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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):

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!

Erel Segal-Halevi
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    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 Apr 12 '15 at 11:50
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    "...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 Apr 22 '15 at 02:21
  • @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 Apr 22 '15 at 09:20

1 Answers1

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$L^{\infty}$: 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 $L^{\infty} = (C_0)^*$.)

Michael
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