Next: Hilbert Spaces Up: Complete Metric Spaces Previous: Cauchy Sequence   Contents   Index

## Cauchy Completeness: Complete Metric Space, Banach Space, and Hilbert Space

Can one turn the last sentence around? In other words, is every Cauchy sequence a convergent sequence? Put differently, if is a Cauchy sequence, is it true that has a limit? What is, in fact, meant by this question is whether has a limit in the same space to which the elements belong.

The answer is this: in a finite dimensional complex vector space, a Cauchy sequence always has a limit in that vector space; in other words, a finite dimensional vector space is complete. However, such a conclusion is no longer true for many familiar infinite dimensional vector spaces.

Example: Consider the inner product space

with inner product .

Claim: is incomplete''.

Discussion: Consider the sequence of continuous functions

From Fig. 1.5 we see that:

1. ; in other words, the sequence is a Cauchy sequence.
2. For fixed

which is a discontinuous function, i.e., . Furthermoe, we say that the sequence of functions converges pointwise to the function .
3. .
4. One can show that any continuous function such that

We say that , the space of continuous square integrable functions , is Cauchy incomplete, or is Cauchy incomplete relative to the given norm . This is so because we have found a Cauchy sequence of functions in the inner product space with the property that

In other words, the limit of the Cauchy sequence does not lie in the inner product space. This is just like the set of rationals which is not extensive enough to accomodate the norm : there are holes (but no gaps) in the space. These holes are the irrational numbers, which are not detected by when it is used to determine whether or not an infinite sequence is convergent.

Example: The -dimensional vector space of rationals over the field of rationals is Cauchy incomplete.

Problem: Why is it that the real line equipped with the distance

is an incomplete metric space?

In order to remove this incompleteness deficiency, one enlarges the space so that it includes the limit of any of its Cauchy sequences. A space which has been enlarged in this sense is said to be Cauchy complete. This enlarged space is called a complete metric space.

The Cauchy completion of the rationals are the reals. The Cauchy completion of an inner product space is a Hilbert space The Cauchy completion of a normed linear space is a Banach space.

Next: Hilbert Spaces Up: Complete Metric Spaces Previous: Cauchy Sequence   Contents   Index
Ulrich Gerlach 2010-12-09