Deduce that (P[0,1], norm(inf)) is not complete.

[ChemEng1]

Homework Statement

Deduce that (P[0,1], norm(inf)) is not complete.

Homework Equations

The Attempt at a Solution

I am not really sure how to start this.

My strongest instinct is to find a polynomial that converges to something that isn't a polynomial (discontinuous, trigonometric, or whatever). I have no tools to find such a polynomial without the 'needle in a haystack' approach.

Every Cauchy sequence is bounded, so trying to find an unbounded element of P[0,1] is out.

Any direction to steer this ship in would be appreciated.

 

[jbunniii]

My strongest instinct is to find a polynomial that converges to something that isn't a polynomial (discontinuous, trigonometric, or whatever). I have no tools to find such a polynomial without the 'needle in a haystack' approach.

Your instinct is good. However, you are not trying to find ONE polynomial that converges to something that is not a polynomial. (Not sure what that even means.) You are trying to find a SEQUENCE of polynomials that converges to something other than a polynomial.

And since you're using the infinity norm, you need the convergence to be uniform, right?

What kinds of theorems do you know about polynomials? E.g. do you know any spaces that they are dense in? What do you know about Taylor series? etc.

 

[ChemEng1]

Your instinct is good. However, you are not trying to find ONE polynomial that converges to something that is not a polynomial. (Not sure what that even means.) You are trying to find a SEQUENCE of polynomials that converges to something other than a polynomial.

Agree. My sloppy use of words here.

And since you're using the infinity norm, you need the convergence to be uniform, right?

Yes.

What kinds of theorems do you know about polynomials? E.g. do you know any spaces that they are dense in?

I would guess that polynomials are dense in space of continuous functions? (This wasn't covered in book or notes however.)

What do you know about Taylor series?

I hadn't thought about Taylor series being a polynomial. So Taylor series that converges to exp^t would be an example to use.

Taylor Expansion of exp^t is 1+t+...+t^n/n!+...

I could use p(t)=1+t+...+t^n/n!. And guess that it converges to p=Taylor expansion.

Need to show that inf[p(t)-p]-->0. Then since p is not an element of P[0,1], the space is not complete.

 

[jbunniii]

I would guess that polynomials are dense in space of continuous functions? (This wasn't covered in book or notes however.)

Yes, they are. This is the Stone-Weierstrass theorem. But if you haven't covered that yet, then don't worry about it - you can handle it using Taylor series.

I hadn't thought about Taylor series being a polynomial.

Each partial sum of a Taylor series has only finitely many terms, and is therefore a polynomial! And the sum of the series is the limit of the sequence of partial sums.

So Taylor series that converges to exp^t would be an example to use.

Taylor Expansion of exp^t is 1+t+...+t^n/n!+...

I could use p(t)=1+t+...+t^n/n!.

Right, although I would call it p_n(t) since you're building a sequence.

And guess that it converges to p=Taylor expansion.

\lim_{n \rightarrow \infty} p_n(t) = p(t) = e^t

By the way, you might want to call the limit something other than p(t) since the notation suggests that it's a polynomial, and the whole point is that you don't want it to be a polynomial. But that's just a nitpick.

Need to show that inf[p(t)-p]-->0.

Don't you mean sup, not inf? And writing it more carefully, it would be

\sup_{t \in [0,1]} [p_n(t) - p(t)] \rightarrow 0

as n \rightarrow \infty

i.e. you need to establish that the convergence is uniform. Do you know any theorems about the uniformity of convergence of Taylor series (or, more generally, power series)?

Then since p is not an element of P[0,1], the space is not complete.

Correct.

Question: where do Cauchy sequences come into this proof?

Good question. They haven't appeared explicitly yet, but having established that p_n \rightarrow p uniformly in the larger space of continuous functions, you immediately know that the sequence p_n is uniformly Cauchy. (If you haven't covered this fact already, you should prove it, but it's an easy proof.)

 

[ChemEng1]

Do you know any theorems about the uniformity of convergence of Taylor series (or, more generally, power series)?

A power series converges uniformly to its limit in the interval of convergence?
^^^ This comes by Google and not text or notes (again). I am not familiar with an 'interval of convergence' though. Is it referring to [0,1] in this case?

 

[jbunniii]

A power series converges uniformly to its limit in the interval of convergence?
^^^ This comes by Google and not text or notes (again). I am not familiar with an 'interval of convergence' though. Is it referring to [0,1] in this case?

Yes, this is correct. Every power series has an interval of convergence, of the form (-R,R), such that the series converges for values of t inside the interval, and diverges for values of t outside the interval. Convergence is uniform on any closed interval of the form [-A,A] where A < R. In the case of the Taylor series for e^t, the interval of convergence is actually (-infinity,infinity), and the series converges uniformly on any bounded closed interval.

If you don't have this theorem to work with, you can prove it directly for your specific series. Do you know any theorems about uniform convergence? How about the Weierstrass M test?

 

[jbunniii]

P.S. If you don't know any theorems about uniform convergence, what do you know about the remainder term of a Taylor series?

 

[ChemEng1]

P.S. If you don't know any theorems about uniform convergence, what do you know about the remainder term of a Taylor series?

The remainder term of a Taylor series would go to zero as the number of terms approaches infinity?

 

[jbunniii]

The remainder term of a Taylor series would go to zero as the number of terms approaches infinity?

It does, but this only proves that the series converges pointwise. Do you know a formula or bound for the remainder term which you could use to show that the convergence is uniform on your interval?