Every Complete Subspace Of A Normed Linear Space Is Closed Banach

Jul 10, 2025 by ADMIN 66 views

In the realm of functional analysis, understanding the properties of subspaces within normed linear spaces is crucial. One significant aspect involves the completeness of these subspaces and its implications. This article delves into the critical theorem stating that every complete subspace of a normed linear space is closed and that if the normed linear space is Banach, the complete subspace is also Banach. We will explore the concepts of normed linear spaces, completeness, and closedness, culminating in a detailed discussion and proof of this fundamental theorem. This exploration is essential for anyone studying functional analysis, as it provides a cornerstone for more advanced topics.

Normed Linear Spaces: The Foundation

To fully grasp the theorem, let's first define a normed linear space. A normed linear space is a vector space V over a field F (where F is typically the real numbers ℝ or the complex numbers ℂ) equipped with a norm. A norm is a function ||·||: V → ℝ that satisfies the following axioms:

Non-negativity: ||x|| ≥ 0 for all x ∈ V, and ||x|| = 0 if and only if x = 0.
Homogeneity: ||αx|| = |α| ||x|| for all x ∈ V and all scalars α ∈ F.
Triangle inequality: ||x + y|| ≤ ||x|| + ||y|| for all x, y ∈ V.

The norm essentially provides a way to measure the 'length' or 'magnitude' of vectors within the space. This measurement allows us to define a metric (a notion of distance) on the space, making it a metric space. The distance d between two vectors x and y in V is defined as d(x, y) = ||x - y||. This metric structure is fundamental because it enables us to discuss concepts like convergence of sequences, continuity of mappings, and the open and closed sets, which are vital in functional analysis.

Examples of Normed Linear Spaces

Some common examples of normed linear spaces include:

Euclidean space ℝⁿ: Equipped with the Euclidean norm ||x|| = √(x₁² + x₂² + ... + xₙ²*), where x = (x₁, x₂, ..., xₙ*).
The space of continuous functions C[a, b]: Defined on a closed interval [a, b], with the supremum norm ||f|| = sup|f(t)| t ∈ [a, b].
The sequence spaces lᵖ: For 1 ≤ p < ∞, consisting of all sequences x = (x₁, x₂, ...) such that Σ|xᵢ|ᵖ < ∞, with the norm ||x||ₚ = (Σ|xᵢ|ᵖ)^(1/p). The sequence space l^∞ consists of all bounded sequences with the supremum norm ||x||_∞ = sup|xᵢ| i ∈ ℕ.

Understanding these examples helps to contextualize the theory and apply it to various mathematical settings. Normed linear spaces provide a rich framework for studying linear operators, functionals, and their properties, making them indispensable in both pure and applied mathematics.

Completeness: Cauchy Sequences and Convergence

Central to our discussion is the concept of completeness in the context of normed linear spaces. To define completeness, we must first understand Cauchy sequences. A sequence (xₙ*) in a normed linear space V is called a Cauchy sequence if for every ε > 0, there exists a positive integer N such that for all m, n > N, we have ||xₘ - xₙ|| < ε. Intuitively, this means that the terms of the sequence become arbitrarily close to each other as n gets larger.

A normed linear space V is said to be complete if every Cauchy sequence in V converges to a limit that is also in V. In other words, if (xₙ*) is a Cauchy sequence in V, then there exists an element x ∈ V such that limₙ→∞ ||xₙ - x|| = 0. The completeness property is crucial because it ensures that the space contains all its limit points, making it well-behaved in many analytical contexts.

Banach Spaces

A Banach space is a complete normed linear space. These spaces are of paramount importance in functional analysis and its applications. Many fundamental theorems, such as the Banach Fixed Point Theorem, the Open Mapping Theorem, and the Closed Graph Theorem, are formulated and proven within the framework of Banach spaces. These theorems provide powerful tools for solving problems in various areas of mathematics and physics.

Examples and Non-Examples of Complete Spaces

The real numbers ℝ and the complex numbers ℂ: These are complete with respect to their usual norms (absolute value and modulus, respectively). This completeness is a fundamental property of the real and complex number systems.
Euclidean space ℝⁿ: This is a Banach space because every Cauchy sequence in ℝⁿ converges to a point in ℝⁿ.
The space of continuous functions C[a, b]: Equipped with the supremum norm, this space is a Banach space. The uniform limit of a sequence of continuous functions that converges uniformly is itself a continuous function.
The sequence spaces lᵖ (1 ≤ p ≤ ∞): These are all Banach spaces. The completeness of these spaces is essential in the study of sequence analysis and operator theory.
The space of rational numbers ℚ: With respect to the usual absolute value norm, ℚ is not complete. For example, a sequence of rational numbers that converges to √2 is a Cauchy sequence in ℚ, but its limit is not in ℚ.
The space of polynomials P[a, b]: Defined on a closed interval [a, b], with the supremum norm, this space is not complete. Polynomials can approximate continuous functions, but the uniform limit of a sequence of polynomials is not necessarily a polynomial.

Understanding which spaces are complete and which are not is crucial for determining the applicability of various theorems and techniques in analysis.

Closed Subspaces: Retaining Limits

Another key concept in our theorem is that of a closed subspace. A subspace U of a normed linear space V is said to be closed if it contains all its limit points. In other words, if (xₙ*) is a sequence in U that converges to a limit x in V, then x must also be in U. Equivalently, a set is closed if its complement is open.

Characterizations of Closedness

There are several ways to characterize the closedness of a set:

Sequential Closedness: A set U is closed if every convergent sequence in U has its limit in U.
Topological Closedness: A set U is closed if it contains its boundary. The boundary of a set consists of all points for which every neighborhood contains points both in the set and in its complement.
Closure: The closure of a set U, denoted by cl(U), is the smallest closed set containing U. A set U is closed if and only if U = cl(U).

The concept of closedness is fundamental in topology and analysis because it ensures that a set is 'complete' in a topological sense. Closed sets are often the natural domains for functions and operators, as they ensure that limits of sequences within the set remain within the set.

Examples of Closed and Non-Closed Subspaces

The subspace {0}: Consisting of only the zero vector, is always closed in any normed linear space. If a sequence of zero vectors converges, it converges to zero.
The entire space V: Is always a closed subspace of itself. Any sequence in V that converges has its limit in V.
Finite-dimensional subspaces: Of a normed linear space are always closed. This is a significant result in functional analysis.
The subspace of polynomials P[a, b]: Is not closed in C[a, b] with the supremum norm. As mentioned earlier, polynomials can approximate continuous functions, but the uniform limit of a sequence of polynomials is not necessarily a polynomial.
The set of sequences with finitely many non-zero terms: Is not closed in l² (the space of square-summable sequences). This is because there are sequences in l² that can be approximated by sequences with finitely many non-zero terms, but which themselves have infinitely many non-zero terms.

The Theorem: Complete Subspaces Are Closed Banach

Now, let's delve into the main theorem: Every complete subspace of a normed linear space is closed. This theorem provides a critical link between completeness and closedness, two fundamental properties in functional analysis.

Proof of the Theorem

Let V be a normed linear space, and let U be a complete subspace of V. We want to show that U is closed. To do this, we need to show that if (xₙ*) is a sequence in U that converges to a limit x in V, then x must be in U.

Assume (xₙ) is a sequence in U that converges to x in V*: This is our starting point. We have a sequence of vectors, each belonging to the subspace U, and this sequence converges to some vector x in the larger space V.
Since (xₙ) converges in V, it is a Cauchy sequence in V*: This is a standard result in analysis: every convergent sequence is a Cauchy sequence. The terms of the sequence become arbitrarily close to each other as n increases.
Since (xₙ) is a Cauchy sequence in U, and U is a subspace of V, (xₙ) is also a Cauchy sequence in U**: This is because the norm in U is inherited from V. The distance between any two terms in the sequence is the same whether we consider them in U or V.
Since U is complete, the Cauchy sequence (xₙ) converges to a limit u in U*: This is the critical step. The completeness of U guarantees that every Cauchy sequence in U has a limit in U. Let's call this limit u.
Thus, we have xₙ → x in V and xₙ → u in U: We have two limits for the same sequence. One is x in the larger space V, and the other is u in the subspace U.
Limits in normed linear spaces are unique, so x = u: This is a fundamental property of normed linear spaces. If a sequence converges, its limit is unique. Therefore, x and u must be the same vector.
Since u ∈ U, we have x ∈ U: Because x and u are the same, and u belongs to the subspace U, it follows that x also belongs to U.

Therefore, we have shown that if (xₙ*) is a sequence in U that converges to a limit x in V, then x is also in U. This is precisely the definition of a closed subspace. Hence, U is closed.

If the normed linear space is Banach

The second part of the theorem implies that if V is a Banach space, then the complete subspace U is also a Banach space. The proof is relatively straightforward. Let (yₙ*) be a Cauchy sequence in the complete subspace U. Since U is a subspace of V, (yₙ*) is also a Cauchy sequence in V. Because V is a Banach space, (yₙ*) converges to some y in V. As we've already proven, the complete subspace U is closed, so the limit y must also belong to U. Thus, every Cauchy sequence in U converges to a limit in U, which means that U is complete and therefore a Banach space.

Implications and Applications

The theorem that every complete subspace of a normed linear space is closed and, if the space is Banach, so is the subspace, has profound implications and numerous applications in functional analysis and related fields. Here are some key implications and applications:

Characterizing Banach Spaces: This theorem provides a powerful tool for characterizing Banach spaces. If a subspace of a normed linear space is complete, it is automatically closed. This simplifies the task of verifying whether a subspace is a Banach space.
Closed Range Theorem: In operator theory, the Closed Range Theorem states that for a bounded linear operator between Banach spaces, the range of the operator is closed if and only if the range of the adjoint operator is closed. This theorem relies heavily on the properties of complete subspaces and their closedness.
Existence and Uniqueness Theorems: Many existence and uniqueness theorems in differential equations, integral equations, and optimization theory are proven using the completeness of certain function spaces. These theorems often rely on finding solutions within complete subspaces, ensuring that the solutions are well-behaved.
Approximation Theory: In approximation theory, the completeness of subspaces plays a crucial role in determining the convergence of approximation schemes. For example, the Stone-Weierstrass Theorem provides conditions under which a subspace of continuous functions is dense in the entire space. The completeness of the space of continuous functions ensures that approximations converge to a well-defined limit.
Functional Analysis Foundations: The theorem is a cornerstone in the study of functional analysis. It is used in the proofs of many other important theorems, such as the Open Mapping Theorem, the Closed Graph Theorem, and the Uniform Boundedness Principle.
Numerical Analysis: In numerical analysis, the completeness of subspaces is essential for the convergence of numerical methods. For example, finite element methods rely on approximating solutions to differential equations in finite-dimensional subspaces. The completeness of the underlying function space ensures that the numerical solutions converge to the true solution as the mesh size decreases.

In summary, the theorem that every complete subspace of a normed linear space is closed and Banach is not just an abstract result; it is a fundamental tool with far-reaching consequences in mathematics and its applications. Understanding this theorem provides a solid foundation for further study in functional analysis and related areas.

Conclusion

In conclusion, the theorem stating that every complete subspace of a normed linear space is closed, and is Banach if the normed linear space is Banach, is a cornerstone of functional analysis. We have explored the foundational concepts of normed linear spaces, completeness, and closedness, culminating in a detailed proof and discussion of the theorem. This result provides a critical link between topological and analytical properties of subspaces, with significant implications for various areas of mathematics and its applications. Understanding this theorem is essential for anyone studying functional analysis, as it provides a framework for more advanced topics and problem-solving techniques in the field.