Question

\(L^{p}\left(\mathbb{R}^{n}, m\right)\) is separable for \(1 \leq p<\infty\). However, \(L^{\infty}\left(\mathbb{R}^{n}, m\right)\) is not separable. (There is an uncountable set \(\mathcal{F} \subset L^{\infty}\) such that \(\|f-g\|_{\infty} \geq 1\) for all \(f, g \in \mathcal{F}\) with \(f \neq g .)\)

Short answer

\(L^p(\mathbb{R}^n, m)\) for \(1 \leq p < \infty\) is separable due to simple functions; \(L^{\infty}(\mathbb{R}^n, m)\) is not separable.

Step-by-step solution

Understanding Separability

A metric space is separable if it contains a countable dense subset. For function spaces like \(L^p\), this involves finding a countable set of functions such that any function in the space can be approximated as closely as desired by functions from this set.

Examining \(L^p\) for \(1 \leq p < \,\infty\)

For spaces \(L^p(\mathbb{R}^n, m)\), where \(1 \leq p < \,\infty\), there exists a dense countable subset: the set of simple functions with rational coefficients and rational cut-off sets is dense in \(L^p\). This is because every \(L^p\) function can be approximated as closely as desired by simple functions, hence these spaces are separable.

Simple Functions as Dense Subset

A simple function takes on a finite number of values and can be expressed as \( \sum a_k \chi_{A_k} \), where \(a_k\) are rational numbers and \(A_k\) are measurable sets with rational-valued characteristic functions. These functions form a countable dense subset in \(L^p\) spaces.

Understanding \(L^{\infty}\)

The space \(L^{\infty}(\mathbb{R}^n, m)\) is not separable. This is due to the existence of an uncountable family of functions where the \(L^{\infty}\) distance \( \|f - g\|_{\infty}\) is at least 1. This indicates no countable dense subset could exist, as densely approximating these functions would violate the distance condition.

Constructing Set \(\mathcal{F}\) in \(L^{\infty}\)

Consider the characteristic functions of disjoint sets, such as the set of characteristic functions \(\chi_{A}\) where \(A\) runs over all subsets of the natural numbers. These functions are in \(L^{\infty}\), and each pair differs by at least 1 in \(L^{\infty}\)-norm, proving non-separability.

Key concepts

Lp Spaces

The concept of \(L^p\) spaces is central to understanding various types of function spaces in mathematics. These function spaces are composed of measurable functions for which the \(p\)-th power of their absolute value is integrable. Here's a closer look at how \(L^p\) spaces function:

• For \(1 \leq p < \infty\), \(L^p\) spaces cover functions \(f\) such that \(\int |f|^p \, dm < \infty\).
• \(L^p\) spaces are metric spaces equipped with the \(L^p\)-norm \(\|f\|_p = \left( \int |f|^p \, dm \right)^{1/p}\).

Understanding these norms is critical because they provide a way to measure the size or magnitude of functions. The nature of the \(L^p\) norm changes with different values of \(p\), influencing convergence and approximation properties.In the context of separable metric spaces, it's essential to know that a space is separable if it contains a dense countable subset. The separability of \(L^p(\mathbb{R}^n, m)\) for \(1 \leq p < \infty\) stems from the ability to approximate any function in the space as closely as desired using a countable subset of simple functions.

Simple Functions

Simple functions are the cornerstone for understanding how more complex functions can be approximated. These functions are composed of a finite combination of characteristic functions and rational coefficients. A typical simple function \(s(x)\) can be expressed as:\[ s(x) = \sum a_k \chi_{A_k}(x) \]where \(a_k\) are rational numbers, and \(\chi_{A_k}\) are characteristic functions over measurable sets \(A_k\). What makes them so important?

• Simple functions take only a finite number of values, and are easy to work with mathematically.
• They act as building blocks for more complicated functions, making analysis manageable.

Approximating functions in \(L^p\) spaces with simple functions is a core method since it leads to simplifying calculations and gaining insights into function behavior. By using a collection of simple functions with rational values and measurable sets, you can form a dense subset within \(L^p\) spaces.

Characteristic Functions

Characteristic functions, often denoted \(\chi_A(x)\), are essential in defining simple functions. A characteristic function for a set \(A\) is defined as:\[ \chi_A(x) = \begin{cases} 1, & \text{if } x \in A \ 0, & \text{if } x otin A \end{cases} \]These functions act like switches, turning on (1) for values within the set \(A\) and off (0) outside it. They play a significant role in creating simple functions because they allow for precise control over which values are taken within a given measure space.

• They help in the decomposition of functions into understandable units.
• Used as building blocks in more complex functional forms such as those in \(L^p\) spaces.

Characteristic functions enable simplification of expressions and calculations in space analysis. Their ability to represent measurable sets precisely helps in the craft of simple functions that form dense subsets in separable spaces.

Dense Subset

A dense subset is crucial in defining the separability of metric spaces. A subset \(S\) of a metric space \(M\) is dense if every point in \(M\) can be approximated as closely as desired by elements of \(S\). In the world of \(L^p\) spaces, dense subsets are usually constructed from simple functions.Here's what makes dense subsets stand out:

• They allow the approximation of any element of the space through elements of the subset.
• In \(L^p\) spaces, dense subsets often consist of simple functions with rational coefficients and rational cutoff sets.

Separable spaces, like many \(L^p\) spaces, contain these dense subsets which ensure that the entire space's structure can be comprehended through a countable set. This concept distinguishes \(L^{\infty}\), which is not separable, marking a contrast in understanding function spaces.