Problem 26 - An Introduction to Abstract Analysis

Question

Let \(A\) and \(B\) be disjoint closed sets in a normed linear space \(X\). Construct a continuous function \(f: X \rightarrow[0,1]\) such that \(f(a)=0\) for all \(a\) in \(A\), and \(f(b)=1\) for all \(b\) in \(B\).

Step-by-Step Solution

Answer

Question: Construct a continuous function \(f: X \rightarrow[0,1]\) on a normed linear space \(X\) that maps every point in the disjoint closed sets \(A\) and \(B\) to 0 and 1, respectively. Answer: The function is defined as \(f(x) = \frac{d_A(x)}{d_A(x) + d_B(x)}\), where \(d_A(x) = \inf_{a \in A} d(x,a)\) and \(d_B(x) = \inf_{b \in B} d(x,b)\) represent the distance from point \(x \in X\) to the sets \(A\) and \(B\), respectively. This function is continuous, maps points in \(A\) to 0 and points in \(B\) to 1.

Step 1: Define distance function

In a normed linear space \(X\), we can define a distance function \(d: X \times X \rightarrow \mathbb{R}\) such that \(d(x,y) = \|x-y\|\), where \(\| \cdot \|\) is the norm on the space \(X\). This distance function gives us a way to measure the distance between any two points in \(X\).

Step 2: Define function to calculate distance to sets

Now, we can define a function \(d_A(x) = \inf_{a \in A} d(x,a)\), which represents the distance from point \(x \in X\) to the set \(A\). Similarly, we can define a function \(d_B(x) = \inf_{b \in B} d(x,b)\) for the distance from point \(x\) to the set \(B\). These functions are continuous and non-negative.

Step 3: Define our function f

Now, we can define our function \(f(x) = \frac{d_A(x)}{d_A(x) + d_B(x)}\). This function maps any point \(x \in X\) to a value between 0 and 1, depending on the ratios of the distance between \(x\) and the sets \(A\) and \(B\).

Step 4: Check the function for the sets A and B

For any point \(a \in A\), we have \(d_A(a) = 0\) (since the distance from a point to itself is 0) and \(d_B(a) > 0\) (because the sets \(A\) and \(B\) are disjoint). Thus, \(f(a) = \frac{0}{0 + d_B(a)}=0\). For any point \(b \in B\), we have \(d_B(b) = 0\), and \(d_A(b) > 0\). Thus, \(f(b) = \frac{d_A(b)}{d_A(b) + 0}=1\).

Step 5: Check the continuity of f

Since both \(d_A\) and \(d_B\) are continuous functions on \(X\), their sum is also continuous. Additionally, since both \(d_A\) and \(d_B\) are non-negative, their sum is non-zero. Thus, the quotient \(f(x) = \frac{d_A(x)}{d_A(x) + d_B(x)}\) is a continuous function on \(X\). Therefore, we have constructed a continuous function \(f: X \rightarrow[0,1]\) such that \(f(a)=0\) for all \(a\) in \(A\), and \(f(b)=1\) for all \(b\) in \(B\).

Key Concepts

Normed Linear Spaces

A normed linear space, often called a normed vector space, is a key concept in functional analysis. It consists of a vector space equipped with a function called a norm. This norm assigns a non-negative length or size to each vector in the space. The norm must satisfy certain properties:

• Non-negativity: For every vector \( x \), the norm \( \|x\| \) is always greater than or equal to zero, and is zero if and only if \( x \) is the zero vector.
• Scalar Multiplication: The norm of a scalar times a vector is the absolute value of the scalar times the norm of the vector, \( \|\alpha x\| = |\alpha| \cdot \|x\| \).
• Triangle Inequality: For any two vectors \( x \) and \( y \), the norm satisfies \( \|x + y\| \leq \|x\| + \|y\| \).

These properties ensure that the concept of “distance” is meaningful in the space, making normed linear spaces a generalization of Euclidean spaces. They provide a framework to discuss and solve various problems in mathematics, especially those involving convergence, continuity, and boundedness.

Distance Function

In a normed linear space, a distance function is crucial for measuring how far apart two elements are. The general form of a distance function \( d(x, y) \) is defined as \( \|x - y\| \), where \( \| \cdot \| \) represents the norm of the space.The properties of this distance function are derived directly from the properties of the norm:

• Non-negativity: The distance \( d(x, y) \) is non-negative, and is zero if and only if \( x = y \).
• Symmetry: The distance is symmetric, meaning \( d(x, y) = d(y, x) \).
• Triangle Inequality: It satisfies the triangle inequality, \( d(x, y) \leq d(x, z) + d(z, y) \) for any point \( z \) in the space.

These properties make the distance function a metric, allowing us to define and analyze the topological and geometric structure of the space. They are particularly useful in determining convergence, which is a key concept in many areas of applied and pure mathematics.

Disjoint Closed Sets

Disjoint closed sets are two or more sets in a space that do not share any common elements and are closed in the topological sense. In the context of a normed linear space, disjoint sets \( A \) and \( B \) have a few important properties:

• Disjointness: A set \( A \) is disjoint from a set \( B \) if their intersection is empty, i.e., \( A \cap B = \emptyset \).
• Closed Sets: A set is closed if it contains all its limit points. This means any convergent sequence entirely contained within the set will converge to a point that is also in the set.
• Distance Consideration: Because the sets are disjoint, there is a positive distance between them. Formally, \( \inf \{ \|a - b\| : a \in A, b \in B \} > 0 \).

Understanding disjoint closed sets is essential when determining the continuity and differentiability of functions between them. They also help in constructing special kinds of functions, as seen in this context where a function was created to separate these sets distinctly in the mapping.

Continuous Mapping

Continuous mapping, or continuity of a function, is a fundamental concept when dealing with functions between normed spaces. A function \( f: X \rightarrow Y \) is continuous if for every point \( x \in X \), and for any given \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that for all \( x' \) in \( X \), if \( \|x - x'\| < \delta \) then \( \|f(x) - f(x')\| < \epsilon \).This concept ensures that small changes in the input result in small changes in the output. Here are key aspects of continuous mappings:

• Preservation of Limits: If a sequence \( \{x_n\} \to x \) in \( X \), then \( \{f(x_n)\} \to f(x) \) in \( Y \).
• Local Behavior: Around any point, the function behaves predictably and won't show sudden jumps.
• Composite Functions: If \( f \) and \( g \) are continuous, then their composition \( g \circ f \) is also continuous.

The exercise here involved creating a continuous function that clearly maps points from one set to another, respecting their disjoint nature. Ensuring continuity is key to transferring structure between spaces without losing information or causing discontinuities.