You are Here: Home >< Maths

# Completeness of polynomials watch

1. Let be the space of polynomials on [0,1] with the supremum norm. Show that is not complete
2. (Original post by gemma331)
Let be the space of polynomials on [0,1] with the supremum norm. Show that is not complete
Work backwards. Instead of trying to think of a cauchy sequence of polynomials that doesn't have a limit in your space, think about approximating other functions with polynomials... for instance you should have a result (else look it up) that tells you that P([0,1]) is dense in C([0,1]).
3. (Original post by gemma331)
Let be the space of polynomials on [0,1] with the supremum norm. Show that is not complete
As Mark85 says, you might have seen results called the Weierstass Approximation theorem, or the Stone-Weierstass theorem, which tell you that the space of polynomials is dense in the space of continuous functions - this answers your problem straight away.

If you want to find an explicit example of a sequence of polynomials which is Cauchy but doesn't converge to a polynomial, my advice would be to think about a particular power series and its truncations.

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: March 28, 2013
Today on TSR

### Tuition fees under review

Would you pay less for a humanities degree?

### Can I get A*s if I start studying now?

Discussions on TSR

• Latest
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

## Groups associated with this forum:

View associated groups
Discussions on TSR

• Latest

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE