The Set Of Finite Linear Combinations With Rational Coefficients Is Dense In A Separable Hilbert Space

In the realm of functional analysis, Hilbert spaces stand as a cornerstone, providing a rich framework for studying infinite-dimensional vector spaces. A Hilbert space is a complete inner product space, and its separability, the existence of a countable dense subset, is a crucial property in various applications. Within this context, we explore a fundamental theorem: the set of finite linear combinations with rational coefficients, denoted as H_Q, is dense in a separable Hilbert space H. This article delves into the proof of this theorem, its implications, and its significance in the broader landscape of mathematical analysis.

Understanding Separable Hilbert Spaces

To fully appreciate the theorem, we must first understand the concept of a separable Hilbert space. A Hilbert space H is considered separable if it contains a countable dense subset. In simpler terms, this means there exists a countable set S within H such that any element in H can be approximated arbitrarily closely by elements from S. This property is vital because it allows us to work with infinite-dimensional spaces using techniques that are more akin to those used in finite-dimensional spaces.

Orthonormal Basis: The Foundation of Separability

The separability of a Hilbert space is intimately linked to the existence of an orthonormal basis. An orthonormal basis {e_n} in a Hilbert space H is a countable set of vectors that are mutually orthogonal (their inner product is zero) and have unit norm (their inner product with themselves is one). Moreover, this set is complete, meaning that any vector in H can be expressed as a linear combination of these basis vectors. This representation is not just any linear combination; it's a series representation that converges in the norm induced by the inner product of the Hilbert space.

The existence of a countable orthonormal basis is both a necessary and sufficient condition for a Hilbert space to be separable. This connection is crucial for the theorem we aim to prove. The orthonormal basis provides a discrete set of building blocks that span the entire space, and the separability ensures that we can approximate any vector in the space using these building blocks.

The Role of Countability

The countability of the orthonormal basis is key. It allows us to enumerate the basis vectors and construct finite linear combinations. This is essential for defining the set H_Q, which consists of finite sums of basis vectors with rational coefficients. The countability, combined with the density of rational numbers in real numbers, forms the backbone of the proof that H_Q is dense in H.

Defining H_Q: Finite Linear Combinations with Rational Coefficients

The central object of our discussion is the set H_Q, defined as the set of all finite linear combinations of the orthonormal basis vectors {e_n} with rational coefficients. Mathematically, this is expressed as:

H_Q = Σ_i=1^N q_ie_i q_i ∈ Q, e_i ∈ {e_n }

Where:

• q_i represents the rational coefficients.
• e_i are the orthonormal basis vectors.
• N is a finite positive integer, indicating that we are considering finite linear combinations.

The set H_Q is a subset of the Hilbert space H, and it possesses several important properties. First, it is a vector space over the field of rational numbers. This means that the sum of any two elements in H_Q is also in H_Q, and the product of any element in H_Q with a rational scalar is also in H_Q. However, H_Q is not a vector space over the field of real numbers because multiplying an element in H_Q by an irrational number will generally result in a vector that is not in H_Q.

Countability of H_Q

A crucial property of H_Q is its countability. Since the set of rational numbers Q is countable, and the orthonormal basis {e_n} is countable, the set of all finite linear combinations with rational coefficients is also countable. This can be understood by recognizing that each element in H_Q is uniquely determined by a finite sequence of rational numbers and corresponding basis vectors. The set of all such finite sequences is countable, which implies that H_Q is countable.

Why Rational Coefficients? The Density of Rationals in Reals

The choice of rational coefficients is not arbitrary. The density of the rational numbers Q in the real numbers R is a fundamental property that plays a critical role in the proof. The density of Q in R means that for any real number, we can find a rational number that is arbitrarily close to it. This property allows us to approximate real coefficients (which arise in the general linear combinations in H) with rational coefficients, which are the building blocks of H_Q.

The use of rational coefficients ensures that H_Q remains a countable set, which is essential for establishing its density in the separable Hilbert space. If we were to use real coefficients instead, the resulting set would be uncountable, and the proof strategy would need to be significantly different.

The Density Theorem: H_Q is Dense in H

The core of our discussion is the theorem stating that H_Q is dense in H. This means that for any vector x in H and any positive real number ε, there exists a vector y in H_Q such that the distance between x and y is less than ε. In mathematical notation:

∀ x ∈ H, ∀ ε > 0, ∃ y ∈ H_Q such that ||x - y|| < ε

Where || || denotes the norm induced by the inner product in the Hilbert space H.

Proof Strategy: Approximating with Finite Sums

The proof of this theorem relies on several key ideas:

1. Expressing vectors as infinite sums: Any vector x in H can be written as an infinite series of the form x = Σ_i=1^∞ c_ie_i, where c_i are real coefficients and {e_n} is the orthonormal basis. This is a direct consequence of the completeness of the orthonormal basis.
2. Truncating the infinite sum: Since the series converges, we can truncate it to a finite sum and make the error arbitrarily small. That is, for any ε > 0, there exists a positive integer N such that ||x - Σ_i=1^N c_ie_i|| < ε/2. This step leverages the convergence of the infinite series.
3. Approximating real coefficients with rational coefficients: For each real coefficient c_i, we can find a rational number q_i that is arbitrarily close to it, thanks to the density of Q in R. We can choose q_i such that |c_i - q_i| < ε/(2N), where N is the number of terms in the truncated sum. This is where the rational coefficients and their density become crucial.
4. Constructing the approximating vector in H_Q: We form the vector y = Σ_i=1^N q_ie_i, which is an element of H_Q by definition.
5. Bounding the error: Finally, we use the triangle inequality and the properties of the norm to show that ||x - y|| < ε. This demonstrates that we can approximate x arbitrarily closely with a vector from H_Q, proving the density of H_Q in H.

Detailed Proof

Let x be an arbitrary vector in H, and let ε > 0 be given. Since {e_n} is an orthonormal basis for H, we can write x as an infinite series:

x = Σ_i=1^∞ c_ie_i

Where c_i = ⟨x, e_i⟩ are the coefficients in the expansion.

Since the series converges, for any ε > 0, there exists a positive integer N such that:

||x - Σ_i=1^N c_ie_i|| < ε/2

Now, for each i = 1, 2, ..., N, since Q is dense in R, we can find a rational number q_i such that:

|c_i - q_i| < ε/(2√N)

Let's define the vector y in H_Q as:

y = Σ_i=1^N q_ie_i

Now we estimate the distance between x and y using the triangle inequality:

||x - y|| = ||(x - Σ_i=1^N c_ie_i) + (Σ_i=1^N c_ie_i - y)||

Using the triangle inequality, we get:

||x - y|| ≤ ||x - Σ_i=1^N c_ie_i|| + ||Σ_i=1^N c_ie_i - y||

We already know that the first term is less than ε/2. For the second term, we have:

||Σ_i=1^N c_ie_i - y|| = ||Σ_i=1^N (c_i - q_i)e_i||

Using the Pythagorean theorem (since the e_i are orthonormal):

||Σ_i=1^N (c_i - q_i)e_i||² = Σ_i=1^N |c_i - q_i|²

Since |c_i - q_i| < ε/(2√N), we have:

Σ_i=1^N |c_i - q_i|² < Σ_i=1^N (ε/(2√N))² = Σ_i=1^N ε²/(4N) = N * ε²/(4N) = ε²/4

Taking the square root, we get:

||Σ_i=1^N (c_i - q_i)e_i|| < ε/2

Therefore:

||x - y|| < ε/2 + ε/2 = ε

This shows that for any x in H and any ε > 0, we can find a y in H_Q such that ||x - y|| < ε. Thus, H_Q is dense in H.

Implications and Significance

The density of H_Q in H has several important implications and highlights the interplay between algebraic structures (Q) and analytic structures (Hilbert spaces).

Countable Dense Subset

The theorem provides a concrete example of a countable dense subset in a separable Hilbert space. This is significant because it explicitly constructs a set with the required properties, demonstrating the nature of separability in these spaces. The existence of a countable dense subset allows us to approximate any element in the Hilbert space using a countable set of elements, which is crucial for computational and approximation techniques.

Approximation Theory

In approximation theory, the theorem is fundamental. It allows us to approximate vectors in a Hilbert space using linear combinations with rational coefficients. This is particularly useful in numerical analysis, where computations are often performed using rational numbers due to their exact representation in computers. The theorem justifies the use of rational approximations in various applications, such as signal processing, image analysis, and numerical solutions of differential equations.

Basis for Numerical Methods

The set H_Q forms a basis for various numerical methods in Hilbert spaces. For instance, in finite element methods, we often approximate solutions to partial differential equations using piecewise polynomial functions. These functions can be expressed as linear combinations of basis functions, and by choosing rational coefficients, we can ensure that the computations are performed with rational numbers, which can be advantageous in certain scenarios.

Connections to Other Areas of Mathematics

The density of H_Q in H also has connections to other areas of mathematics, such as harmonic analysis and operator theory. In harmonic analysis, we often work with Fourier series, which are expansions of functions in terms of trigonometric functions. The coefficients in these expansions are typically complex numbers, but we can approximate them using rational numbers to obtain a dense subset of functions with rational coefficients. In operator theory, the theorem can be used to approximate bounded linear operators on Hilbert spaces using operators with rational matrix entries, which can simplify computations and analysis.

Conclusion

The theorem that the set of finite linear combinations with rational coefficients H_Q is dense in a separable Hilbert space H is a cornerstone result in functional analysis. It demonstrates the interplay between algebraic and analytic structures, highlighting the importance of separability and the density of rational numbers in real numbers. This theorem has far-reaching implications in approximation theory, numerical analysis, and various other areas of mathematics, providing a powerful tool for approximating vectors and functions in infinite-dimensional spaces. By understanding and applying this theorem, we can gain deeper insights into the structure and properties of Hilbert spaces and their applications in the broader mathematical landscape.