Equivalence Of Field Norms A Detailed Proof And Analysis

In the realm of real analysis, metric spaces, and normed spaces, the concept of norm equivalence plays a pivotal role. This article delves into the equivalence of field norms, a fundamental idea with significant implications in various areas of mathematics. Specifically, we will explore the conditions under which two norms on a field can be considered equivalent, focusing on the Cauchy sequence criterion. We will also provide a detailed explanation and proof of the theorem that establishes the equivalence, along with examples and applications to enhance understanding. This discussion is inspired by Koblitz's "p-adic Numbers, Ultrametric Analysis, and Applications," Chapter 1, Exercise 5, and aims to provide a thorough treatment of the topic.

Understanding Field Norms

Before diving into the equivalence of field norms, it's crucial to define what a field norm is. A field norm, denoted as $||\*||$, is a function that maps elements of a field $F$ to non-negative real numbers, satisfying the following properties:

1. Non-negativity: $||x|| \geq 0$ for all $x \in F$, and $||x|| = 0$ if and only if $x = 0$.
2. Multiplicativity: $||xy|| = ||x|| \cdot ||y||$ for all $x, y \in F$.
3. Triangle Inequality: $||x + y|| \leq ||x|| + ||y||$ for all $x, y \in F$.

These properties ensure that the norm behaves in a manner consistent with our intuition about size or magnitude. Familiar examples of field norms include the absolute value on the field of real numbers ($\mathbb{R}$) and the complex numbers ($\mathbb{C}$). In the context of $p$-adic numbers, the $p$-adic norm is a crucial example, offering a different way to measure the "size" of a number based on its divisibility by a prime number $p$.

The concept of Cauchy sequences is central to understanding norm equivalence. A sequence $(x_n)$ in a field $F$ equipped with a norm $||\*||$ is said to be a Cauchy sequence if, for every $\epsilon > 0$, there exists a positive integer $N$ such that $||x_m - x_n|| < \epsilon$ for all $m, n > N$. In simpler terms, the terms of the sequence become arbitrarily close to each other as $n$ increases. This notion is crucial for defining completeness in metric spaces, which is essential for many analytical arguments.

Defining Equivalence of Field Norms

Two field norms, $||\*||_1$ and $||\*||_2$, on the same field $F$ are said to be equivalent if a sequence is Cauchy with respect to $||\*||_1$ if and only if it is Cauchy with respect to $||\*||_2$. This definition implies that the two norms induce the same notion of convergence and completeness in the field. In other words, if a sequence "squeezes together" under one norm, it also "squeezes together" under the other norm.

A more practical way to characterize norm equivalence is through the existence of positive constants $C_1$ and $C_2$ such that:

$$C_1 ||x||_1 \leq ||x||_2 \leq C_2 ||x||_1$$

for all $x$ in the field $F$. This inequality provides a direct comparison between the magnitudes of elements as measured by the two norms. If such constants exist, the norms are equivalent because they essentially scale each other. This condition ensures that small values in one norm correspond to small values in the other, and vice versa, thereby preserving the Cauchy property.

To further clarify, let's delve deeper into the implications of this inequality. The inequality $C_1 ||x||_1 \leq ||x||_2$ indicates that if $||x||_1$ is small, then $||x||_2$ must also be small. Conversely, the inequality $||x||_2 \leq C_2 ||x||_1$ implies that if $||x||_2$ is small, then $||x||_1$ must be small. Together, these inequalities establish a proportionality between the two norms, ensuring that convergence and Cauchy sequences are preserved. This proportionality is the cornerstone of norm equivalence, and it allows us to switch between norms without altering the fundamental analytic properties of the field.

For example, consider the usual absolute value norm $|x|$ and the norm $2|x|$ on the field of real numbers. These norms are equivalent because we can choose $C_1 = 2$ and $C_2 = 2$, satisfying the inequality $2|x| \leq 2|x| \leq 2(2|x|)$. This simple example illustrates how a constant scaling factor does not change the essential characteristics of the norm, and the concept of equivalence captures this invariance. In more complex scenarios, the constants $C_1$ and $C_2$ may not be as straightforward, but the principle remains the same: equivalent norms provide a consistent measure of magnitude, preserving the analytical structure of the underlying field.

The Equivalence Theorem and Its Proof

The central theorem we aim to prove states the equivalence between the Cauchy sequence definition and the constant scaling definition of norm equivalence. Specifically, the theorem states:

Theorem: Suppose $||\*||_1$ and $||\*||_2$ are two field norms on a field $F$. Then $||\*||_1$ and $||\*||_2$ are equivalent (a sequence is Cauchy in $||\*||_1$ if and only if it is Cauchy in $||\*||_2$) if and only if there exist positive constants $C_1$ and $C_2$ such that $C_1 ||x||_1 \leq ||x||_2 \leq C_2 ||x||_1$ for all $x \in F$.

Proof:

Part 1: Assuming the Existence of Constants, Prove Equivalence based on Cauchy Sequences

Assume that there exist positive constants $C_1$ and $C_2$ such that $C_1 ||x||_1 \leq ||x||_2 \leq C_2 ||x||_1$ for all $x \in F$. We need to show that a sequence $(x_n)$ is Cauchy in $||\*||_1$ if and only if it is Cauchy in $||\*||_2$.

First, suppose $(x_n)$ is Cauchy in $||\*||_1$. This means that for any $\epsilon > 0$, there exists an integer $N$ such that for all $m, n > N$, we have $||x_m - x_n||_1 < \epsilon$. We want to show that $(x_n)$ is also Cauchy in $||\*||_2$. Using the given inequality, we have:

$$||x_m - x_n||_2 \leq C_2 ||x_m - x_n||_1$$

Now, let $\epsilon' = \epsilon / C_2$. Since $(x_n)$ is Cauchy in $||\*||_1$, there exists an integer $N$ such that for all $m, n > N$, we have $||x_m - x_n||_1 < \epsilon'$. Substituting this into the inequality, we get:

$$||x_m - x_n||_2 \leq C_2 ||x_m - x_n||_1 < C_2 (\epsilon / C_2) = \epsilon$$

Thus, $(x_n)$ is Cauchy in $||\*||_2$.

Conversely, suppose $(x_n)$ is Cauchy in $||\*||_2$. This means that for any $\epsilon > 0$, there exists an integer $N$ such that for all $m, n > N$, we have $||x_m - x_n||_2 < \epsilon$. We want to show that $(x_n)$ is also Cauchy in $||\*||_1$. Using the given inequality, we also have:

$$C_1 ||x_m - x_n||_1 \leq ||x_m - x_n||_2$$

Dividing both sides by $C_1$, we get:

$$||x_m - x_n||_1 \leq (1/C_1) ||x_m - x_n||_2$$

Now, let $\epsilon' = C_1 \epsilon$. Since $(x_n)$ is Cauchy in $||\*||_2$, there exists an integer $N$ such that for all $m, n > N$, we have $||x_m - x_n||_2 < \epsilon'$. Substituting this into the inequality, we get:

$$||x_m - x_n||_1 \leq (1/C_1) ||x_m - x_n||_2 < (1/C_1) \epsilon' = (1/C_1) (C_1 \epsilon) = \epsilon$$

Thus, $(x_n)$ is Cauchy in $||\*||_1$.

Therefore, if there exist positive constants $C_1$ and $C_2$ such that $C_1 ||x||_1 \leq ||x||_2 \leq C_2 ||x||_1$, then the two norms are equivalent in the Cauchy sequence sense.

Part 2: Assuming Equivalence based on Cauchy Sequences, Prove the Existence of Constants

Now, assume that $||\*||_1$ and $||\*||_2$ are equivalent in the Cauchy sequence sense. We need to show that there exist positive constants $C_1$ and $C_2$ such that $C_1 ||x||_1 \leq ||x||_2 \leq C_2 ||x||_1$ for all $x \in F$.

We will prove this part by contradiction. Suppose that there does not exist a constant $C_2 > 0$ such that $||x||_2 \leq C_2 ||x||_1$ for all $x \in F$. This means that for every positive integer $n$, there exists an element $x_n \in F$ such that $||x_n||_2 > n ||x_n||_1$. Define a sequence $y_n = x_n / ||x_n||_2$. Then, $||y_n||_2 = 1$ for all $n$, and $||y_n||_1 = ||x_n / ||x_n||_2||_1 = ||x_n||_1 / ||x_n||_2 < 1/n$. Thus, $||y_n||_1$ converges to 0 as $n$ goes to infinity.

Now, consider the sequence $z_n = y_n$. Since $||y_n||_1$ converges to 0, the sequence $z_n$ converges to 0 in $||\*||_1$, and hence it is a Cauchy sequence in $||\*||_1$. By the assumed equivalence of the norms, $z_n$ must also be a Cauchy sequence in $||\*||_2$. This implies that $||z_n||_2$ must converge to 0. However, $||z_n||_2 = ||y_n||_2 = 1$ for all $n$, which is a contradiction. Therefore, there must exist a constant $C_2 > 0$ such that $||x||_2 \leq C_2 ||x||_1$ for all $x \in F$.

Similarly, suppose that there does not exist a constant $C_1 > 0$ such that $C_1 ||x||_1 \leq ||x||_2$ for all $x \in F$. This means that for every positive integer $n$, there exists an element $x_n \in F$ such that $||x_n||_2 < (1/n) ||x_n||_1$. Define a sequence $w_n = x_n / ||x_n||_1$. Then, $||w_n||_1 = 1$ for all $n$, and $||w_n||_2 = ||x_n / ||x_n||_1||_2 = ||x_n||_2 / ||x_n||_1 < 1/n$. Thus, $||w_n||_2$ converges to 0 as $n$ goes to infinity.

Now, consider the sequence $v_n = w_n$. Since $||w_n||_2$ converges to 0, the sequence $v_n$ converges to 0 in $||\*||_2$, and hence it is a Cauchy sequence in $||\*||_2$. By the assumed equivalence of the norms, $v_n$ must also be a Cauchy sequence in $||\*||_1$. This implies that $||v_n||_1$ must converge to 0. However, $||v_n||_1 = ||w_n||_1 = 1$ for all $n$, which is a contradiction. Therefore, there must exist a constant $C_1 > 0$ such that $C_1 ||x||_1 \leq ||x||_2$ for all $x \in F$.

Combining these results, we conclude that if $||\*||_1$ and $||\*||_2$ are equivalent in the Cauchy sequence sense, then there exist positive constants $C_1$ and $C_2$ such that $C_1 ||x||_1 \leq ||x||_2 \leq C_2 ||x||_1$ for all $x \in F$.

This completes the proof of the theorem.

Examples and Applications

To solidify our understanding, let's consider some examples and applications of equivalent field norms.

Example 1: The Usual Absolute Value and a Scaled Norm

Consider the field of real numbers $\mathbb{R}$ with the usual absolute value norm $|x|$ and another norm defined as $||x|| = 2|x|$. We can easily see that these norms are equivalent. Let $||\*||_1 = |\*|$ and $||\*||_2 = 2|\*|$. Then, we can choose $C_1 = 2$ and $C_2 = 2$, satisfying the inequality:

$$2|x| \leq 2|x| \leq 2(2|x|)$$

Thus, these norms are equivalent, as we discussed earlier.

Example 2: Equivalence in Finite-Dimensional Vector Spaces

In finite-dimensional vector spaces, all norms are equivalent. This is a crucial result in functional analysis, simplifying many arguments because we can choose a norm that is most convenient for the problem at hand. For example, in $\mathbb{R}^n$, the Euclidean norm ($||x||_2$), the taxicab norm ($||x||_1$), and the maximum norm ($||x||_{\infty}$) are all equivalent. This means that convergence in one norm implies convergence in all other norms, which is a powerful tool for analysis.

Application: Completeness

The equivalence of norms plays a significant role in determining the completeness of a field or a vector space. A field (or vector space) is said to be complete with respect to a norm if every Cauchy sequence converges to a limit within the field (or vector space). If two norms are equivalent, then the completeness with respect to one norm implies completeness with respect to the other. This is because Cauchy sequences and convergence are preserved under norm equivalence.

For example, the field of real numbers $\mathbb{R}$ is complete with respect to the usual absolute value norm. Since any norm equivalent to the absolute value norm will also preserve Cauchy sequences, $\mathbb{R}$ will also be complete with respect to any norm equivalent to the absolute value norm. This principle extends to more general settings, such as Banach spaces, where the completeness of the space is crucial for the existence and uniqueness of solutions to differential equations and other analytical problems.

Example 3: Non-Equivalent Norms

To appreciate the concept of equivalence, it's helpful to consider an example of non-equivalent norms. Consider the field of rational numbers $\mathbb{Q}$. Let $||\*||_1$ be the usual absolute value norm and $||\*||_2$ be the $p$-adic norm for some prime $p$. These norms are not equivalent. The $p$-adic norm measures the divisibility of a number by $p$, while the absolute value norm measures the distance from zero in the usual sense. Sequences that are Cauchy in the $p$-adic norm may not be Cauchy in the absolute value norm, and vice versa.

For instance, consider the sequence $x_n = p^n$. In the usual absolute value norm, $||x_n||_1 = |p^n| = p^n$, which tends to infinity as $n$ increases. However, in the $p$-adic norm, $||x_n||_2 = ||p^n||_p = p^{-n}$, which tends to zero as $n$ increases. This stark contrast demonstrates that the two norms induce very different notions of convergence and are therefore not equivalent. This non-equivalence is fundamental in the study of $p$-adic analysis, which explores the unique properties of fields equipped with $p$-adic norms.

Conclusion

In conclusion, the concept of equivalence of field norms is a cornerstone in the study of real analysis, metric spaces, and normed spaces. The equivalence theorem provides a powerful tool for relating different norms on the same field, allowing us to switch between norms while preserving the essential analytical properties. The Cauchy sequence criterion offers a fundamental definition of equivalence, while the existence of scaling constants provides a practical way to check for equivalence. Understanding this concept is crucial for tackling advanced topics in analysis and related fields, such as functional analysis and $p$-adic analysis. The examples and applications discussed in this article serve to illustrate the significance and versatility of norm equivalence in various mathematical contexts. By grasping these principles, one can gain a deeper appreciation for the rich structure and interconnectedness of mathematical concepts.