Equivalent Norms Induce The Same Topology In Linear Spaces

In the realm of functional analysis, a core concept revolves around the equivalence of norms defined on a linear space. This article delves into the profound implications of two norms being equivalent on a linear space X, particularly focusing on the topological structures they induce. Specifically, we aim to elucidate why equivalent norms give rise to the same topology. Understanding this equivalence is pivotal as it directly impacts the convergence of sequences, the continuity of mappings, and the overall structural properties of the space. This exploration will not only solidify the theoretical underpinnings but also highlight the practical significance in various applications of functional analysis.

Defining Norms and Their Equivalence

At the heart of our discussion lies the definition of a norm. A norm, denoted as ||·||, on a linear space X is a real-valued function that satisfies several key properties: non-negativity (||x|| ≥ 0 for all x in X, and ||x|| = 0 if and only if x = 0), absolute homogeneity (||αx|| = |α| ||x|| for all scalars α and x in X), and the triangle inequality (||x + y|| ≤ ||x|| + ||y|| for all x, y in X). The norm effectively provides a measure of the “length” or “magnitude” of vectors within the space. Now, when we introduce a second norm, say ||·||₂, on the same linear space X, the notion of norm equivalence comes into play. Two norms, ||·||₁ and ||·||₂, are said to be equivalent if there exist positive constants C₁ and C₂ such that for all x in X, the following inequalities hold: C₁||x||₁ ≤ ||x||₂ ≤ C₂||x||₁. This seemingly simple condition has far-reaching consequences, particularly regarding the topological structures induced by these norms.

The essence of norm equivalence lies in the fact that these constants bound the norms relative to each other. What this means in practical terms is that no norm grows unboundedly larger or smaller than the other. The constants C₁ and C₂ act as scaling factors, ensuring that the two norms remain proportional. To fully grasp the significance of this equivalence, consider sequences of vectors within the linear space. If a sequence converges to a limit under one norm, the equivalence ensures that it also converges to the same limit under the other norm. This is a crucial property, as it indicates that the convergence behavior within the space is preserved across equivalent norms. Furthermore, norm equivalence has profound implications for the continuity of linear operators. A linear operator that is continuous with respect to one norm will also be continuous with respect to any norm equivalent to it. This preservation of continuity is fundamental in many areas of analysis, including the study of differential equations and approximation theory. In essence, the concept of equivalent norms allows us to transition between different ways of measuring vector magnitudes without altering the fundamental analytical properties of the space. This provides a robust and flexible framework for mathematical analysis, where the choice of norm can be tailored to the specific problem at hand without sacrificing the underlying structure of the space. The constants C₁ and C₂ act as the bridge between these different perspectives, ensuring that the essential features of the space remain invariant.

The Topology Induced by a Norm

A norm on a linear space naturally induces a topology. This induced topology is crucial because it provides the framework for discussing concepts like open sets, closed sets, convergence, and continuity. In simpler terms, it provides the necessary structure to define which points are “close” to each other. The topology induced by a norm is referred to as the norm topology or the metric topology. To understand how a norm induces a topology, we first need to define the metric or distance function. Given a norm ||·|| on a linear space X, the distance d(x, y) between two points x and y in X is defined as ||x - y||. This distance function satisfies the axioms of a metric: it is non-negative, symmetric, and satisfies the triangle inequality.

The metric thus allows us to quantify the separation between points in the space. With the distance function in hand, we can define open balls. An open ball centered at a point x with radius r > 0 is the set of all points y in X such that d(x, y) < r, or equivalently, ||x - y|| < r. These open balls serve as the building blocks for defining open sets in the norm topology. A set U in X is considered open if, for every point x in U, there exists an open ball centered at x that is entirely contained in U. In other words, an open set is a set where every point has a “neighborhood” around it that is also within the set. This definition of open sets forms the basis of the norm topology. Once we have defined open sets, we can define closed sets as the complements of open sets. A set is closed if it contains all its limit points. This means that if a sequence of points in a closed set converges, the limit of that sequence must also be in the set. The concepts of open and closed sets are fundamental in analysis, as they allow us to define continuity of functions and convergence of sequences. A sequence (xₙ) in X is said to converge to a limit x if, for every open set U containing x, there exists an index N such that xₙ is in U for all n > N. This definition of convergence is closely tied to the norm, as the open sets are defined in terms of the norm. Similarly, a function f from X to another normed space Y is continuous if the inverse image of every open set in Y is an open set in X. This means that small changes in the input x result in small changes in the output f(x). The norm topology thus provides the essential framework for defining and studying analytical properties of linear spaces. It allows us to discuss convergence, continuity, and other topological properties in a rigorous and precise manner. The fact that a norm induces a topology is a cornerstone of functional analysis, linking the algebraic structure of a linear space with the analytical properties derived from the norm.

Equivalent Norms Induce the Same Topology: The Proof

The central theorem we aim to prove is that if two norms, ||·||₁ and ||·||₂, are equivalent on a linear space X, then they induce the same topology. This means that a set is open with respect to the topology induced by ||·||₁ if and only if it is open with respect to the topology induced by ||·||₂. To prove this, we need to show that the open sets generated by the two norms are the same. Recall that two norms ||·||₁ and ||·||₂ are equivalent if there exist positive constants C₁ and C₂ such that for all x in X, C₁||x||₁ ≤ ||x||₂ ≤ C₂||x||₁. Let τ₁ be the topology induced by ||·||₁ and τ₂ be the topology induced by ||·||₂. We want to show that τ₁ = τ₂. To do this, we will show that every open set in τ₁ is also open in τ₂, and vice versa.

First, let U be an open set in τ₁. This means that for every point x in U, there exists an r₁ > 0 such that the open ball B₁(x, r₁) = y ∈ X ||x - y||₁ < r₁ is contained in U. We need to show that U is also open in τ₂, which means we need to find an r₂ > 0 such that the open ball B₂(x, r₂) = y ∈ X ||x - y||₂ < r₂ is also contained in U. Since ||x - y||₂ ≤ C₂||x - y||₁, if we choose r₂ = r₁/C₂, then for any y in B₂(x, r₂), we have ||x - y||₂ < r₁/C₂. Multiplying by C₂ gives us ||x - y||₂ C₂ < r₁, which implies ||x - y||₁ < r₁. Thus, y is in B₁(x, r₁), and since B₁(x, r₁) is contained in U, we have y in U. This shows that B₂(x, r₁/C₂) ⊆ B₁(x, r₁) ⊆ U. Therefore, for every x in U, there exists an r₂ > 0 (specifically, r₂ = r₁/C₂) such that B₂(x, r₂) ⊆ U, which means U is open in τ₂. Now, we need to show the converse: that every open set in τ₂ is also open in τ₁. Let V be an open set in τ₂. This means that for every point x in V, there exists an r₂ > 0 such that the open ball B₂(x, r₂) = y ∈ X ||x - y||₂ < r₂ is contained in V. We need to find an r₁ > 0 such that the open ball B₁(x, r₁) = y ∈ X ||x - y||₁ < r₁ is also contained in V. Since C₁||x - y||₁ ≤ ||x - y||₂, if we choose r₁ = r₂/C₁, then for any y in B₁(x, r₁), we have ||x - y||₁ < r₂/C₁. Multiplying by C₁ gives us C₁||x - y||₁ < r₂, which implies ||x - y||₂ < r₂. Thus, y is in B₂(x, r₂), and since B₂(x, r₂) is contained in V, we have y in V. This shows that B₁(x, r₂/C₁) ⊆ B₂(x, r₂) ⊆ V. Therefore, for every x in V, there exists an r₁ > 0 (specifically, r₁ = r₂/C₁) such that B₁(x, r₁) ⊆ V, which means V is open in τ₁. Since we have shown that every open set in τ₁ is open in τ₂ and every open set in τ₂ is open in τ₁, we conclude that τ₁ = τ₂. This completes the proof that equivalent norms induce the same topology on a linear space.

Implications of Topological Equivalence

The fact that equivalent norms induce the same topology has several important implications in functional analysis. These implications extend to the behavior of sequences, the continuity of functions, and the completeness of the space. Understanding these implications is crucial for effectively working with normed spaces and their applications.

Convergence of Sequences

One of the most direct implications of topologically equivalent norms is the preservation of convergence. If a sequence (xₙ) in X converges to a limit x under the norm ||·||₁, then it also converges to the same limit under any norm ||·||₂ that is equivalent to ||·||₁. This is because convergence is a topological property, and since the two norms induce the same topology, the notion of convergence remains the same. Formally, a sequence (xₙ) converges to x in the norm ||·|| if for every ε > 0, there exists an N such that ||xₙ - x|| < ε for all n > N. If ||·||₁ and ||·||₂ are equivalent, then there exist constants C₁ and C₂ such that C₁||x||₁ ≤ ||x||₂ ≤ C₂||x||₁ for all x in X. Suppose (xₙ) converges to x in ||·||₁. Then for any ε > 0, there exists an N such that ||xₙ - x||₁ < ε/C₂ for all n > N. This implies that ||xₙ - x||₂ ≤ C₂||xₙ - x||₁ < C₂(ε/C₂) = ε for all n > N. Thus, (xₙ) converges to x in ||·||₂. Conversely, if (xₙ) converges to x in ||·||₂, then for any ε > 0, there exists an N such that ||xₙ - x||₂ < C₁ε for all n > N. This implies that C₁||xₙ - x||₁ ≤ ||xₙ - x||₂ < C₁ε, so ||xₙ - x||₁ < ε for all n > N. Thus, (xₙ) converges to x in ||·||₁. This preservation of convergence is crucial in many analytical contexts, as it allows us to choose the most convenient norm for a particular problem without altering the fundamental convergence properties of the space.

Continuity of Mappings

Another significant implication is the preservation of continuity of mappings between normed spaces. If f: X → Y is a function from a normed space X with norm ||·||₁ to a normed space Y with norm ||·||, and if ||·||₂ is a norm on X equivalent to ||·||₁, then f is continuous with respect to ||·||₁ if and only if it is continuous with respect to ||·||₂. Continuity is also a topological property, and since equivalent norms induce the same topology, the continuity of mappings is preserved. A function f is continuous at a point x₀ if for every open set V in Y containing f(x₀), there exists an open set U in X containing x₀ such that f(U) ⊆ V. In terms of norms, this means that for every ε > 0, there exists a δ > 0 such that if ||x - x₀||₁ < δ, then ||f(x) - f(x₀)|| < ε. If ||·||₂ is equivalent to ||·||₁, then there exist constants C₁ and C₂ such that C₁||x||₁ ≤ ||x||₂ ≤ C₂||x||₁. If f is continuous with respect to ||·||₁, then for any ε > 0, there exists a δ > 0 such that if ||x - x₀||₁ < δ, then ||f(x) - f(x₀)|| < ε. Now, choose δ' = δ/C₂. If ||x - x₀||₂ < δ', then ||x - x₀||₂ < δ/C₂, which implies ||x - x₀||₁ ≤ (1/C₁) ||x - x₀||₂ < (1/C₁)(δ/C₂) < δ. Thus, ||f(x) - f(x₀)|| < ε, which means f is continuous with respect to ||·||₂. The converse can be shown similarly, demonstrating that continuity is preserved under equivalent norms. This result is particularly useful in the study of linear operators, where continuity is a fundamental property. If a linear operator is continuous with respect to one norm, it is continuous with respect to any equivalent norm, simplifying the analysis and application of these operators.

Completeness and Banach Spaces

Completeness is another crucial property in functional analysis. A normed space is complete if every Cauchy sequence converges to a limit within the space. A complete normed space is called a Banach space. If two norms are equivalent, they preserve completeness. This means that if a linear space X is complete under one norm, it is also complete under any equivalent norm. A sequence (xₙ) is Cauchy in the norm ||·|| if for every ε > 0, there exists an N such that ||xₙ - xₘ|| < ε for all n, m > N. If ||·||₁ and ||·||₂ are equivalent, then there exist constants C₁ and C₂ such that C₁||x||₁ ≤ ||x||₂ ≤ C₂||x||₁ for all x in X. Suppose (xₙ) is Cauchy in ||·||₁. Then for any ε > 0, there exists an N such that ||xₙ - xₘ||₁ < ε/C₂ for all n, m > N. This implies that ||xₙ - xₘ||₂ ≤ C₂||xₙ - xₘ||₁ < C₂(ε/C₂) = ε for all n, m > N. Thus, (xₙ) is Cauchy in ||·||₂. Conversely, if (xₙ) is Cauchy in ||·||₂, then for any ε > 0, there exists an N such that ||xₙ - xₘ||₂ < C₁ε for all n, m > N. This implies that C₁||xₙ - xₘ||₁ ≤ ||xₙ - xₘ||₂ < C₁ε, so ||xₙ - xₘ||₁ < ε for all n, m > N. Thus, (xₙ) is Cauchy in ||·||₁. If X is complete under ||·||₁, then every Cauchy sequence in ||·||₁ converges to a limit x in X. Since Cauchy sequences are preserved under equivalent norms, (xₙ) is also Cauchy in ||·||₂, and it converges to the same limit x in X. Therefore, X is complete under ||·||₂. This preservation of completeness is crucial in functional analysis, particularly in the study of Banach spaces. Many important results, such as the open mapping theorem and the closed graph theorem, rely on the completeness of the space. If two norms are equivalent, we can choose the norm that is most convenient for a particular problem without affecting the completeness of the space.

Examples of Equivalent Norms

To further illustrate the concept of equivalent norms, let's consider some concrete examples. These examples will help solidify the understanding of how norms can be equivalent and what this equivalence implies in different contexts.

Euclidean Space (ℝⁿ)

In the Euclidean space ℝⁿ, there are several commonly used norms, including the p-norms, defined as follows: For x = (x₁, x₂, ..., xₙ) ∈ ℝⁿ, the p-norm is given by ||x||ₚ = (Σᵢ|xᵢ|ᵖ)^(1/p) for 1 ≤ p < ∞, and ||x||∞ = maxᵢ|xᵢ. A well-known result is that all p-norms on ℝⁿ are equivalent. This means that for any two p, q ∈ [1, ∞], there exist positive constants C₁ and C₂ such that C₁||x||ₚ ≤ ||x||<sub style=