Problem 11 - Advanced Linear Algebra

Question

Prove that if a Hilbert space \(H\) has infinite Hilbert dimension, then no Hilbert basis for \(H\) is a Hamel basis.

Step-by-Step Solution

Answer

In conclusion, if a Hilbert space \(H\) has infinite Hilbert dimension, then no Hilbert basis for \(H\) can be a Hamel basis. We proved this by assuming that a Hilbert basis is a Hamel basis for an infinite-dimensional Hilbert space and then deriving a contradiction based on this assumption. We constructed a vector \(v\) as an infinite sum of elements from the basis \(B\), which should be uniquely expressible as a finite linear combination of elements in \(B\) if it were a Hamel basis. However, since the sum defining \(v\) has infinitely many terms in an infinite-dimensional Hilbert space, this contradicts the uniqueness property of a Hamel basis, showing that it is impossible for a Hilbert basis to be a Hamel basis for an infinite-dimensional Hilbert space.

Step 1: Hilbert Space Definition

A Hilbert space is a complete inner product space, which means that it is endowed with an inner product and that every Cauchy sequence of its vectors converges to a vector in the space. Completeness ensures that the space has a solid geometric structure, allowing us to use concepts like orthogonality and orthogonal projection.

Step 2: Hilbert Dimension Definition

The Hilbert dimension of a Hilbert space is the cardinality of a Hilbert basis, which is a set of orthogonal vectors that any vector in the space can be expressed as a linear combination of with convergence in the norm.

Step 3: Hilbert Basis Definition

A Hilbert basis is a set of orthogonal vectors in a Hilbert space that is dense in the space, meaning that any vector in the Hilbert space can be approximated arbitrarily closely by linear combinations of the basis elements.

Step 4: Hamel Basis Definition

A Hamel basis is a set of linearly independent vectors such that every vector in the space can be uniquely expressed as a finite linear combination of basis elements.

Step 5: Assume Hilbert Basis is a Hamel Basis

Now, let's assume that a Hilbert basis \(B\) for an infinite-dimensional Hilbert space \(H\) is a Hamel basis.

Step 6: Derive a Contradiction

We know that the Hilbert dimension of \(H\) is infinite since \(H\) is infinite-dimensional. Therefore, \(B\) must contain infinitely many elements. However, since \(B\) is a Hamel basis, every vector in \(H\) can be uniquely expressed as a finite linear combination of basis elements. Let \(b_1, b_2, \dots, b_n, \dots\) be the elements of our Hilbert (and Hamel) basis \(B\). Consider the following vector: \[v = \sum_{n=1}^{\infty} \frac{1}{2^n} b_n\] We can see that this sum converges in the norm since it is a convergent geometric series for each coordinate. This means that \(v \in H\). Because \(B\) is a Hamel basis, it must be the case that \(v\) can be uniquely expressed as a finite linear combination of elements in \(B\). However, this is a contradiction, since the sum defining \(v\) has infinitely many terms for an infinite-dimensional Hilbert space. Therefore, the assumption that a Hilbert basis can be a Hamel basis for an infinite-dimensional Hilbert space is incorrect.

Step 7: Conclusion

In conclusion, we have shown that if a Hilbert space \(H\) has infinite Hilbert dimension, then no Hilbert basis for \(H\) can be a Hamel basis. We achieved this by deriving a contradiction based on the assumption that a Hilbert basis is a Hamel basis for an infinite-dimensional Hilbert space, which means it cannot be true.

Key Concepts

Hilbert basis

In a Hilbert space, a Hilbert basis is a crucial concept. It is essentially a collection of orthogonal vectors that form the building blocks of the space itself.
A particularly important property of a Hilbert basis is its density, meaning that with the vectors in the basis, we can approximate any vector in the Hilbert space as closely as desired.
This approximating ability is done using linear combinations of the basis vectors with infinite terms, aligned with the space's completeness. Some key points about a Hilbert basis include:

• It consists of orthogonal vectors.
• It can approximate any vector in the space through convergence.
• The number of elements is defined by the Hilbert dimension, which can be infinite in some spaces.

Understanding a Hilbert basis helps us grasp the geometric structure of the space and paves the way for advanced concepts like functional analysis.

Hamel basis

The concept of a Hamel basis is quite different from a Hilbert basis. Whereas a Hilbert basis allows for infinite linear combinations, a Hamel basis is restricted to finite combinations. This does not take away from its utility; however, it does change how it functions in certain types of vector spaces.
A Hamel basis makes it possible for every vector in the space to be expressed in one unique way using a finite sum of basis vectors.
Here's more about a Hamel basis:

• Vectors are linearly independent.
• Every vector in the space is represented by a finite combination of basis vectors.
• Hamel bases are more intuitive in finite-dimensional vector spaces.

While this is useful in finite-dimensional spaces, it becomes impractical in infinite-dimensional spaces like the Hilbert spaces because infinitely many elements cannot be captured by finite sums alone.

infinite-dimensional vector spaces

An infinite-dimensional vector space, as the name suggests, is a space that has an infinite number of dimensions. This can be quite complex, as it means there is no finite limit to the number of basis vectors needed to span the space.
Such spaces are often encountered in functional spaces, such as spaces of functions, sequences, or even in quantum mechanics.
Characteristics of infinite-dimensional vector spaces include:

• They cannot be spanned by a finite basis of vectors.
• Typically require bases that allow infinite linear combinations, like a Hilbert basis.
• Extend beyond simple geometric intuition.

Understanding these spaces requires a shift in thinking. We move from finite, manageable collections of vectors to infinite collections that need more abstract mathematical tools to handle. These spaces often challenge our traditional understanding but are essential for various advanced fields of science and mathematics.