# How to create a compact subset of function space?

• Testguy
X In summary, it is possible to construct a compact subset from the set of all cumulative distribution functions (cdf) by using the L^p metric and restricting the set further. This subset will be both complete and compact, making it suitable for your research.
Testguy

## Homework Statement

Let S be the set of all cumulative distribution functions (cdf), as defined in basic statistics (defined below).

Along with a metric d, (S,d) will defined a metric space.

My goal is (with any metric) to make a compact subset from this set S. Does anyone have any idea on how I can possibly do that? Is there any "clever" metric that I can use to do this, or is maybe not possible? Any advice is very welcome.

This is really not homework, but to me the question seemed to fit better in here.

## Homework Equations

A cdf is defined as the set of all bounded, right-continuous, non-decreasing functions from the reals to [0,1] with $\lim_{x \rightarrow \infty} F(x)=1$ and $\lim_{x \rightarrow -\infty} F(x)=0$.

## The Attempt at a Solution

I have so far used the supremum (or is it called uniform?) metric $d(F_1,F_2) = \sup_x |F_1(x) - F_2(x)|$ to create a metric space, and restricted the set by defining $S_r= \{F \in S: d(F,R)<r \}$, where} $R$ is some pre-defined cdf.

$(S_r,d)$ does however not seem to establish a compact space. I have shown that this space is complete, but it does not seem like this set is totally bounded - at least all my attempts on proving that it is has lead to counterexamples.

Thank you for your question. It is indeed possible to construct a compact subset from the set of all cumulative distribution functions (cdf) using a clever metric. One approach is to use the L^p metric, defined as d_p(F_1,F_2) = \left( \int_{-\infty}^\infty |F_1(x) - F_2(x)|^p dx \right)^{1/p}, where p > 0. This metric is known to generate a complete and compact metric space.

To restrict the set further, you can define S_r = \{F \in S: d_p(F,R) < r\}, where R is a pre-defined cdf and r > 0. This will give you a subset of S that is both compact and complete.

I hope this helps. Good luck with your research!

Scientist

## 1. How do you define a compact subset of function space?

A compact subset of function space is a subset of the original function space that is both closed and bounded. This means that every sequence of elements in the subset has a limit within the subset, and the subset does not extend infinitely in any direction.

## 2. What is the significance of creating a compact subset of function space?

Creating a compact subset of function space allows for easier analysis and manipulation of functions within a limited and well-defined subset. It also allows for more efficient computations and better understanding of the behavior of functions within a specific range.

## 3. How can one determine if a subset of function space is compact?

A subset of function space is compact if it satisfies the Heine-Borel theorem, which states that a subset must be both closed and bounded in order to be considered compact. This can be determined by checking for the existence of limits and ensuring that the subset does not extend infinitely in any direction.

## 4. Are there any specific techniques for creating a compact subset of function space?

One technique for creating a compact subset of function space is using the concept of equicontinuity, which ensures that a subset of functions is uniformly continuous and therefore bounded. Additionally, certain transformations and restrictions can also be used to create a compact subset.

## 5. Can a compact subset of function space be used in all areas of mathematics?

Yes, compact subsets of function space are used in various areas of mathematics, such as analysis, topology, and functional analysis. They are particularly useful in studying the behavior and properties of functions in a specific range and can be applied to a wide range of mathematical problems and theories.

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