Question

Let \(X\) be an infinite-dimensional Banach space. Show that \(X\) admits no countable Hamel (algebraic) basis. Therefore, \(c_{00}\) cannot be normed to become a Banach space. Hint: If \(\left\\{e_{i}\right\\}\) is a countable infinite Hamel basis of a Banach space \(X\), put \(F_{n}=\operatorname{span}\left\\{e_{1}, \ldots, e_{n}\right\\} . F_{n}\) are closed and thus, by the Baire category theorem, at least one \(F_{n_{0}}\) has a nonempty interior; that is, there is \(x \in X\) and a ball \(B=\delta B_{X}\) such that \(x+B \subset F_{n_{0}} .\) Using linearity of \(F_{n_{0}}\), we have that \(-x+B \subset F_{n_{0}}\), so \(B \subset(x+B)+(-x+B) \subset F_{n_{0}} .\) Thus 0 is an interior point of \(F_{n_{0}} .\) This would mean that \(F_{n_{0}}=X\), a contradiction.

Short answer

An infinite-dimensional Banach space cannot have a countable Hamel basis, meaning \( c_{00} \) cannot be normed to become a Banach space.

Step-by-step solution

Define Constructed Sets

Assume, for contradiction, that the Banach space \( X \) has a countable Hamel basis \( \{e_{i} \} \). Define the finite-dimensional sets \( F_{n}=\text{span}(\{e_{1}, \ldots, e_{n} \}) \) for each \( n \).

Apply Baire Category Theorem

The sets \( F_{n} \) are closed. By the Baire category theorem, since \( X \) is a Banach space (and thus a complete metric space), at least one \( F_{n_{0}} \) must have a nonempty interior.

Identify Nonempty Interior

Since \( F_{n_{0}} \) has a nonempty interior, there exists \( x \in X \) and a ball \( B=\text{int}(B_X) \) such that \( x+B \subset F_{n_{0}} \). Due to the linearity of \( F_{n_{0}} \), we have that \(-x+B \subset F_{n_{0}} \), so combining these gives \( B \subset (x+B) + (-x+B) \subset F_{n_{0}} \).

Deduce Interior Point

Therefore, the ball \( B \) containing the zero vector implies that 0 is an interior point of \( F_{n_{0}} \).

Conclude Contradiction

The fact that \( F_{n_{0}} \) contains an interior point 0 implies that \( F_{n_{0}} = X \) because a proper subspace of an infinite-dimensional Banach space cannot have an interior point. This leads to a contradiction because \( F_{n_{0}} \) is finite-dimensional by construction.

Conclude that \( c_{00} \) is Not Normable

From the contradiction, it follows that \( X \) cannot have a countable Hamel basis. Consequently, \( c_{00} \) cannot be normed to become a Banach space because \( c_{00} \) has a countable Hamel basis by definition.

Key concepts

Banach space

A Banach space is a type of vector space that is complete with respect to its norm. This means that every Cauchy sequence in a Banach space converges to a limit within the space. Here are a few important properties:

• **Normed Space:** All Banach spaces are normed spaces, but not all normed spaces are Banach spaces. The norm gives a way to measure the size or length of vectors.


• **Completeness:** This ensures the validity of many crucial mathematical techniques, including analysis and approximation. Completeness implies that every Cauchy sequence (a sequence where points get arbitrarily close to each other as the sequence progresses) converges within the space.


• **Infinite-Dimensionality:** The exercise highlights that if a Banach space is infinite-dimensional, certain properties like having a countable Hamel basis become impossible.

Understanding Banach spaces is critical in functional analysis, a field that extends linear algebra to infinite dimensions and underlies various mathematical and engineering applications.

Baire category theorem

The Baire category theorem is a fundamental result in topology and analysis. It states that in a complete metric space (like a Banach space), the intersection of countably many dense open sets is also dense. Here's how it relates:

• **Closed Sets in Banach spaces:** In the exercise, the sets \( F_n \) are described as closed. According to the Baire category theorem, in a Banach space, one of these closed sets will have a nonempty interior.


• **Interior Points:** If a closed set in a Banach space has an interior point, it must be the whole space. This helps prove the contradiction that an infinite-dimensional Banach space cannot have a countable Hamel basis.


• **Applications:** The theorem is widely used in many areas like functional analysis and probability theory. It is crucial for proving the structure of various function spaces and understanding the behavior of sequences of functions.

To summarize, the Baire category theorem helps us understand how sets behave in Banach spaces and is pivotal in proving many fundamental results.

Infinite-dimensional spaces

Infinite-dimensional spaces, such as Banach spaces, have unique properties that distinguish them from finite-dimensional spaces.

• **Hamel Basis:** In finite-dimensional spaces, any vector can be expressed as a finite linear combination of basis vectors. However, this does not hold in infinite-dimensional spaces, as shown in the exercise.


• **Complexity:** The structure of infinite-dimensional spaces is much more complex. For instance, the existence of a countable Hamel basis in an infinite-dimensional Banach space leads to contradictions.


• **c_{00} Space:** The space \(c_{00}\), consisting of sequences that are eventually zero, is an example of a space that cannot be normed to become a Banach space. This space has a countable Hamel basis by definition, which is incompatible with the properties of a Banach space.


• **Functional Analysis:** Understanding infinite-dimensional spaces is crucial for various fields, including quantum mechanics, signal processing, and more. They require more sophisticated mathematical tools and deeper understanding than finite-dimensional spaces.

Grasping the concept of infinite-dimensional spaces is key to advancing in higher mathematics and its applications.