What is a Complete Metric Space in Mathematics?

[Amok]

Can someone help me understand the notion of complete metric space? I've read the definition (the one involving metrics that go to 0), but I can't really picture what it is. Does anyone have any examples that could help me understand this?

 

[micromass]

I'm not sure if there even is something you can picture. Completeness is simply a quite technical condition that has a lot of benifits. Intuitively, one can say that a space is complete if every sequence that should converge, also converges.
What is a sequence that should converge? Well a sequence who's terms lie closer and closer together. For example, the sequence (1/n) should converge, because the terms are closer and closer. But (n) does not converge, because the terms both have distance 1 from each other.

The space $$\mathbb{R}$$ is complete: every sequence that should converge converges, but $$\mathbb{Q}$$ is incomplete, indeed a rational sequence that converges to $$\pi$$ does not converge in $$\mathbb{Q}$$.

 

[Amok]

But cam you show that a Hilbert space or a $$\mathbb{R}$$ space converges? Using the definition of distance, for example? Can you show that every Cauchy sequence in a certain space converges?

 

[micromass]

But cam you show that a Hilbert space or a $$\mathbb{R}$$ space converges? Using the definition of distance, for example?

What do you mean with "a Hilbert spaces converges"?

Can you show that every Cauchy sequence in a certain space converges?

Yes, one can show that for a lot of spaces, so it's certainly not an impossible condition to check. The only space for which it is really hard to check is for $$\mathbb{R}$$, but that's because the definition of $$\mathbb{R}$$ is quite complicated...

 

[CellCoree]

A complete metric space is a mathematical concept that describes a set of points that have a defined distance between them and satisfy certain properties. This notion is important in fields such as analysis, topology, and geometry.

To better understand this concept, let's break down the definition. A metric space is a set of points where the distance between any two points is defined by a metric. This metric can be thought of as a measure of how far apart two points are from each other.

Now, the completeness of a metric space refers to the idea that the space contains all of its limit points. In other words, if we have a sequence of points in the space that get closer and closer together, the limit of that sequence must also be in the space. This ensures that there are no "missing" points in the space.

One way to visualize a complete metric space is to think of it as a set of points on a line. Each point on the line has a defined distance from every other point, and there are no gaps or missing points. This is similar to how a ruler has evenly spaced markings with no gaps.

Another example is the set of real numbers, which is a complete metric space. Every real number has a defined distance from every other real number, and there are no missing numbers in between. This is why the real numbers are often used as a basis for understanding metric spaces.

I hope this helps you understand the notion of a complete metric space. It is a fundamental concept in mathematics and has many applications in various fields.