Problem 5 - Basic Analysis: Introduction to Real Analysis

Question

Suppose \((X, d)\) is a metric space and \(\varphi:[0, \infty) \rightarrow \mathbb{R}\) is an increasing function such that \(\varphi(t) \geq 0\) for all \(t\) and \(\varphi(t)=0\) if and only if \(t=0 .\) Also suppose \(\varphi\) is subadditive, that is, \(\varphi(s+t) \leq\) \(\varphi(s)+\varphi(t) .\) Show that with \(d^{\prime}(x, y):=\varphi(d(x, y)),\) we obtain a new metric space \(\left(X, d^{\prime}\right)\).

Step-by-Step Solution

Answer

\(d'(x, y)\) is a valid metric because it satisfies all metric space conditions.

Step 1: Verify Non-negativity

To establish that \(d'(x, y) = \varphi(d(x, y))\) is a metric, we begin by verifying non-negativity. Since \(\varphi(t)\) is given to be non-negative, i.e., \(\varphi(t) \geq 0\) for all \(t\), it follows that \(d'(x, y) = \varphi(d(x, y)) \geq 0\). So the non-negativity condition is satisfied.

Step 2: Verify Identity of Indiscernibles

For the identity of indiscernibles, we need \(d'(x, y) = 0\) if and only if \(x = y\). Since \(\varphi(t) = 0\) if and only if \(t = 0\) and \(d(x, y) = 0\) if and only if \(x = y\), it follows that \(d'(x, y) = \varphi(d(x, y)) = 0\) if and only if \(x = y\). Thus, this condition is also satisfied.

Step 3: Verify Symmetry

To confirm symmetry, we need \(d'(x, y) = d'(y, x)\). Since \(d(x, y) = d(y, x)\) by the properties of metrics and \(\varphi\) only depends on the non-negative real number, we have \(d'(x, y) = \varphi(d(x, y)) = \varphi(d(y, x)) = d'(y, x)\). Thus, symmetry holds.

Step 4: Verify Triangle Inequality

For the triangle inequality, we need to show \(d'(x, z) \leq d'(x, y) + d'(y, z)\). By the subadditivity of \(\varphi\), i.e., \(\varphi(s + t) \leq \varphi(s) + \varphi(t)\), and using the triangle inequality for \(d\), i.e., \(d(x, z) \leq d(x, y) + d(y, z)\), we have: \(\varphi(d(x, z)) \leq \varphi(d(x, y) + d(y, z)) \leq \varphi(d(x, y)) + \varphi(d(y, z)) = d'(x, y) + d'(y, z)\). Therefore, the triangle inequality holds.

Key Concepts

Subadditive Function

In the realm of mathematics, a function is described as subadditive if it satisfies a specific property. A function \( \varphi \) is subadditive when for any two non-negative values \( s \) and \( t \), the inequality \( \varphi(s + t) \leq \varphi(s) + \varphi(t) \) holds true.
This property is important because it ensures the value of the function when applied to the sum of two arguments does not exceed the sum of the function values of these two arguments separately.
In the context of metric spaces, the subadditive property is crucial in establishing properties like the triangle inequality. It helps ensure that a new metric, when redefined using a subadditive function, maintains the essential qualities of a distance measure.

Triangle Inequality

The triangle inequality is a fundamental attribute of metric spaces. It states that for any three points \( x, y, \) and \( z \), the distance from \( x \) to \( z \) must be less than or equal to the sum of the distances from \( x \) to \( y \) and \( y \) to \( z \). In formula form, this is:
\[ d(x, z) \leq d(x, y) + d(y, z) \]
When transforming a metric\( d \) with a function \( \varphi \) to form a new metric \( d' \), the subadditivity of \( \varphi \) becomes essential. It ensures that the new metric \( d' \) still respects this property:

• \( \varphi(d(x, z)) \leq \varphi(d(x, y) + d(y, z)) \)
• \( \leq \varphi(d(x, y)) + \varphi(d(y, z)) \)

This proves that the new metric satisfies the triangle inequality, showing how subadditivity bridges this requirement.

Symmetry in Metrics

In metric spaces, symmetry is an intuitive concept describing how the measured distance between two points should be the same regardless of the order in which these points are considered. Formally, a metric \( d \) is symmetric if for any two points \( x \) and \( y \), we have \( d(x, y) = d(y, x) \).
This symmetry is a fundamental quality of any valid distance measurement. When using a function like \( \varphi \) to define a new metric \( d' \) such that \( d'(x,y) = \varphi(d(x,y)) \), the symmetry property naturally carries over. This happens because the function \( \varphi \) acts on the non-negative real numbers, which are inherently symmetric in terms of equality and manipulation.
Thus, if \( d(x, y) = d(y, x) \), then \( \varphi(d(x, y)) = \varphi(d(y, x)) \), ensuring \( \varphi \) preserves symmetry in the new metric.

Non-negativity in Metrics

Non-negativity is the simplest property of a metric, declaring that the distance between any two points must not be negative. In metric terms, for any points \( x \) and \( y \), \( d(x, y) \geq 0 \). This ensures that distance, a concept rooted in reality, reflects values that make logical sense in everyday life.
When defining a new metric \( d' \) using an increasing and non-negative function \( \varphi \) (i.e., \( \varphi(t) \geq 0 \) for all \( t \)), we automatically maintain the non-negativity condition in \( d'(x, y) = \varphi(d(x, y)) \). Since \( d(x, y) \) is already non-negative, and \( \varphi \) is defined to output non-negative values, the resulting metric \( d' \) will naturally adhere to non-negativity.
This preserves the intuitive understanding of distance, keeping the structure consistent with real-world expectations.