(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider P[0,1] the linear space of C[0,1] consisting of all polynomials. Show that the sequence {p_{n}} where p_{n}(t)=t^{n}has the property that its coefficient sequences converge but the sequence {p_{n}} does not converge in (P[0,1], ∞-norm).

2. Relevant equations

Observation: Graphing p_{n}(t)=t^{n}, as n→∞, p_{n}(t)→0 for t=[0,1). p_{n}(t)→1 for t=1.

3. The attempt at a solution

1. Does this approach work?

2. Should infinity norm be used to determine the coefficient convergence?

3. Should you always use the limit of the coefficient covergence to "test" the {p_{n}} convergence?

4. Is it enough to show that {p_{n}} doesn't converge to 0 in P[0,1] with ∞-norm?

Part 1: p_{n}(t) work.

For t=[0,1)

Observe t>t^{2}>t^{3}>t^{4}>...>t^{∞}>0. As n→∞, t^{n}→0.

For t=1,

p_{n}(t)=1 for all n.

Part 2: {p_{n}} work.

Guess: norm[{p_{n}}]_{∞}→0.

To show: norm[{p_{n}}-0]_{∞}→0.

For a given t with 0≤t≤1, then norm[{p_{n}}-0]_{∞}=sup[|p_{n}(t)-0|]=sup{|t^{1}|, |t^{2}|, ... , |t^{n}|}=1 (≠0). Therefore {p_{n}} does not converge to 0 in P[0,1] with ∞-norm.

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# Homework Help: Show coefficient sequences converge.

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