C([a, b])

Definition

The notation c([a, b]) refers to the space of continuous functions defined on the closed interval [a, b]. This space is significant because it serves as the foundation for many concepts in approximation theory, particularly regarding how well continuous functions can be approximated by simpler functions like polynomials. Understanding c([a, b]) is crucial when discussing the Weierstrass approximation theorem, which states that any continuous function on this interval can be uniformly approximated by polynomials.

5 Must Know Facts For Your Next Test

1. c([a, b]) is a complete normed vector space, which means it is closed under limits and every Cauchy sequence converges to a limit within the space.
2. The Weierstrass approximation theorem guarantees that for every continuous function f in c([a, b]), there exists a polynomial P such that the maximum difference between f and P over [a, b] can be made arbitrarily small.
3. Functions in c([a, b]) are not only continuous but also bounded since they are defined over a closed interval.
4. The space c([a, b]) is often equipped with the supremum norm, defined as ||f|| = sup{|f(x)| : x ∈ [a, b]}, providing a way to measure the 'size' of functions within this space.
5. c([a, b]) includes not only polynomials but also trigonometric functions and other forms that can be uniformly approximated by polynomials, emphasizing its broad applicability in analysis.

Review Questions

• How does the structure of c([a, b]) facilitate the application of the Weierstrass approximation theorem?
  • c([a, b]) consists of all continuous functions on the interval [a, b], which are inherently well-behaved. The properties of this space ensure that any continuous function can be uniformly approximated by polynomials due to the completeness and compactness characteristics of closed intervals. This makes c([a, b]) an ideal setting for applying the Weierstrass theorem as it directly deals with continuous functions and guarantees approximation.
• Discuss the implications of using the supremum norm on c([a, b]) for understanding uniform convergence.
  • Using the supremum norm on c([a, b]) helps to define uniform convergence clearly since it measures how close functions get to each other across the entire interval. When we say a sequence of functions converges uniformly to a function f, it means that their supremum distance from f over [a, b] approaches zero. This is essential for applying results like the Weierstrass approximation theorem because it ensures that approximations remain consistently close throughout the interval rather than just at individual points.
• Evaluate how compactness of [a, b] impacts the behavior of continuous functions in c([a, b]) when approximated by polynomials.
  • The compactness of [a, b] plays a crucial role in establishing that every sequence of continuous functions in c([a, b]) has a uniformly convergent subsequence. This property is vital for applying tools like Arzelà-Ascoli theorem which ensures that these continuous functions can be approximated effectively by polynomials without losing uniformity. Consequently, this highlights why compact intervals are essential in approximation theory since they guarantee convergence behaviors that allow us to confidently use polynomial approximations as stated by Weierstrass.

Related terms

Uniform Convergence:

A type of convergence for sequences of functions where the speed of convergence does not depend on the point in the domain, ensuring that all functions in the sequence stay close together uniformly.

Polynomial Approximation:

The process of approximating a given function using polynomial functions, which is a key concept in approximation theory, especially highlighted by the Weierstrass theorem.

Compactness:

A property of a space that allows every open cover to have a finite subcover; in the context of c([a, b]), compactness relates to the fact that closed and bounded intervals are compact in Euclidean space.