4. The denseness of sets. Does the function being approximated have to be a unif

Question

4. The denseness of sets. Does the function being approximated have to be a uniformly continuous, or just continuous or does it matter?

Definition. A set E is said to be nowhere dense in a normed space X if E contains no open set The classic example of a dense set is the set of rationals in the real line. Another example, provided by the well-known Weierstrass approximation theorem, is that the space of polynomials is dense in the space C[a, b] (a) Does the function being approximated have to be uniformly continuous, or just continuous, or does it not matter? (b) Outline the simplest proof that you can find. It's OK to comment that certain steps are incomprehensible. (c) Prove rigorously, using the theorem, that the space of polynomials is dense in the space Cla,b

Explanation / Answer

The function must be continuous on [a,b]. This can be easily seen in the proof of Weierstrass approximation theorem.. Uniform Continuity is not required.

PS: If you want the proof of theorem then you can comment.

Related Questions

Navigate

• Browse (All)
• Browse 4
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.