A Non-Separable Finite-Dimensional Inner Product Space

In the realm of functional analysis, the properties of inner product spaces play a crucial role in various mathematical theories and applications. While it is well-known that a finite-dimensional inner product space is always separable and complete, the opposite examples where these properties fail can be intriguing and insightful. This article explores the concept of a non-separable finite-dimensional inner product space and provides a clear example to illustrate this phenomenon.

Introduction to Inner Product Spaces

Before delving into the specifics, it is essential to define the terms and concepts involved. An inner product space is a vector space over the field of real or complex numbers equipped with an inner product, which is a notion of quantity and direction for vectors. The properties we will be discussing—separability and completeness—are foundational in functional analysis and are closely related to the structure and behavior of these spaces.

Separability and Completeness in Inner Product Spaces

Separability refers to the property of a space having a countable dense subset. In simpler terms, it means that there exists a countable set of points such that every point in the space can be approximated arbitrarily closely by points in this countable set. Completeness, on the other hand, means that every Cauchy sequence in the space converges to a point within the space. A space that is both separable and complete is often referred to as a "separable and complete space," which is a desirable property in many mathematical contexts.

Finite-Dimensional Inner Product Spaces

When dealing with finite-dimensional inner product spaces, the Riesz Representation Theorem and other fundamental results in functional analysis guarantee that these spaces are indeed separable and complete. However, this raises an interesting question: can a finite-dimensional inner product space fail to be separable? This article answers this question with a clear example.

Counterexample: A Non-Separable Finite-Dimensional Inner Product Space

To construct a counterexample, we need to consider the relationship between finite-dimensional spaces and the structures that can arise in infinite-dimensional spaces. Let us consider the vector space V over the field of real numbers with a basis {e_1, e_2, e_3, ...}. We can define an inner product on V by setting the inner product of any two basis vectors e_i and e_j to be the Kronecker delta function, where (langle e_i, e_j rangle delta_{ij}).

The Setup

Let us define the space V as the vector space of all sequences of real numbers (x_1, x_2, x_3, ...) such that x_i 0 for all sufficiently large i. In other words, V consists of sequences that are eventually zero. We can equip V with the inner product defined by:

[langle (x_1, x_2, x_3, ...), (y_1, y_2, y_3, ...) rangle sum_{i1}^{infty} x_i y_i]

This inner product is well-defined because each sequence in V is eventually zero.

The Non-Separability Argument

Now, let us show that the space V is not separable. To do this, we need to show that there is no countable dense subset in V. Consider the set of all sequences of the form (0, 0, ..., 1, 0, ...), where the 1 is in the n-th position for some fixed n. Each such sequence has an inner product of 1 with the sequence where the 1 is in the n-th position and 0 elsewhere. If we have a countable dense subset, then there must be a sequence in this subset that can approximate any sequence in V arbitrarily closely. However, since each sequence in this countable dense subset can have only finitely many non-zero terms, it is impossible to approximate a sequence with infinitely many non-zero terms, thus proving that V is not separable.

The Completeness Argument

It is also easy to show that V is complete. Consider a Cauchy sequence in V. Since each sequence in V is eventually zero, the sequences in the Cauchy sequence must agree on increasingly large indices. Thus, the Cauchy sequence converges to a sequence in V, so V is complete.

Conclusion

The counterexample we have presented demonstrates that it is possible for a finite-dimensional inner product space to fail to be separable. However, it is important to note that this space is still a finite-dimensional inner product space, as required. This example highlights the distinction between finite-dimensional and infinite-dimensional spaces and the unique properties that can arise in each.

Key Takeaways

1. Finite-dimensional inner product spaces are always separable and complete. 2. It is possible for a space to fail to be separable while still being finite-dimensional and having other properties like completeness. 3. The construction of non-separable finite-dimensional inner product spaces requires a careful understanding of the inner product and the structure of the underlying vector space.