Proving Relatively Compactness in C([a, b]) using Arzelá-Ascoli Theorem

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In summary, we have shown that a set $M \subset C([a, b])$ is relatively compact if there exist $m>0$ and $L>0$ such that $|f(x_0)| \leq{} m$ for all $f \in M$ and $|f(x)-f(y)| \leq{} L |x-y|$ for all $f\in M$ and for all $x,y \in [a,b]$. We have used the Arzelá-Ascoli theorem to establish this result.
  • #1
fabiancillo
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I don't know how to solve this proof

Prove that a set $M \subset C([a, b])$ for which there exist $m. L> 0$ and $x_0 \in [a; b]$ such that $|f(x_0)| \leq{} m$ for all $f \in M$ and $|f(x)-f(y)| \leq{} L |x-y|$ for all $f\in M$ and for all $x,y \in [a,b]$ is relatively compact in $C([a, b])$

My attemptSince $L >0$ and $ f \in M $ we can conclude that M it is equicontinuous and we have that $ |f(x)-f(y)| \leq{} L |x-y| $and $f(x)$ is continuous and $[a,b]$ compact.
By Arzela-Acoli Theorem Then is relatively compact in$ C([a, b]$
How can I apply the Arzelá-Ascoli theorem?
Thanks

 
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  • #2
The Arzelà-Ascoli theorem asserts that $M$ is relatively compact in $C[a,b]$ if $M$ is equicontinuous and pointwise bounded.

To see that $M$ is equicontinuous, fix positive number $\varepsilon$ and set $\delta = \varepsilon/L$. For all $x,y\in[a,b]$, $|x-y| < \delta$ and $f\in M$ implies $|f(x)-f(y)| \le L|x-y| < L(\varepsilon/L) = \varepsilon$. This shows, in fact, that $M$ is uniformly equicontinuous.

As for pointwise boundedness, fix an $x\in [a,b]$. For all $f\in M$, $|f(x)| \le |f(x_0)| + |f(x)-f(x_0)| \le m + L|x-x_0|$. Thus $\sup\limits_{f\in M} |f(x)| \le m + L|x-x_0|$.
 
  • #3
for your question! You are on the right track with your attempt. Here is how you can apply the Arzelá-Ascoli theorem to solve this proof:

First, recall that the Arzelá-Ascoli theorem states that a subset $M \subset C([a, b])$ is relatively compact if and only if it is equicontinuous and uniformly bounded.

In this case, we have already established that $M$ is equicontinuous, since $|f(x)-f(y)| \leq{} L |x-y|$ for all $f\in M$ and for all $x,y \in [a,b]$.

Next, we need to show that $M$ is uniformly bounded. We know that there exists $m>0$ such that $|f(x_0)| \leq{} m$ for all $f \in M$. This means that all the functions in $M$ have a maximum absolute value of $m$. Therefore, $M$ is uniformly bounded by $m$.

Since $M$ is both equicontinuous and uniformly bounded, it follows that $M$ is relatively compact in $C([a, b])$ by the Arzelá-Ascoli theorem. This completes the proof.
 

What is the Arzelá-Ascoli Theorem?

The Arzelá-Ascoli Theorem is a fundamental theorem in functional analysis that provides a necessary and sufficient condition for a set of functions to be relatively compact in the space of continuous functions on a compact interval.

Why is the Arzelá-Ascoli Theorem important in proving relatively compactness?

The Arzelá-Ascoli Theorem is important because it provides a powerful tool for proving relatively compactness in the space of continuous functions. This is useful in many areas of mathematics, including analysis, differential equations, and functional analysis.

What is the definition of relatively compactness?

Relatively compactness is a property of a set of functions that means the set is "almost" compact, meaning it is bounded and has a limit point. This is a weaker condition than compactness, but still very useful in analysis.

What are the key steps in using the Arzelá-Ascoli Theorem to prove relatively compactness in C([a, b])?

The key steps in using the Arzelá-Ascoli Theorem to prove relatively compactness in C([a, b]) are: 1) verifying that the set of functions is bounded, 2) showing that the set is equicontinuous, and 3) proving that the set is pointwise bounded. If all three conditions are met, then the set is relatively compact.

Can the Arzelá-Ascoli Theorem be applied to spaces other than C([a, b])?

Yes, the Arzelá-Ascoli Theorem can be applied to other spaces, such as the space of continuous functions on a non-compact interval or the space of differentiable functions. However, the conditions for relatively compactness may differ in these spaces.

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