Answer :
Answer:
The Arzelà-Ascoli Theorem states that a sequence of functions is relatively compact in C([a, b]) (the space of continuous functions on [a, b] with the uniform norm) if and only if:
B) The sequence is uniformly bounded and uniformly equicontinuous.
This means that for a sequence of functions to have a convergent subsequence (i.e., to be relatively compact), it must be both uniformly bounded (there exists a single bound that works for all functions in the sequence over the entire interval) and uniformly equicontinuous (the degree of continuity does not vary significantly across the functions in the sequence).
Step-by-step explanation:
The Arzelà-Ascoli Theorem states that a sequence of functions is relatively compact in C([a, b]) if and only if:
- The sequence is uniformly bounded: There exists a constant M such that for all functions [tex]f_n[/tex] in the sequence and all [tex]\(x \in [a, b]\), \(|f_n(x)| \leq M\).[/tex]
- The sequence is uniformly equicontinuous: For every [tex]\(\epsilon > 0\), there exists a \(\delta > 0\) such that for all \(f_n\) in the sequence and all \(x, y \in [a, b]\), if \(|x - y| < \delta\), then \(|f_n(x) - f_n(y)| < \epsilon\).[/tex]
These conditions ensure the sequence does not "blow up" and does not oscillate too wildly, guaranteeing the existence of a uniformly convergent subsequence. So the correct answer is:
B) The sequence is uniformly bounded and uniformly equicontinuous.
The Arzelà-Ascoli Theorem states - B) The sequence is uniformly bounded and uniformly equicontinuous.
The Arzelà-Ascoli Theorem is a fundamental result in the theory of functional analysis, specifically in the study of function spaces. It states that a sequence of functions is relatively compact in the space of continuous functions C([a, b]) if and only if the sequence is uniformly bounded and equicontinuous.
This condition ensures that any sequence meeting these criteria has a subsequence that converges uniformly. To break it down:
- A sequence of functions {f_n} is uniformly bounded if there exists a constant M such that |f_n(x)| ≤ M for all functions in the sequence and for all x in [a, b].
- A sequence of functions {f_n} is equicontinuous if for every ε > 0, there exists a δ > 0 such that for all x, y in [a, b] with |x - y| < δ, we have |f_n(x) - f_n(y)| < ε for all n.
The correct answer is: B) The sequence is uniformly bounded and uniformly equicontinuous.
