Stone–Weierstrass Theorem
Polynomials are dense in continuous functions.
Key idea
Polynomials are dense in continuous functions.
In depth
Every continuous function on a closed interval can be uniformly approximated by polynomials (Weierstrass); the general Stone version covers algebras of functions on compact spaces that separate points. It justifies approximating complicated functions by simple ones.
Worked example
|x| on [−1, 1] can be approximated arbitrarily well by polynomials, although it is not smooth.
Real-world uses
- Justifying polynomial approximations of any continuous signal in numerical methods
Common mistake
Applying it to non-compact domains.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Stone–Weierstrass Theorem?
Polynomials are dense in continuous functions. It is a topic at the expert level in Analysis & Topology.
Where is Stone–Weierstrass Theorem used in real life?
Real-world examples: Justifying polynomial approximations of any continuous signal in numerical methods.
What should I learn before Stone–Weierstrass Theorem?
Make sure you are comfortable with Analytic Functions, the previous topic in Analysis & Topology.
What is a common mistake with Stone–Weierstrass Theorem?
Applying it to non-compact domains.
How can I practise Stone–Weierstrass Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Stone–Weierstrass Theorem?
Move on to Fourier Series Convergence next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.