How to show completeness of C(K)?

  • Thread starter minibear
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In summary, to show that (C(K), d_infinity) is complete, one must show that all Cauchy sequences in this metric space converge. This can be done by considering the sequence of continuous functions {fn} and showing that {fn(x)} is a Cauchy sequence for each x in K. The proof may involve using Cauchy or triangle inequality.
  • #1
minibear
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The thing is: if you have (K, d) to be a compact metric space, then how to show (C(K) , d_infinity) is complete ?( d_infinity is uniform metric)

I know maybe I should use Cauchy or triangle inequality, but I am just not clear how to construct the proof.

Anyone who has any idea about that, please help. THX!
 
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  • #2
To show that any metric space is complete, you must show that all Cauchy sequences converge, the definition of "complete".

Now, since C(k) is the set of continuous functions on K, with "sup" metric, each member of C corresponds to a continuous function and a Cauchy sequence on C(k) is a sequence of continuous functions on K, say {fn}. I would look at the sequence {fn(x)} for each x in K. Try to show that is a Cauchy sequence in K.
 

Related to How to show completeness of C(K)?

What is C(K)?

C(K) is a set of functions that map the set K to real numbers. It is commonly used in mathematical analysis and functional analysis.

Why is it important to show completeness of C(K)?

Completeness is a fundamental property of a mathematical space that indicates the presence of all its limit points. In the case of C(K), completeness is necessary to ensure that the space contains all possible continuous functions on the set K.

How can completeness of C(K) be shown?

Completeness of C(K) can be shown by proving that every Cauchy sequence in the space converges to a function in C(K). This can be done using various techniques such as the Banach fixed point theorem or the Baire category theorem.

What are some applications of completeness of C(K)?

The completeness of C(K) is essential in many areas of mathematics, including functional analysis, differential equations, and topology. It also has practical applications in physics, engineering, and economics.

Are there any counterexamples to the completeness of C(K)?

Yes, there are counterexamples to the completeness of C(K) in certain cases. For example, if the set K is not totally bounded, then C(K) may not be complete. It is important to carefully consider the properties of K when determining the completeness of C(K).

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