Showing a metric space is complete

In summary, the conversation discusses the completeness of the metric space of all continuous real-valued functions on the interval [0, a] with the metric d(x,y)=sup_{0\leq t\leq a}e^{-Lt}|x(t)-y(t)|. The attempt at a solution involves trying to prove the completeness of this space by showing that every Cauchy sequence in it has a limit, and suggests using a pointwise convergence approach.
  • #1
chipotleaway
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Homework Statement


Show the space of all space of all continuous real-valued functions on the interval [0, a] with the metric [itex]d(x,y)=sup_{0\leq t\leq a}e^{-Lt}|x(t)-y(t)|[/itex] is a complete metric space.

The Attempt at a Solution


Spent a few hours just thinking about this question, trying to prove it directly from the definition that says a complete metric space is one where every Cauchy sequence in it has a limit in the space.

I started with an arbitrary Cauchy sequence of functions [itex]d(x_m, x_n)=sup_{0\leq t\leq a}e^{-Lt}|x_m(t)-x_n(t)|[/itex]...that's it! I don't know how to find this limit and show that the sequence converges to that.
 
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  • #2
Try to show that your sequence converges pointswise. So show that for every ##t##, the sequence ##(x_n(t))_n## converges to some real number which I denote by ##x(t)##. Then show that ##x## defined like this is continuous and that ##x_n\rightarrow x## in your metric.
 

1. What is a complete metric space?

A complete metric space is a metric space in which every Cauchy sequence converges to a point within the space. In other words, every sequence of points in a complete metric space has a limit point within the space.

2. How do you show that a metric space is complete?

To show that a metric space is complete, you must prove that every Cauchy sequence in the space has a limit point within the space. This can be done by showing that the sequence converges to a point within the space or by showing that the sequence is a Cauchy sequence using the definition of a Cauchy sequence.

3. What is a Cauchy sequence?

A Cauchy sequence is a sequence of points in a metric space in which the distance between any two points in the sequence approaches zero as the sequence goes to infinity. In other words, the points in a Cauchy sequence get closer and closer together as the sequence goes on.

4. Can a subset of a complete metric space be complete?

Yes, a subset of a complete metric space can be complete. This is because a subset of a complete metric space inherits all of the properties of the larger space, including completeness.

5. Is completeness a necessary condition for a metric space?

No, completeness is not a necessary condition for a metric space. There are metric spaces that are not complete, such as the space of rational numbers. However, completeness is a desirable property as it ensures that all Cauchy sequences have a limit point within the space.

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