "embedded" just means viewing those indicator functions as representatives of their classes. The weak-$^\*$ topology on $L^{\infty}$ is given by the identification $L^{\infty} = (C\_0)^\*$.)


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